Main

The scattering of light is ubiquitous and, one might argue, the fundamental mechanism by which we observe light in nature1,2,3,4. It is the reason the sky is blue and sunsets are red, and it is also the reason snow is white and apparently opaque5. In the presence of weak scattering, the full-wave equation approach can be used for simulating and understanding light propagation and, specifically, also for accounting for coherent effects and formation, for example, of speckle patterns6. However, for very strong scattering (the strong diffusive regime), coherent effects play a minor or negligible role, speckle patterns are no longer visible and a more useful description is provided by the radiative transport equation. This describes energy transport through a series of scattering processes and therefore also describes other seemingly unrelated regimes such as neutron diffusion7 that, very much like light, also have important imaging applications8,9,10.

In the strong diffusive regime, the thickness of the medium, L, is much larger than the transport mean free path (over which a ray loses all memory of its original direction, equal to the inverse of the reduced scattering coefficient, \({\mu }_{\mathrm{s}}^{{\prime} }\)), \(L\gg 1/{\mu }_{\mathrm{s}}^{{\prime} }\) and the absorption coefficient is considerably smaller than the scattering coefficient, \({\mu }_{\mathrm{s}}^{{\prime} }\gg {\mu }_{\mathrm{a}}\). The radiative transport equation can then be approximated by a diffusion equation that continues to describe diverse phenomena such as heat, neutron and light diffusion.

However, scattering renders materials opaque. This was one of the main challenges overcome with the invention of the optical fibre—the development of high-purity glass that can transport light over large distances without suffering from attenuation due to absorption and scattering. Optical fibres and waveguides also rely on a refractive index contrast between an inner core and an outer cladding region such that the waves undergo total internal reflection and a consequent exponential localization in space11,12. It is also possible then to take a more heuristic approach where the requirement of exponential mode localization is used as a physical signature for guiding electromagnetic waves. Returning to the case in point, a notable feature of light propagation upon entering a scattering medium is that it will be exponentially attenuated and will therefore be largely scattered backwards. This simple observation can therefore be used heuristically to raise the question as to whether this exponential decay of the photon density distribution can give rise to confinement or guiding.

Here, we demonstrate a waveguiding mechanism for the photon density whereby a thin cylindrical element within a uniform strongly scattering medium (for example, a cylinder of lower diffusion along the propagation direction that we can identify as a ‘core’, analogous to standard optical fibres) effectively guides the energy flow. The governing equations support the existence of guided modes, supported also by Monte Carlo numerical simulations. Given the generality of the underlying equations, these results apply also to neutron transport, therefore providing an effective means for the guiding of particles and not only electromagnetic waves. We also perform experiments that show evidence of ‘scatter guiding’. Light is seen to be confined to a broad region that is shaped by the core and is transmitted with more than two orders of magnitude more efficiency compared with the case without a core.

Analytical model for photon density modes

In the limit in which the propagation distance in the scattering medium is \(L\gg 1/{\mu }_{\mathrm{s}}^{{\prime} }\) and \({\mu }_{\mathrm{s}}^{{\prime} }\gg {\mu }_{\mathrm{a}}\), the equation for the fluence rate Φ(r, t) (that we also refer to as the ‘photon density’ in this work) is1,2,4,13

$${c}^{-1}\frac{\partial \varPhi }{\partial t}+\nabla \times {\bf{J}}+{\mu }_{\mathrm{a}}({\bf{r}})\varPhi =S({\bf{r}},t),$$
(1)

where J(r, t) = −D(r)∇Φ(r, t) is the flux, S is the source, c is the speed of light in the medium and \(D\approx 1/(3{\mu }_{\mathrm{s}}^{{\prime} })\) is the photon diffusion coefficient. Physically, the fluence rate has units of watts per square metre and is the angle-integrated radiance incident onto a small volume centred on position r at time t. A description in terms of the real-valued fluence rate is warranted owing to scattering in the media composing the system, which scrambles the light phase so that it does not play a role on average, and the light undergoes diffusion as opposed to the more familiar diffraction in non-scattering media. The photon density equation (1) accounts for this light diffusion (the term proportional to D) in addition to absorption (the term proportional to μa) and light generation via the source S. The photon density equation (1) therefore bears a resemblance to a broad range of diffusion equations including the heat equation, neutron and electron diffusion equations in condensed media, and the Fokker–Planck equation of statistical mechanics.

