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[Pathik Sahoo](https://orcid.org/0000-0002-5102-9482), [Pushpendra Singh](https://orcid.org/0000-0002-7274-6683), Jhimli Manna, Ravindra P. Singh, [Jonathan P. Hill](https://orcid.org/0000-0002-4229-5842), [Tomonobu Nakayama](https://orcid.org/0000-0001-9696-475X), Subrata Ghosh, [Anirban Bandyopadhyay](https://orcid.org/0000-0002-8823-4914)

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[A Third Angular Momentum of Photons](https://mdr.nims.go.jp/datasets/78945726-b5d6-4398-b3af-50e1c94fea6f)

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A Third Angular Momentum of PhotonsCitation: Sahoo, P.; Singh, P.; Manna,J.; Singh, R.P.; Hill, J.P.; Nakayama, T.;Ghosh, S.; Bandyopadhyay, A. AThird Angular Momentum ofPhotons. Symmetry 2023, 15, 158.https://doi.org/10.3390/sym15010158Academic Editor: GiovanniAngiulliReceived: 18 December 2022Revised: 30 December 2022Accepted: 3 January 2023Published: 5 January 2023Copyright: © 2023 by the authors.Licensee MDPI, Basel, Switzerland.This article is an open access articledistributed under the terms andconditions of the Creative CommonsAttribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).symmetryS SArticleA Third Angular Momentum of PhotonsPathik Sahoo 1,2, Pushpendra Singh 1,2 , Jhimli Manna 2, Ravindra P. Singh 3, Jonathan P. Hill 1 ,Tomonobu Nakayama 1 , Subrata Ghosh 4,5 and Anirban Bandyopadhyay 1,2,*1 International Center for Materials Nanoarchitectronics (MANA), NIMS, 1-2-1 Sengen,Tsukuba 3050047, Ibaraki, Japan2 Research Center for Advanced Measurement and Characterization (RCAMC), NIMS, 1-2-1 Sengen,Tsukuba 3050047, Ibaraki, Japan3 Quantum Science & Technology Laboratory, Physical Research Laboratory, Navrangpura,Ahmedabad 380009, Gujarat, India4 Chemical Science and Technology Division, CSIR-North East Institute of Science and Technology, NEIST,Jorhat 785006, Assam, India5 Academy of Scientific and Innovative Research (AcSIR), Ghaziabad 201002, Uttar Pradesh, India* Correspondence: anirban.bandyo@gmail.comAbstract: Photons that acquire orbital angular momentum move in a helical path and are observed asa light ring. During helical motion, if a force is applied perpendicular to the direction of motion, anadditional radial angular momentum is introduced, and alternate dark spots appear on the light ring.Here, a third, centrifugal angular momentum has been added by twisting the helical path furtheraccording to the three-step hierarchical assembly of helical organic nanowires. Attaining a thirdangular momentum is the theoretical limit for a photon. The additional angular momentum convertsthe dimensionless photon to a hollow spherical photon condensate with interactive dark regions. Astream of these photon condensates can interfere like a wave or disintegrate like matter, similar to thebehavior of electrons.Keywords: light–matter interaction; phase singularity; optical vortex; helical nanowireMain TextVortices are ripples in the homogeneous distribution of fields. They diminish graduallyafter giving insights into 4D spacetime in astrophysics [1], form lattices in superconduc-tors [2], operate algorithms using electromechanical nested rhythms [3], etc. All vorticesdissipate and eventually disappear. Here, we have reshaped vortices of photons [4–10]into a 3D particle, addressing the problem of their transient existence. Analogous to 3Dobjects with spatial dimensions on three orthogonal axes, we have added three angularmomenta along three orthogonal axes of an electromagnetic energy packet to create a 3Dshape of fields. Design optimization of synthetic organic helical nanowires was used tocontrol the passage of incident photons so that three mutually perpendicular dipolar forcescreate a hollow sphere of light by interference [11–15]. These condensate-like photonicstructures interfere like waves and disintegrate into constituent photonic structures; lightstructure does break into pieces. Dimensionless photons acquire dimensions and form andcreate non-dissipative ripples in spacetime curvature that might be tuned using an appliedelectromagnetic signal, such as radio or microwaves. Vortices are known to interfere,which could be used to build complex structures, including knots of darkness [16–18] andcomplex phase structures for quantum computing [19–21], however, the engineering ofoptical vortices has been primarily limited to two dimensions [22–25]. Here, an explicittheory is built and experimentally verified [26] to universally design a 3D matrix of threeinterconnected helical nanowires that naturally generates a standalone 3D vortex. Thesame design protocol can be used to create system invariant virtual particles of differentfields, from turbulences in the atomic knots [27] to the quantum fluids [28].Symmetry 2023, 15, 158. https://doi.org/10.3390/sym15010158 https://www.mdpi.com/journal/symmetryhttps://doi.org/10.3390/sym15010158https://doi.org/10.3390/sym15010158https://creativecommons.org/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://www.mdpi.com/journal/symmetryhttps://www.mdpi.comhttps://orcid.org/0000-0002-7274-6683https://orcid.org/0000-0002-4229-5842https://orcid.org/0000-0001-9696-475Xhttps://orcid.org/0000-0002-9104-517Xhttps://orcid.org/0000-0002-8823-4914https://doi.org/10.3390/sym15010158https://www.mdpi.com/journal/symmetryhttps://www.mdpi.com/article/10.3390/sym15010158?type=check_update&version=1Symmetry 