Spiral galaxy is one of the most beautiful natural patterns of our mysterious universe. To understand the formation of spiral galaxies, density wave theory [1] was proposed approximately six decades ago, which claimed that the celestial bodies in a galaxy are constrained by gravitational forces or potential. Namely, the celestial bodies in a galaxy are more likely to appear at the local maxima of the density wave, and also in the local minima or the wells of the gravitational potential, which is similar to the principle of optical trapping [2].

On the other hand, structured light has attracted increasing attention for its unique properties of amplitude, phase, and polarization, and it has found applications in various fields, such as optical manipulation [3–5], optical communications [6], and optical imaging [7]. Common types of structured light include vortex beams [8,9], Bessel beams [10], and Airy beams [11]. Among them, vortex beams are characterized by hollow-shaped intensity, helical phase, and twisted wavefront structure.

In this work, inspired by the beautiful structure of spiral galaxies, we propose a method for constructing a novel structured light, which has divergent spiral arms like a spiral galaxy. For the unique potential well [12] of such structured light, we will employ it in an optical trapping experiment to demonstrate its capability in trapping nanoparticles. Finally, we discuss the inspiration drawn from spiral galaxies and transformation optics [13].

Now, inspired by the beautiful patterns of spiral galaxies, we show how to generate such a structured light with a similar spiral pattern based on the interference of two beams, viz., a vortex wave and a trumpet wave. A common vortex beam, Laguerre–Gaussian mode, at the source plane can be written as [9] LGpl(ρ,φ;z=0)=Apl(ρ)exp(ilφ),(1)Apl(ρ)∝(ρ2w)|l|Lp|l|[2ρ2w2]exp(−ρ2w2),(2)where ρ,φ,z are the cylindrical coordinates, l is the azimuthal index (topological charge), p is the radial index of the LG beam, w is the beam waist size, and Lp|l|[x] is the associated Laguerre polynomial. The intensity profile and wavefront of LG beams at z=0 for l=2,3 and p=0 are shown in Figs. 1(a) and 1(b).

Let us suppose there is a trumpet wave, with Gaussian amplitude and trumpet phase, and its complex amplitude at z=0 can be written as Utrumpet(ρ,φ;z=0)=B(ρ)exp[iΔΦ(ρ)],(3)B(ρ)=exp(−ρ2w′2),(4)where w′ is the transverse size of the trumpet wave and ΔΦ(ρ) is the trumpet shaped radial phase. We can write the trumpet phase ΔΦ(ρ) as an integral form, ΔΦ(ρ)=k0∫0z0n(ρ,z)dz,(5)where n(ρ,z) is the equivalent refractive index of a medium, k0=2π/λ is the vacuum wave number, λ is the wavelength, and z0 is the longitudinal length of the medium. Considering the spherically symmetric refractive index n(ρ,z)=rg/(ρ2+z2−rg), where rg is the critical parameter with the length dimension, the trumpet phase can then be simplified to ΔΦ(ρ)=k0rgln(z02+ρ2+z0ρ).(6)

A sample of the trumpet wave we constructed is shown in Fig. 1(c), which has a trumpet-shaped phase. We will discuss the origin and details of this trumpet phase later, in the Section 4.

Then, the coaxial interference field and the intensity of a vortex wave and a trumpet wave at z=0 plane can be written as U(ρ,φ;z=0)=LGpl(ρ,φ)+Utrumpet(ρ,φ),(7)I(ρ,φ;z=0)=Imax+Imin2+Imax−Imin2×cos[−lφ+ΔΦ(ρ)],(8)where Imax=[Apl(ρ)+B(ρ)]2 and Imin=[Apl(ρ)−B(ρ)]2. The interference patterns for the vortex wave with topological charge l=2,3 and the trumpet wave are shown in Figs. 1(d) and 1(e), which show two and three spiral arms, respectively. This indicates that the topological charge of the vortex wave determines the number of spiral arms. The spiral arms are characterized by the divergence as the increasing of the radial coordinate ρ, just like the spiral galaxies in many astronomical observations [14].