The geometry we consider is a cylindrical core of radius Rcore with coefficients \({\mu }_{\mathrm{s}{({\mathrm{core}})}}^{{\prime} }\), μa(core) that is, surrounded by a coaxial diffusive material cladding of radius Rclad, with coefficients \({\mu }_{\mathrm{s}({\mathrm{clad}})}^{{\prime} }\), μa(clad) and with air outside. We remove the source (S = 0) to look for modal solutions of the form that decay exponentially Φ(r, z) = e−γzϕ(r) along the z propagation direction. The steady-state photon density equation for modal solutions with cylindrical symmetry then becomes

$${\nabla }_{r}^{2}\phi (r)+({\gamma }^{2}-{\gamma }_{x}^{2})\phi (r)=0,$$
(2)

where \({\nabla }_{r}^{2}\) is the transverse Laplacian, x indicates the core or cladding (that is, there is an equation each for the core and cladding regions) and \({\gamma }_{x}=\scriptstyle\sqrt{{\mu }_{\mathrm{a}(x)}/D}\approx \sqrt{3{\mu }_{\mathrm{a}(x)}{\mu }_{\mathrm{s}(x)}^{{\prime} }}\) is the extinction rate in the core/cladding material.

Equation (2) has the form of the well-known Helmholtz equations from fibre optics. Here, however, the Laplacian term \({\nabla }_{r}^{2}\) describes the effects of photon diffusion as opposed to diffraction. The three key parameters that determine the light propagation regimes, as we will see below, are γ (the modal extinction coefficient), γcore and γclad, which can be likened to the propagation constant and the core and cladding refractive index, respectively, in the case of traditional optical fibres. By analogy, equation (2) will therefore have a full set of modal solutions, that is, ground plus higher-order modes, and these can serve as a basis for the mode expansion of the fluence rate. On the other hand, ϕ(r) represents a photon fluence rate that is real and positive, as opposed to the complex electric field envelope in the fibre case. This means that the higher-order modes cannot be excited in isolation or represent a physical fluence rate as they, by necessity, exhibit field nodes and hence negative regions. The fluence rate must, in general, comprise a superposition of modes that is everywhere positive valued. Here, our focus is on the ground-state mode that experiences the lowest loss under propagation and thus survives at long distances, analogous to finding the lowest-order mode of an optical fibre. These lowest-loss modes will preserve their transverse profile with increasing distance, although their overall amplitude will decay exponentially. Finally, we note that a waveguiding structure is obtained by creating a core with γcore < γclad. This can be achieved by reducing the core scattering coefficient, \({\mu }_{\mathrm{s}({\mathrm{core}})}^{{\prime} } < {\mu }_{\mathrm{s}({\mathrm{clad}})}^{{\prime} }\), by reducing the core absorption coefficient, μa(core) < μa(clad) or by reducing both.

Mode solutions

As in the case of standard fibres, we can distinguish between the solution in the core and the solution in the cladding. For all cases where γ > γcore, the core solution is (see Supplementary Information for full details)

$$\phi (r)={\phi }_{0}\,{J}_{0}\left(\sqrt{{\gamma }^{2}-{\gamma }_\mathrm{core}^{2}}r\right),$$
(3)

where ϕ0 is a constant and J0 is the zero-order Bessel function. As a special case, we consider an air core; for this case, we take the double limit Dcore → 0 and μa(core) → 0, in such a way that \({\gamma }_\mathrm{core}=\sqrt{{\mu }_\mathrm{core}/{D}_\mathrm{core}}\to 0\). We then obtain the solution \(\phi (r)={\phi }_{0}\,{J}_{0}\left(\gamma r\right)\).