2023, 15, 158 2 of 10Here, we have transformed a dimensionless photon into a cavity-free standalone vortexstructure under ambient conditions using a light–matter interaction that condenses photonsto acquire a volume, which can be superposed to obtain both wave-like interference andparticle-like disintegration. A 3D template is used where electron density distributes in anintricate 3D path [22] wherein incident photons experience three orthogonal fields as theyinteract with matter. Templates for light–matter interaction have so far been constructedusing only 2D surfaces, with curves having dimensions of the order of wavelength toreshape waveform using only the reflection of light [23,24]. For the 3D semitransparentcavity used here, photons do not reflect but refract and transmit through the overlappingcavity boundaries. In contrast to the surfaces of the existing 2D templates, the entire cavityis engaged in reshaping the emitted photon. If designed suitably, the emitted photon’sstructure resembles the shape of the template it passes through. The unprecedentedthree-mode interactions of photons with three dimensions of matter make possible thedesign of a cavity that resembles the expected singularity region of the reshaped photonpost-light–matter interaction.A phase singularity is neither zero nor an astronomically large value; it is an undefinedentity when the phase gradient varies arbitrarily in space. Adding an angular momentumrequires encoding a phase singularity in the wavefront in a particular direction [11–13].However, the simultaneous encoding of three-phase singularities in the three perpendic-ular directions is not known. Light interacts with the electron density distribution of amaterial [14] so that the object’s surface morphology modulates the vortices of the emittedphotons and affects the singularity regions [15].Figure 1A shows a helical nanowire wherein energy would take a spiral path, leavinga central core similar to an empty space encoded as a phase singularity region when anincident photon adopts a spiral route during transmission through the helical electrondensity distribution path. Projected light is then not a dot with orbital angular momentumθ; it is a ring of light known as an optical vortex. Figure 1B shows a spiral of a spiral wherethe singularity region is a helical tube. Incident photons split into two entangled photonsources and interfere, creating alternate dark and bright spots on the projected ring of light,and we obtain a radial angular momentum φ. Figure 1C illustrates the seminal work ofNye, who demonstrated mathematically that a twisted helix or a helix of a helix couldemit particle-like standalone optical vortices [25] (see Visualization 1). For this reason, weopted to use a twisted helix, similar to that naturally abundant in microtubules or DNA.Here, incident photons experience three orthogonal electron density distributions; threeperpendicular forces rotate the optical vortex centrifugally to obtain the third centrifugalangular momentum ω as a hollow sphere of light. The dark regions on the particle-likesphere of light are singularity regions; system points move along the borderline betweendark and bright regions as constituents of final optical condensate, represented by a singlewave function Ψ(θ, φ, ω). Face-to-face collision between a pair of vortices generated inwater forms small vortices at the ring perimeter [26]. We have repeated that experimenthere using light vortices to disintegrate the wave function of the entire condensate andcreate a ring of light with dark and bright spots. The light condensate is a single particlethat exhibits both interference like a wave and disintegration like a particle.Since the shape of a cavity or template and cavity-modified photon is similar, wehave used a three-level self-assembly in which molecules first form a spiral structurewith a diameter of ~15–20 nm (Methods, online). The resulting spiral structure thenself-assembles into a larger helix with a diameter of ~120 nm, which finally forms as ahierarchically structured helix with a diameter of ~12 µm. We have also studied thisreaction environment using computer simulations [28,29]. With increasing numbers ofmolecules in the self-assembly simulations, three orthogonal phase transitions occur asconstituent molecular structures bind (Figure 2A, see Visualization 2). Incorporatingsuch 3D dynamics requires a tensor to express all states Ψ on a spherical optical surface,Ψ(θ, φ, ω) = CosθSinω + eiφSinθCosω. A plot of this function has the form of a tripletof teardrops oriented on 120◦ solid angular planes (Figure 2B, right). The optical surfaceSymmetry 2023, 15, 158 3 of 10is the phase surface. We insert a pair of electrodes into the reaction vessel to measuresignals during the three phase transitions. The connected spectrum analyzer showsa triplet of resonance frequencies. To estimate the dimensions of the