Figure 2 is plotted to see the characteristics of the trumpet wave and how spiral structures depend on the parameters more clearly. The blue solid line in Fig. 2(a) shows the calculated trumpet phase. For ρ≪z0, the term (z02+ρ2+z0)/ρ in Eq. (6) can be approximated as 2z0/ρ; thus, we get the asymptotic line 1 in Fig. 2(a): ΔΦ(ρ)=k0rgln(2z0/ρ), which is a logarithmic function. Similarly, for ρ≫z0, we can get the asymptotic line 2 in Fig. 2(a), which is a reciprocal function.

With the calculated form of the trumpet-shaped phase ΔΦ(ρ), we show how the critical parameter rg affects ΔΦ(ρ) and leads to different forms of the spiral arms. The interferograms corresponding to different values of the parameter rg are shown in Figs. 2(b)–2(d). From the figures, we can see that, with the increasing of rg, the spiral arms take more turns and become more tightly wound. We further consider the spiral line formed by local maxima in the light intensity. Letting −lφ+ΔΦ(ρ)=0, we can derive the spiral equation for the spiral arms in the interferograms as φ=k0rglln(z02+ρ2+z0ρ).(9)

The derived spiral equation is plotted in the white dashed line for each pattern with different rg, shown in Figs. 2(b)–2(d). We know that, for ρ≪z0, ΔΦ(ρ) can be approximated to a logarithmic asymptotic line; therefore, the spiral arms can be further simplified to the logarithmic spirals φ=φ0−k0rg/lln(ρ), where φ0=k0rg/lln(2z0). The spiral equation is consistent with many astronomical observations [1,15,16], where the spirals in spiral galaxies can be well fitted into the logarithmic functions. To study the relationship between spiral arms and rg, a critical parameter, the spiral pitch angle, is introduced, which describes the winding of spiral arms used in astronomical research. Visually, the spiral pitch angle decreases in Figs. 2(b)–2(d). As shown in Fig. 2(e), it is defined as the angle between tangent lines of the spiral and a circle at a certain point, where the circle passes through [17]. This local property can describe the global winding characteristics of spiral arms. The relation between pitch angle p of the spirals and rg is given by Fig. 2(e), with different values of the topological charge l. Figures 2(b)–2(d) lie in the l=2 line in Fig. 2(e), respectively. Therefore, the negative correlation between spiral arm pitch angle p and rg is demonstrated.

Now we set up an experiment to generate the structured light inspired by spiral galaxies. We use a phase-only spatial light modulator (SLM) to generate the trumpet shaped phase hologram and a forked grating (FG) to generate vortex beams with designed topological charge. A Fourier transform lens L3 generates a vortex beam at the Fourier plane where pinhole filter PF1 is placed to filter out unwanted diffraction orders. A 4-f system composed of L6,L7, and PF2 images the SLM modulated light to the CMOS plane. A Mach–Zehnder interferometer is set up for the two waves’ summation, performed by beam splitter BS2 in Fig. 3(a). We present the original spiral galaxy images and the simulations for the holograms in Fig. 3(b). The spiral galaxy in the image shown in Fig. 3(b1) is NGC 1566, in the Dorado constellation. Figure 3(b2) presents the image of the spiral galaxies NGC 1300. The spiral galaxy NGC 5194 (M51) image is shown in Fig. 3(b3). Mostly, spiral galaxies have two spiral arms, but an image of spiral galaxy NGC 2008 with four arms is shown in Fig. 3(b4). Using our experimental setup, we generate different spiral galaxy patterns by changing the parameters. The number and rotation direction of spiral arms can be mimicked by the magnitude and sign of the topological charge l of the vortex wave. The divergence of the spiral is controlled by rg in our model. The corresponding simulations and experimental results are shown in Fig. 3(b).

The arms of spiral galaxies consist of stars, dust, and gas, constrained by gravitational potential. It will be fascinating to perform optical trapping using the generated spiral-structured light. The technique using the radiation force of light beam to manipulate particles is widely known as optical tweezers [2], and the particles can be trapped by forces generated by optical intensity gradients [12]. The average gradient force is proportional to the relative permittivity of the medium ε and the real part of the polarizability α of the particles in the electric field and is written as ⟨F⟩=14εε0Re{α}∇I,(10)where ε0 is the vacuum permittivity and I=|E|2 is the intensity of the light. Then the optical potential can be expressed as ⟨V⟩=−∫⟨F⟩dr=−1/4εε0Re{α}I=ϵI. The intensity of the interference field in Eq. (8) leads to the potential ⟨V⟩ of optical force, ⟨V⟩=V0(ρ)+V1(ρ)×cos[−lφ+ΔΦ(ρ)],(11)where V0(ρ)=ϵ(Imax+Imin)/2, V1(ρ)=ϵ(Imax−Imin)/2, and ϵ=−1/4εε0Re{α}.