In the cladding, the solution depends on the relative value of γ compared with γclad. For γ < γclad, we have

$$\phi (r)\approx A{K}_{0}\left(\sqrt{{\gamma }_\mathrm{clad}^{2}-{\gamma }^{2}}\,r\right),$$
(4)

where A is a constant and K0 is the exponentially decaying modified Bessel function of the second kind. Here, we have assumed the large cladding limit for illustration (see Supplementary Information for more details). We therefore have a guided mode in the strict sense, that is, the mode spatial profile is exponentially localized and indeed has the same functional mode profile as found in standard optical fibres.

Some of the results below will involve curved waveguides where, similarly to traditional waveguides, we observe higher modal losses, that is, γ > γclad. In this case, the cladding solution will take the form (see Supplementary Information)

$$\phi (r)=A{J}_{0}\left(\sqrt{{\gamma }^{2}-{\gamma }_\mathrm{clad}^{2}}\,r\right),$$
(5)

that is, the cladding mode profile switches from a convex exponential decay to a concave Bessel profile.

Experiments

To study the main features of photon density waveguiding, we performed experiments in three-dimensional (3D)-printed resin structures. The resin used in these experiments was characterized using time-of-flight measurements (see, for example, refs. 14,15) and was measured to have a reduced scattering coefficient of \({\mu }_{\mathrm{s}}^{{\prime} }\simeq 35\,\mathrm{cm}^{-1}\) and absorption coefficient of μa ≃ 0.04 cm−1; therefore, γ = 2 cm−1 is in the same range as for some biological tissues16. Light from a pulsed or continuous wave laser is coupled into the structures via a fibre tip that is placed up against the material or core. A first set of measurements was performed by filling the core with scattering material made of 1–6 mg of TiO2 in 30 ml glycerol with \({\mu }_{\mathrm{s}}^{{\prime} }\) that varies linearly with the TiO2 concentration14 in the range of 0.05–0.3 cm−1. The resulting core extinction coefficients are therefore γcore = 0.09–0.2 cm−1.

Figure 1 shows the measured photon density mode profile at the output, as captured by a complementary metal–oxide–semiconductor camera from which a line-out is plotted along the horizontal axis. The mode profile in the core region is shown in the inset and does not depend on the core/cladding details and only changes (decreases) in amplitude as γcore → γclad. The dashed blue curve shows the expected Bessel profile (equation (3)).

Fig. 1: Photon density mode-guiding experiments.
figure 1

Experimental profiles of light transmitted through a resin cladding structure with a straight 0.5-mm-radius core filled with a TiO2:glycerol solution. Cladding mode profiles are shown for two different TiO2:glycerol solutions (1 mg and 1.5 mg TiO2 in 30 ml glycerol) with fits from the analytical model with the Bessel K0 function, where γ/γclad = 0.66 and 0.99, respectively (dashed curves). Inset: the profile in the core mode, measured separately owing to the large dynamic range. This shows a good fit to the predicted Bessel J0 profile that, aside from a vertical scaling factor, was independent of core/cladding details (shown for the 1 mg TiO2 in 30 ml glycerol case). The transmitted energy is measured to be 100-fold larger compared with a solid (no core) resin cylinder.

Full size image

The cladding mode profile is shown for two cases of low TiO2 concentrations, that is, γcore = 0.09 and 0.12. We observe exponentially localized mode profiles that are well fitted (dashed curves) by equation (4) with γ/γclad = 0.99 and 0.66, as indicated in Fig. 1. This demonstrates that it is indeed possible to excite (exponentially localized) photon density modes in a scattering medium.

Figure 2a shows, for reference, three beam profiles at the exit of a uniform cylinder (no core) with the relative Bessel J0 theoretical curves (cladding radii indicated in centimetres). Figure 2b then shows the case in which we have an air core with a radius of 0.5 mm or 1 mm (graph inset). Our theory, despite the approximation made in deriving the solutions in this case, correctly predicts an exponentially localized solution that is very well reproduced by the theoretical curve (dashed lines) with γ/γclad = 0.44 and 0.33, respectively (the same data on a logarithmic scale together with the full theory solution that accounts for the cladding external boundary conditions are shown in Supplementary Fig. 2).