self-assembledstructures as they evolve through the three transmission modes, we focused a camera onthe reaction vessel to detect diffracted nano-micro-particles at nucleation centers as theprecursor solution cooled. Simultaneously, a polarized He-Ne laser (beam cross-section:1 mm2) was transmitted through the nucleation centers and projected on a screen, wherea projection camera captured the sequence of events. First, birefringence—or a pair oflight spots—is observed above ~10 nm particle size [30] (Figure S2), then fast flashing oflight spheres occurs for 0.01 s, followed by the appearance of an interference pattern ofparallel beams. This is the first optical vortex based on an orbital angular momentum.Subsequently, the initially formed vortex disappears, with no transmission visible at theprojection screen, and no diffused particles are detected in the reaction vessel camera.We refer to this dark state as a metastable state. Following this state, particles reappearand converge, and we observe photons with a radial angular momentum. A second sub-sequent metastable state is followed finally by forming the third condensate, consistingof a sphere of light. From scanning electron micrographs (SEM), we match the size ofobjects generating light diffraction to assess the most probable appearance of nanowireassemblies that generate photon structures with one, two, or three angular momenta(see Visualization 3 and its description (Text S4 online. Visualization 3)).Symmetry 2023, 15, x FOR PEER REVIEW 3 of 10    Figure 1. A generic engineering principle to synthesize optical structures having three angular momenta. The transition from (A) a scalar vortex (𝜃) to (B) a vector vortex (𝜃, 𝜙) then to (C) a tensor vortex or a 3D optical sphere Ψ(𝜃,𝜙, 𝜔) is depicted. In each panel, the top row shows vortices pro-jected on a screen, the middle row indicates the photon transmission pathway, and the bottom row shows the electron density distribution path in a template that regulates light-matter interaction to generate a particular vortex shape. Three complex functions describe the 1D, 2D, and 3D optical vortices. In panel C (lower part), a black dot on the helix indicates the added periodicity in the electron density distribution to create the helix of a helix described by Nye [25]. Since the shape of a cavity or template and cavity-modified photon is similar, we have used a three-level self-assembly in which molecules first form a spiral structure with a diameter of ~15–20 nm (Methods, online). The resulting spiral structure then self-assem-bles into a larger helix with a diameter of ~120 nm, which finally forms as a hierarchically structured helix with a diameter of ~12 μm. We have also studied this reaction environ-ment using computer simulations [28,29]. With increasing numbers of molecules in the self-assembly simulations, three orthogonal phase transitions occur as constituent molec-ular structures bind (Figure 2A, see Visualization 2). Incorporating such 3D dynamics re-quires a tensor to express all states Ψ  on a spherical optical surface, Ψ(𝜃, 𝜙, 𝜔) =𝐶𝑜𝑠𝜃𝑆𝑖𝑛𝜔 + 𝑒𝑖𝜙𝑆𝑖𝑛𝜃𝐶𝑜𝑠𝜔. A plot of this function has the form of a triplet of teardrops oriented on 120° solid angular planes (Figure 2B, right). The optical surface is the phase surface. We insert a pair of electrodes into the reaction vessel to measure signals during the three phase transitions. The connected spectrum analyzer shows a triplet of resonance frequencies. To estimate the dimensions of the self-assembled structures as they evolve through the three transmission modes, we focused a camera on the reaction vessel to de-tect diffracted nano-micro-particles at nucleation centers as the precursor solution cooled. Simultaneously, a polarized He-Ne laser (beam cross-section: 1 mm2) was transmitted through the nucleation centers and projected on a screen, where a projection camera cap-tured the sequence of events. First, birefringence—or a pair of light spots—is observed above ~10 nm particle size [30] (Figure S2), then fast flashing of light spheres occurs for 0.01 s, followed by the appearance of an interference pattern of parallel beams. This is the first optical vortex based on an orbital angular momentum. Subsequently, the initially formed vortex disappears, with no transmission visible at the projection screen, and no diffused particles are detected in the reaction vessel camera. We refer to this dark state as a metastable state. Following this state, particles reappear and converge, and we observe photons with a radial angular momentum. A second subsequent metastable state is fol-lowed finally by forming the third condensate, consisting of a sphere of light. From scan-ning electron micrographs (SEM), we match the size of objects generating light diffraction to assess the most probable appearance of nanowire assemblies that generate photon structures with one, two, or three angular momenta (see Visualization 3 and its descrip-tion (text S4 online. Visualization 3)). Figure 1. A generic engineering principle to synthesize optical structures having three angularmomenta. The transition from (A) a scalar vortex (θ) to (B) a vector vortex (θ, φ) then