Figure 4(a) shows the vector distribution of optical forces with a background of the optical potential, and the equipotential lines are also presented. To simulate the optical trapping of nanoparticles in the medium, here we choose distilled water, and within the light field we consider the following kinetic equation [19]: γdrdt=F(r)+2ηkBTW(t),(12)where γ is the friction coefficient of water, F(r) is the gradient force, 2ηkBTW(t) is the white noise term used to model random collisions from water molecules (temperature dependent), and the mass of the nanoparticles is neglected. For water at temperature T=298K, the strength of the noise term is 2.7 pN. Our numerical simulation is based on the discrete form of Eq. (12), which we implement as a random-walk-style algorithm to model dynamics of nanoparticles in optical trapping. In the simulation, the initial positions of the nanoparticles are equally distributed and the initial speed is zero, and the wavelength of incident light is λ=532nm. The result for the distribution of the particles after t=1s is presented in Fig. 4(b), showing two spiral arms formed by the particles represented by the white dots.

We have conducted an experiment to validate the optical trapping of the polystyrene particles, with the radius of 100 nm, of maximum compliance with simulation settings. We generated the structured light field with the Mach–Zehnder interferometer. Then the light field was coupled to the high NA objective, which was used to focus the light field to the sample chamber, where we placed the water suspension of the nanoparticles. We carefully controlled the power of the laser to acquire the suitable gradient force on the particle. A CMOS camera in a microscope system was set to image the trapped particles, and the results are shown in Fig. 4(c). The particles are concentrated at the areas where the optical potential is relatively small, and we can calculate the planar density distribution σ(ρ,θ) of the particles. We use the Fourier transform to express the density distribution as σ(ρ,φ)=∑l=0∞clcos[−lφ+Φ(ρ)] and the inverse transform to decompose the spiral spectrum cl. This decomposition extracts different spiral sets in the mass density pattern from experimental and simulated results. From Fig. 4(d), the fundamental order of the spiral set is l=2 clearly, with some decreasing harmonics.

We will further study the time evolution of the nanoparticles trapped in the structured light field. In Fig. 5(a), the nanoparticles are equally distributed in the vicinity of the beam center, where the time is set to be t=0s. At t=1s, the nanoparticles spiral arms emerge, and there still exist some particles outside the potential wells, shown in Fig. 5(b). In Fig. 5(c), the spiral arm is clearly formed after t=2s, and the particles are distributed in the designed potential wells. From Fig. 5(d), we can see that the matter arms become stable.

Now, we discuss the rationale for introducing a vortex wave and a trumpet wave to produce a spiral-galaxies-inspired structured light in Section 2. We will reveal the origin of the trumpet phased wave in transformation optics in Section 4.A, and we will further discuss the optical potential wells as an optical analog for gravitational potential in Section 4.B.

Nowadays, researchers have found that black holes always exist in the center of spiral galaxies [20–22], and vortex light around rotating black holes has been demonstrated as well [23]. Inspired by such a fact, we introduced a vortex wave. Besides, as a supermassive celestial body, black holes will bend the spacetime [24], and this is the origin of the trumpet wave we introduced. In general relativity, curved spacetimes are characterized by metrics gij and the corresponding line element ds2=gijdxidxj governed by Einstein’s field equations, i,j=1,2,3,4, which could have solutions of the following isotropic type [24,25]: ds2=g00(r)[−dt2+n2(r)(dr2+r2dΩ2)],(13)r2=ρ2+z2,(14)where dΩ2=dθ2+sin2θdφ2,(t,r,θ,φ) are the (3+1)-dimensional spherical coordinates, g00(r) is to be determined by solving the field equation, and n(r) is the equivalent refractive index of the metric. We can derive that the metric for the static (Schwarzschild-like) and slow-rotating gravitating body (Kerr-like) could have an equivalent refractive index [13], n(r)=1+rgr−rg,(15)where rg is the event horizon of the central black hole. For the propagation of electromagnetic waves, the isotropic curved spacetime can be equivalently replaced by the media with a spatially dependent refractive index [13,26]. This is brought by the invariance of Maxwell’s equations under coordinate transformations [27]. Applying this transformation optics approach, dielectric material with this spatial refractive index distribution n(r) can well mimic the curved spacetime. Next, for a plane electromagnetic wave propagating in the z direction in this equivalent medium, the phase shift of the plane wave is ΔΦ(ρ)=k0×∫0z0n(ρ,z)dz, where z0 is the longitudinal length of the transformed medium and k0 is the wave number in the free space.