Fig. 2: Photon density mode experiments.
figure 2

a, Top: a resin cylinder with no core. Bottom: the measured spatial profiles (horizontal line-outs from the full two-dimensional image), for cylinders with a radius of 1, 2 and 2.5 cm (the photograph shows an example of a 2.5-cm-radius cylinder). The solid line shows the data, and the dashed lines show parameter-free fits with Bessel functions. b, Top: a resin cylinder (2.5 cm radius) with a straight (empty) core. Bottom: an experimental line-out for rcore = 0.5 mm (solid curve) and a fit with a modified Bessel function of the second kind (dashed curve). Inset: the same data for rcore = 1 mm. c, Top: a resin cylinder (2.5 cm radius) with a bent (empty) core. Bottom: results for three different bend radii of 5.5 cm, 3.5 cm and 2.8 cm and also a ‘no core’ sample. Note that the 5.5 cm core intensity is ~2× higher than the 3.5 cm core intensity, while the cladding profiles are nearly identical and barely distinguishable in the figure. The top photographs in all figures show the resin structures illuminated with a red laser. The insets in the main graphs show a schematic ‘cut-out’ of the resin waveguide structures to visualize the internal structure.

Full size image

Figure 2c shows the results with curved air cores for various curvatures (indicated in centimetres). In all cases, we see that the photon density mode profile in the cladding region has now switched from the convex profile of the exponential decay seen in Fig. 2b to the concave, Bessel function shape, as predicted by our theory when γ > γclad. In analogy with traditional waveguides, the additional losses that increase the value of γ can be attributed to the bending of the core. Despite these higher losses, we note that the transmitted power is considerably larger compared with the light transmitted without a core structure. For the case of the straight scattering core in Fig. 1 and for the case of the lowest curvature in the air core waveguides in Fig. 2c, we found an increase of ~110× and ~5×, respectively (see Supplementary Figs. 6 and 7 for more details and numerical simulations of propagation loss).

To gain more insight into what is happening inside the medium, we performed Monte Carlo simulations using the same structure geometries described above and medium parameters \({\mu }_{\mathrm{s}}^{{\prime} }=15\,\text{cm}^{-1}\) and μa = 0.016 cm−1 (the qualitative features of the results do not depend on these exact values). Figure 3a,b shows the results for the case of a solid cylinder with no core, and Fig. 3c,d shows the case of a curved core, when selecting only 10 rays or 50 rays. In the absence of a core, most of the rays are back-reflected, while the presence of the core clearly provides a guiding mechanism as the rays accumulate around the curved core and exit the distal end. We note that guiding still takes place despite the photon density modes no longer having a spatial exponential decay profile (compared with the straight core case). Figure 3e shows a schematic view of a structure that consists of a D-shaped cylinder (cladding) with a curved core (bend radius of 28 mm) that is placed 1 mm beneath the flat surface (Fig. 3f). We simulate this experimentally (50 million rays) in Fig. 3g and compare this with an experimental measurement in Fig. 3h (a full set of simulations for various bend curvatures is shown in Supplementary Fig. 6). The experimental profile agrees well with the simulated profile and demonstrates that, despite the dependence of the photon density mode on the boundary conditions, these modes are robust even to strong modifications of the structure geometry and boundary conditions.

Fig. 3: Photon density modes in bent waveguides.
figure 3

a,b, Numerical simulations (Methods) of photon paths in a solid (no core) resin cylinder (radius 2.5 cm), with 10 rays (a) and 50 rays (b) plotted. c,d, The same simulations as in a and b with a 0.5-mm-radius, empty core and 10 rays (c) and 50 rays (d) that are detected at the resin structure exit facet (z = 5 cm). e, A schematic of the geometry tested in experiments to observe photon density mode guiding in a bent waveguide in a D-shaped cladding to allow visual access to the core. f, A lateral-view photograph of the 3D-printed waveguide structure. g, A numerical simulation (50 million rays) of the structure in f. h, The experimentally measured light profile of the structure in f.