to (C) a tensorvortex or a 3D optical sphere Ψ(θ, φ, ω) is depicted. In each panel, the top row shows vorticesprojected on a screen, the middle row indicates the photon transmission pathway, and the bottomrow shows the electron density distribution path in a template that regulates light-matter interactionto generate a particular vortex shape. Three complex functions describe the 1D, 2D, and 3D opticalvortices. In panel (C) (lower part), a black dot on the helix indicates the added periodicity in theelectron density distribution to create the helix of a helix described by Nye [25].Since the cavity shape for light–matter interaction and a reshaped photon with threeangular momenta are similar, both emitted photon condensate and organic condensateresponsible for its formation have the same energy expression or Hamiltonian H. Fourlight–matter interactions dominate refraction and transmission in a twisted helix to ren-der volume to a photon by adding a third angular momentum. Synthesis of self-similarhelical-symmetry-based superstructures have been grown spontaneously using self-assembly from the nano-scale to the visible scale [31]. The organic twisted helix structurethat generates photon condensate was recreated in a Computer Science and Technology(CST) simulator. Four key features were noted and included in H = H1 + H2 + H3 + H4(Figure S1a,b, Text S1A). An electromagnetic field flowing in a loop on the helix with mul-tiple parallel pathways of electron density loops maintaining an energy gap. Quantizedflux exchanges between the loops, where each loop acts as a distinct energy level (H1). H1Symmetry 2023, 15, 158 4 of 10regulates the refraction of photons. The orientations of helical nanowires in three layersgrown from a triplet of precursor molecules are directed in different directions, leadingto the anisotropy of energy flow in transmission (H2). Refracted and transmitted partsgoverned by geometric parameters of the twisted helix, length, pitch, diameter, and latticearea act as two distinct coherent sources, which interfere and generate a 3D vortex (H3).The additional twist in the coil, which Nye has proposed would generate free 2D lightstructures, is implemented here by using an organic structure containing screw and edgedislocations [25] and eventually governs the shape of singularity regions (H4, Figure S1c).The function Γi(r, t) = 3 ∑i cosxi + 4 ∏i cosxi includes three periodic variations of electrondensity of states along three orthogonal axes, which could have at most 12 singularityregions on the hollow sphere (Figure S1b). Complex singularity regions have alreadybeen realized using femtosecond pulses [32]. For θ, a photon’s azimuthal angular mo-mentum is l}; for φ, its radial angular momentum is p}; for ω, centrifugal angularmomentum is o}; here, (l, p, o) are topological charges. Since all twelve values on a spherefollow −π ≤ Γi(r, t) ≤ +π, the third angular momentum of the largest dark region is~0.2π} kg m2/sec. This suggests a complex knot structure [33].Symmetry 2023, 15, x FOR PEER REVIEW 4 of 10   Since the cavity shape for light–matter interaction and a reshaped photon with three angular momenta are similar, both emitted photon condensate and organic condensate responsible for its formation have the same energy expression or Hamiltonian Η. Four light–matter interactions dominate refraction and transmission in a twisted helix to render volume to a photon by adding a third angular momentum. Synthesis of self-similar heli-cal-symmetry-based superstructures have been grown spontaneously using self-assembly from the nano-scale to the visible scale [31]. The organic twisted helix structure that gen-erates photon condensate was recreated in a Computer Science and Technology (CST) simulator. Four key features were noted and included in Η = Η1 + Η2 + Η3 + Η4 (Figure S1a,b, text S1A). An electromagnetic field flowing in a loop on the helix with multiple parallel pathways of electron density loops maintaining an energy gap. Quantized flux exchanges between the loops, where each loop acts as a distinct energy level (H1). H1 regulates the refraction of photons. The orientations of helical nanowires in three layers grown from a triplet of precursor molecules are directed in different directions, leading to the anisotropy of energy flow in transmission (H2). Refracted and transmitted parts gov-erned by geometric parameters of the twisted helix, length, pitch, diameter, and lattice area act as two distinct coherent sources, which interfere and generate a 3D vortex (H3). The additional twist in the coil, which Nye has proposed would generate free 2D light structures, is implemented here by using an organic structure containing screw and edge dislocations [25] and eventually governs the shape of singularity regions (H4, Figure S1c). The function Γ𝑖(𝑟, 𝑡) = 3∑ 𝑐𝑜𝑠𝑥𝑖 + 4∏ 𝑐𝑜𝑠𝑥𝑖𝑖𝑖  includes three periodic variations of elec-tron density of states along three orthogonal axes, which could have at most 12 singularity regions on the hollow sphere (Figure S1b). Complex singularity regions have already been realized using femtosecond pulses [32]. For 𝜃, a photon’s azimuthal angular momentum is 𝑙ℏ; for 𝜙, its radial angular momentum is 𝑝ℏ; for 𝜔, centrifugal angular momentum is 𝑜ℏ ; here, ( 𝑙, 𝑝, 𝑜 ) are topological charges. Since all twelve values on a sphere follow −𝜋 ≤ Γ𝑖(𝑟, 𝑡) ≤ +𝜋, the third angular momentum of the largest dark region is ~0.2𝜋ℏ kg m2/sec. This suggests a complex knot structure [33].  