Existing literature [17,28] shows that the black hole mass is negatively correlated with the spiral pitch angle. We know that the event horizon rg=2GM/c2 in Schwarzschild spacetime, where G is the gravitational constant and c is the speed of light in vacuum, is proportional to the total mass M in the central celestial body. This reveals the negative correlation between rg and spiral arm pitch angle p of spiral galaxies, which agrees well with our inference drawn from Fig. 2.

Density wave theory is a classical theory to investigate the dynamics of spiral galaxies [1]. Celestial bodies and other matter within spiral galaxies are influenced by gravitational fields, forming distinctive spiral arm patterns. Density wave theory solved the dynamics equations in phase space, and a spiral-type solution is admitted for the surface mass density of the galaxy disk [29], σ(ρ,φ,t)=S(ρ)cos[ωt−mφ+Φ(ρ)],(16)where φ is the azimuthal angle, m is an integer, Φ(ρ) and S(ρ) are the stable radial phase and amplitude, respectively, and ω is the angular pattern speed. This spiral mass distribution explains the spiral structure of the galaxies in the density wave theory. Then the gravitational potential V can be obtained by solving the Poisson equation ∇2V=σ(ρ,φ), where σ(ρ,φ) is the mass density of the galaxy disk. Under asymptotic approximation, the total gravitational potential Vgr(ρ,φ,t) can be expressed as the summation of a stationary part V0(ρ) and a non-stationary part [1,29], Vgr(ρ,φ,t)=V0(ρ)+V1(ρ)cos[ωt−mφ+Φ(ρ)].(17)

This potential itself contains a spiral structure with two spiral arms, formed at the local maxima of mass density and minima of potential.

Now we look back on our spiral-galaxies-inspired structured light and demonstrate that the optical potential is the analog of a gravitational potential, while the mass density of the trapped nanoparticles is the analog of the mass density wave. The optical potential wells are shaped by light intensity through gradient forces. Thus, we have Eq. (11) for the optical potential, and we showed in Eq. (17) the gravitational potential for spiral galaxies. If we perform the substitution (Imax+Imin)/2→V0(ρ), (Imax−Imin)/2→V1(ρ), l→m, ΔΦ(ρ)→Φ(ρ) for the corresponding terms, the optical potential can exactly simulate the gravitational potential in the density wave theory. Moreover, the asymptotic logarithmic form of ΔΦ(ρ) also matches the selection of the radial phase in the density wave theory Φ(ρ)=φ0−(l/tanp)ln(ρ) [15], where p is the pitch angle. For nanoparticles trapped in potential wells, they form a spiral mass density distribution. Figure 4(d) shows the spiral set decomposition of the mass density, which demonstrates an l=2 spiral as the fundamental one. This agrees with the mass density wave in Eq. (16) for m=2. Therefore, we can mimic density waves through the structured light.

In conclusion, inspired by spiral galaxies, we have generated spiral-structured light using the interference of a vortex wave and a trumpet phased wave, and we have performed an optical trapping experiment to show the forming of the spiral-galaxy-like structures using nanoparticles. We demonstrate that the number of spiral arms is determined by the topological charge of the vortex field. We discussed the optical analog of density wave theory and the physical nature of the trumpet wave based on transformation optics. The trumpet light is modulated by the gradient refractive index medium to mimic spacetime curvature induced by black holes. Our model shows that the trumpet phase will affect the pitch angle of spiral arms through wavefront curvature, drawing a similarity to the influence of Schwarzschild radius on spiral arms in astronomical observations. Our work, from an optical analogy, might be helpful to understand the mechanism of spiral galaxy formation.