Full size image

In Fig. 4, we investigate the effect of an inclusion in the cladding. We 3D-printed a structure that has a curved core (curvature radius of 2.8 cm), thus supporting a broad mode, with a 5-mm-radius hole that traverses the whole structure perpendicularly to the plane of the core and is displaced internally from the core by 5 mm (Fig. 4, inset). The photon density mode maintains its Bessel-like structure but can sense the difference between a hole filled with air or water, and it is only when we insert a totally opaque object (metal) that we start to also observe a slight distortion of the mode profile.

Fig. 4: Sensitivity to cladding ‘inclusions’.
figure 4

Experimentally measured output profiles (line-outs from two-dimensional images) with various materials filling the ‘inclusion’. Inset: the cylindrical structure used, with a cut-out of the full cylinder to illustrate its internal structure.

Full size image

Interestingly, and partly the inspiration for this work, photon density waveguiding can also occur naturally. Traditional optical waveguiding from a refractive index contrast was similarly first reported in 1842 by Jean-Daniel Colladon to occur naturally in a thin jet of water17. In the case of photon density modes, light propagating through the human head, for example, is strongly affected by the presence of a relatively transparent cerebrospinal fluid (CSF) that is contained between otherwise dense, scattering layers of bone and grey/white matter. The emphasis in the past has been on the role played by the CSF in limiting light penetration into the brain grey/white matter or as an indicator of neurodegeneration18,19,20,21,22,23, but this could in the future be used as a route to transfer light across larger regions or even across the whole brain, as seen in Monte Carlo simulations (Supplementary Fig. 4). Similarly, other areas of the human body, such as tendons, can also conduct light (see Supplementary Fig. 4 for a photograph of a human forearm tendon guiding light over several centimetres).

Conclusions

By inserting a core structure inside an opaque scattering medium, it is possible to excite exponentially localized modes that survive even in the presence of perturbations such as bending, and which improve light transmission by orders of magnitude. We underline that the light guiding discussed here is fundamentally different from previous light-guiding mechanisms, including the process of guiding light in highly anisotropic, that is, fibrous scattering media, that has been identified in dentin24.

We have highlighted some interesting parallels with standard optical fibre modes that can provide any readers who are familiar with optical fibres with an intuition as to how the photon density modes might behave and also provide ideas for potential applications and uses. For example, looking at the role of the CSF or other structures in the human body from the perspective of a light-guiding problem might offer new insights into how to control and harness photon density modes to access deep body locations. It is also possible to clear thin channels using spatially shaped beams in scattering fluids25 or in fog with light filaments from high-power lasers, with applications, for example, in free-space telecommunications26,27,28. The results presented here would suggest the possibility of a photon density mode that follows the optically cleared channel.

Finally, we have underlined that the same equations that govern the propagation of the photon density apply also to neutron transport, implying that the proposed mechanism provides a mechanism to guide also particles and not just light.

Methods

Monte Carlo numerical simulations were performed using Ansys Zemax Optic Studio, which has a non-sequential ray-tracing mode that allows for Monte Carlo scattering events to be modelled. The simulations shown in Fig. 3 were performed with the following common software settings:

Material size: cylinder R = 25 mm, L = 50 mm

Refractive index: 1.37

Mode: non-sequential ray tracing

Bulk scattering: Henyey–Greenstein model

g (anisotropy factor): 0.87

Mean path scattering length: 1/(μa + μs) = 0.08665 mm−1 (actual Zemax input)

Transmission: μs/(μa + μs) = 0.99986 (actual Zemax input)

Corresponding absorption and scattering coefficients: μa = 0.0016 mm−1 and \({\mu }_{\mathrm{s}}^{{\prime} }=1.5\,\text{mm}^{-1}\)

Input source: radial beam of 1 mm diameter (filling the core)

Figure 3a–d shows random samples of a few rays (10 and 50) showing their ray paths, where every colour depicted is a new scattering event or segment. Figure 3g is a run of 50 million input rays, and the detector is a longitudinal slice along the X/2−Z plane, thus showing the light intensity distribution inside the resin. The detector is not absorbing, and so the rays are unaffected as they propagate and pass through the detector, appearing as an intensity hit.