Figure 2. Triplet resonance bands drive molecular self-assembly suitable for three angular mo-menta of a photon. (A) Density of states of the single, double, and triple molecular precursors (IU-PAC Name (S) −Phenyl-hexadecanoylamino-acetic acid methyl ester) are plotted as a function of energy (THz) as molecular dynamics simulation runs in a potential box (see Visualization 2). Reso-nance frequencies are the highest−energy density peaks derived at HOMO and LUMO levels. The band for interacting with available free thermal noise (5−6 THz) is highlighted in orange. Three arrows indicate HOMO + 1, HOMO + 2, and HOMO + 3 from the kT band. (B) Screenshot depicts a naturally forming helical triplet from Visualization 2 (left). From this structure, higher−order triplet self-assembly is observed on the micrometer to millimeter scale (immediate right). At most right, Figure 2. Triplet resonance bands drive molecular self-assembly suitable for three angular mo-menta of a photon. (A) Density of states of the single, double, and triple molecular precursors(IUPAC Name (S) −Phenyl-hexadecanoylamino-acetic acid methyl ester) are plotted as a functionof energy (THz) as molecular dynamics simulation runs in a potential box (see Visualization 2).Resonance frequencies are the highest−energy density peaks derived at HOMO and LUMO levels.The band for interacting with available free thermal noise (5−6 THz) is highlighted in orange. Threearrows indicate HOMO + 1, HOMO + 2, and HOMO + 3 from the kT band. (B) Screenshot depicts anaturally forming helical triplet from Visualization 2 (left). From this structure, higher−order tripletself-assembly is observed on the micrometer to millimeter scale (immediate right). At most right,function CosθCosω + iejφSinθSinω is plotted for three variables θ, φ, ω. Three loops are at 120◦ solidangles and symmetrically oriented in a virtual 3D sphere. (C) Spectrum analyzer measures transmis-sion coefficients (S21−S12) in situ across the helical nanowire superstructure. (D) Two time profiles ofpure light diffracted (top) and reflected, transmitted, and refracted by growing structure as precursormolecules form 1D nanowire to helically twisted structures. Live visualization of four distinct phasetransitions and metastable states as observed using the reaction vessel camera (top, Visualization 3)and optical vortex condensate formation observed using the projection camera (bottom, Visualization4). The actual size of optical structures is 5 µm × 5 µm in the beaker, and when they are magnifiedusing a lens and projected on a wall and presented here, they are 10 cm × 10 cm (each window sizeof Figure 2D).We rotated the reaction vessel in which helical nanowires form and reorganize, countedthe dark or singularity regions on the light sphere, and estimated their shapes (Figure 3A).Symmetry 2023, 15, 158 5 of 10On the projected hollow light sphere, there exist either polygons, whose vertices arecomposed of a framework of light columns or circular dark regions whose perimeters arecomposed of light rings. Two parts, the front and back of the light structure, are detectedseparately; taken together, we confirm that it is a standalone structure (Figure 3B top). Threetime profiles of magnetic flux variations on the projected vortex along three orthogonaldirections of its posterior reveal streams of spherical pulses (Figure 3B bottom). Thus,the gel produces a linear chain of free optical structures under ambient conditions. Werepeatedly dissolved the gel superstructure by heating it at 75 ◦C, cooled it to reform,and observed the number of dark regions, n, their areas, and the average distances 〈x〉between regional centers (see Visualization 4, Figure 3C, Text S1 B). This process wasrepeated exhaustively for different gels, revealing that the geometric arrangements of darkregions on the light sphere are not random but are specific to gel precursor molecules(Table S1). The n vs. pitch/diameter plot for helical nanowires shows that the darkregions of 3D light structures are linked to the structural symmetry of the helical nanowireassembly (Figure 3D). We also used mirrors to reflect the optical superstructure with thegel-containing reaction vessel fixed; each dark region is the constituent photon’s phasesphere (Figure 3D). Incident photons interact with the characteristic length, pitch, anddiameter of the hierarchical nanowire network, and the resulting dark regions map Blochspheres connected by geometric shapes. Table S1 (Text S2) shows the 3D assembly of Blochspheres or topology of multiple vortices representing each gel [34].Symmetry 2023, 15, x FOR PEER REVIEW 5 of 10   function 𝐶𝑜𝑠𝜃𝐶𝑜𝑠𝜔 + 𝑖𝑒𝑗𝜙𝑆𝑖𝑛𝜃𝑆𝑖𝑛𝜔 is plotted for three variables 𝜃, 𝜙, 𝜔. Three loops are at 120° solid angles and symmetrically oriented in a virtual 3D sphere. (C) Spectrum analyzer measures transmission coefficients (S21−S12) in situ across the helical nanowire superstructure. (D) Two time profiles of pure light diffracted (top) and reflected, transmitted, and refracted by growing structure as precursor molecules form 1D nanowire to helically twisted structures. Live visualization of four distinct phase transitions and metastable states as observed using the reaction vessel camera (top, Visualization 3) and optical vortex condensate formation observed using the projection camera (bot-tom, Visualization 4). The actual size of optical structures is 5 μm × 5 μm in the beaker, and when they are magnified using a lens and projected on a wall and presented here, they are 10 cm × 10 cm (each window size of Figure 2D). We rotated the reaction vessel in which helical nanowires form and reorganize, counted the dark or singularity regions on the light sphere, and estimated their shapes (Figure 3A). On the projected hollow light sphere, there exist either polygons, whose ver-tices are composed of a framework of light columns or circular dark regions whose pe-rimeters are composed of light rings. Two parts, the front and back of the light structure, are detected separately; taken together, we confirm that it is a standalone structure (Figure 3B top). Three time profiles of magnetic flux variations on the projected vortex along three orthogonal directions of its posterior reveal streams of spherical pulses (Figure 3B bot-tom). Thus, the gel produces a linear chain of free optical structures under ambient con-ditions. We repeatedly dissolved the gel superstructure by heating it at 75 °C, cooled it to reform, and observed the number of dark regions, n, their areas, and the average distances < 𝑥 > between regional centers (see Visualization 4, Figure 3C, text S1 B). This process was repeated exhaustively for different gels, revealing that the geometric arrangements of dark regions on the light sphere are not random but are specific to gel precursor mole-cules (Table S1). The 𝑛 vs. pitch/diameter plot for helical nanowires shows that the dark regions of 3D light structures are linked to the structural symmetry of the helical nanowire assembly (Figure 3D). We also used mirrors to reflect the optical superstructure with the gel-containing reaction vessel fixed; each dark region is the constituent photon’s phase sphere (Figure 3D). Incident photons interact with the characteristic length, pitch, and di-ameter of the hierarchical nanowire network, and the resulting dark regions map Bloch spheres connected by geometric shapes. Table S1 (text S 2) shows the 3D assembly of Bloch spheres or topology of multiple vortices representing each gel [34].  Figure 3. Confirmation of particle-like 3D morphology of the optical vortex condensate. (A) Var-iation in the vortex shape observed using the projection camera, as helical nanowire assembly is rotated by rotating the vessel. White dot placed at a selected light loop on the sphere confirms its return to origin following a 360° rotation (see Visualization 4). (B) Optical vortex condensate is pro-jected on a screen 7 m distant (top), in order to measure positional values (x,y,z) and its Figure 3. Confirmation of particle-like 3D morphology of the optical vortex condensate. (A) Varia-tion in the vortex shape observed using the projection camera, as helical nanowire assembly is rotatedby rotating the vessel. White dot placed at a selected light loop on the sphere confirms its returnto origin following a 360◦ rotation (see Visualization 4). (B) Optical vortex condensate is projectedon a screen 7 m distant (top), in order to measure positional values (x,y,z) and its correspondingmagnetic flux values (red, green, yellow) in microtesla (bottom). The scanner is moved periodicallyback and forth across the structure, and the magnetic flux values capture five periods. Extent of asingle period is indicated by the arrow. (C) Ratio of pitch and diameter for six gels with the numberof dark domains on the sphere. (D) Vortex images captured by rotating sample and mirror reflectionseparately to eliminate the possibility that the 3D vortex is the half-sphere of a Gaussian wave packet.360◦ rotation of the sphere regenerates the same pattern in input and output. 3D assembly of Blochspheres shown adjacent to a rotational phase hologram is an invariant 3D sphere networks formedby theoretically simulating the holographic laser projection from H. Geometrically connected Blochspheres measures phase on the surface of a host Bloch sphere. The actual size of optical structures is5 µm × 5 µm in the beaker, and when they are magnified using a lens and projected on a wall andpresented here, they are 10 cm × 10 cm (each window size of all panels).Symmetry 2023, 15, 158 6 of 10To provide further evidence that the sphere of light is a unit photon structure withthree angular momenta, we passed two entangled photons at 90◦ out-of-phase throughtwo gel solutions (see Figure 4A,B). The two gels emit two optical superstructures as vortexspheres. We interfered the optical spheres at a beam splitter junction by two methods:first at 45◦, then by orienting the splitter further by 90◦. A Canada balsam layer joins thetwo parts of the beam splitter. Here, the layer thickness was adjusted to the He-Ne laserwavelength to split the photon into two parts, one reflected and the other transmitted dueto evanescent wave coupling as frustrated total internal reflection (FTIR). We could selectwhether vortex spheres superpose at the junction on the same or opposite side duringinterference. In the first protocol, two spherical vortices pass through one common part ofthe beam splitter and interfere at the junction surface. Since two vortex spheres do not crossthe junction, this is a reflection mode superposition (Figure 4A). Second, two vortices enterfrom two sides of the beam splitter and arrive at the same point inside the Canada balsamjunction. Since the vortex spheres must cross the junction for interference, we achievetransmission mode superposition (Figure 4B). Due to FTIR, these two interferences usedifferent energy regions of the same vortex sphere.Symmetry 2023, 15, x FOR PEER REVIEW 7 of 10    Figure 4. Wave-particle duality of the photon condensate or optical 3D sphere. In panels A and B, an interferometer where He-Ne laser source S is split into two entangled components by a beam splitter (two β-Barium Borate or BBO crystals fixed together) was used. Two 90° out-of-phase po-larizers filter out two different parts, which pass through different gel solutions 𝐺𝐴 and 𝐺𝐵, gener-ating two vortices with +𝜔 and −𝜔 parts (green clockwise, red counter-clockwise), interfered at the beam splitter BS acting as interferometer I. For Panel A, the output is taken as an interference pattern, and for panel B, the output is taken as a 2D vortex. The only difference between the exper-imental setup for panel A and panel B is the orientation of the junction of the two parts of the beam splitter assigned as interferometer I. The actual size of optical structures is 5 μm × 5 μm in the beaker, and when they are magnified using a lens and projected on a wall and presented here, they are 10 cm × 10 cm (each window size all panels). Using the third angular momentum, one can reshape the photon’s waveform into a standalone structure. By tuning the light sphere or ellipsoid’s diameter or distance be-tween two endpoints at which the intensity of the waveform becomes zero, it is possible to precisely regulate the pressure exerted by a 3D photon stream. Furthermore, by adjust-ing the singularity area, particles might be encapsulated in the hollow sphere for trans-portation. Mass transport using light could be possible in the future using 3D optical struc-tures. The free particle-like complex photon structure derived by Nye is used here to cre-ate a template for light–matter interaction [25]. However, using that unique light–matter interaction, the existing idea of using a 2D template that reflects a photon is replaced here by a 3D template that refracts and transmits a look-alike photon. Introduction to 3D-tem-plate-based engineering of light would open the door to more complex structures of light that could memorize codes to instruct and operate machines at a remote distance and build circuit-less, algorithm-free computers [35]. One of the most important outcomes of the 3D template engineering is the use of birefringence, which generates entangled pho-tons in a hydrogen-bonded nanowire-made cavity that vibrates to condense photons into an integrated form. Entangled photons are fused to create a single photon structure; this is the basic engineering for organic-gel-based quantum computers, quantum cryptog-raphy, reverse engineering of polyatomic time crystals [36], brain inspired computing [37], etc. Therein, geometric shapes made of coupled quantum states or Bloch spheres are editable in a particle-like photonic structure whose shape would be fairly similar to the cavity that the photon encounters. Therefore, one would design templates, and the result-ant photon structure would look like the template.  Figure 4. Wave-particle duality of the photon condensate or optical 3D sphere. In panels (A,B), aninterferometer where He-Ne laser source S is split into two entangled components by a beam splitter(two β-Barium Borate or BBO crystals fixed together) was used. Two 90◦ out-of-phase polarizersfilter out two different parts, which pass through different gel solutions GA and GB, generating twovortices with +ω and −ω parts (green clockwise, red counter-clockwise), interfered at the beamsplitter BS acting as interferometer I. For Panel (A), the output is taken as an interference pattern,and for panel (B), the output is taken as a 2D vortex. The only difference between the experimentalsetup for panel (A) and panel (B) is the orientation of the junction of the two parts of the beamsplitter assigned as interferometer I. The actual size of optical structures is 5 µm × 5 µm in thebeaker, and when they are magnified using a lens and projected on a wall and presented here, theyare 10 cm × 10 cm (each window size all panels).In reflection mode, both spheres arrive at the same side of the splitter and interfere.A wave-like interference is observed with parallel grids of constructive and destructiveinterference visible (Figure 4A). However, when both beams of Ψ(θ, φ, ω)—namely, A(θ, φ, ω) and B (θ, φ,−ω)—arrive at the same point following the transmission modethrough the different sides of a beam splitter, then one of the angular momenta is eliminatedSymmetry 2023, 15, 158 7 of 10and the output returns as the optical vortex beam (A + B = Ψ(θ, φ); Figure 4B). Therefore,we can derive a photon with two angular momenta from a photon having three angularmomenta. Disintegrating 3D photons into 2D photon-fragments is quite similar to thecollision of water ripples [26]. It is a new type of wave–particle duality. To create A and B,the entangled photons are made 90◦ out-of-phase using paired BBO crystals before beingmade incident on the gel solution. Polarization of photons does not affect scalar and vectorangular momenta. However, it changes the direction of the twist of twists, i.e., the tensorangular momentum ω, resulting in two spherical vortices or photon condensates, ω and−ω. Since the twist of a twist effect resulting in ±ω is canceled, one of the three emittedvortices is observed as a vector vortex beam φ. It is possible to fine-tune the collapsed 2Dvortices obtained by colliding the optical spheres. We can then select and edit one of thetwelve topological quantum states, each with a distinct clocking direction embedded in theoptical sphere, using Γi(r, t).Using the third angular momentum, one can reshape the photon’s waveform intoa standalone structure. By tuning the light sphere or ellipsoid’s diameter or distancebetween two endpoints at which the intensity of the waveform becomes zero, it is possibleto precisely regulate the pressure exerted by a 3D photon stream. Furthermore, by adjustingthe singularity area, particles might be encapsulated in the hollow sphere for transportation.Mass transport using light could be possible in the future using 3D optical structures. Thefree particle-like complex photon structure derived by Nye is used here to create a templatefor light–matter interaction [25]. However, using that unique light–matter interaction,the existing idea of using a 2D template that reflects a photon is replaced here by a 3Dtemplate that refracts and transmits a look-alike photon. Introduction to 3D-template-based engineering of light would open the door to more complex structures of light thatcould memorize codes to instruct and operate machines at a remote distance and buildcircuit-less, algorithm-free computers [35]. One of the most important outcomes of the3D template engineering is the use of birefringence, which generates entangled photonsin a hydrogen-bonded nanowire-made cavity that vibrates to condense photons into anintegrated form. Entangled photons are fused to create a single photon structure; this isthe basic engineering for organic-gel-based quantum computers, quantum cryptography,reverse engineering of polyatomic time crystals [36], brain inspired computing [37], etc.Therein, geometric shapes made of coupled quantum states or Bloch spheres are editablein a particle-like photonic structure whose shape would be fairly similar to the cavity thatthe photon encounters. Therefore, one would design templates, and the resultant photonstructure would look like the template.Supplementary Materials: The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/sym15010158/s1. S1. Materials and Methods; A. Preparation ofGelators; B. Birefringence and optical vortex study. S2. Supplementary Texts; 2.1. Hamiltonian of thehelical nanowire; 2.2. Optical interaction with the helical nanowire: 2.3. Measuring third angularmomentum for seventeen gels: S3. Supplementary Figures; Figure S1. Hamiltonian for gel’s light–matter interaction; Figure S2. Birefringence of a helical nanowire. Table S1; Nested Bloch spheresor Hilbert spaces for six organic gel precursors. S4. Four visualization descriptions; Visualization 1:Basic concept for third angular momentum, Visualization 2: Molecular dynamics of gel precursors.Visualization 3: Four phase transitions for vortex condensate, Visualization 4: Statistical database ofrepeated experiments. S5. References [38–58].Author Contributions: Conceptualization, A.B.; data curation, P.S. (Pathik Sahoo); formal analysis,P.S. (Pathik Sahoo) and P.S. (Pushpendra Singh); funding acquisition, A.B.; investigation, A.B.;methodology, P.S. (Pathik Sahoo) and J.M.; software, P.S. (Pushpendra Singh); supervision, A.B.;writing—original draft, A.B.; writing—review and editing, P.S. (Pathik Sahoo), P.S. (PushpendraSingh), J.M., R.P.S., J.P.H., T.N., S.G. and A.B. All authors have read and agreed to the publishedversion of the manuscript.https://www.mdpi.com/article/10.3390/sym15010158/s1https://www.mdpi.com/article/10.3390/sym15010158/s1Symmetry 2023, 15, 158 8 of 10Funding: The authors acknowledge the Asian Office of Aerospace R&D (AOARD), a part of theUnited States Air Force (USAF), for Grant no. FA2386-16-1-0003 (2016–2019) on electromagneticresonance-based communication and intelligence of biomaterials.Data Availability Statement: See Supplemental Document for supporting content.Acknowledgments: The authors acknowledge the Asian Office of Aerospace R&D (AOARD), a partof the United States Air Force (USAF), for Grant no. FA2386-16-1-0003 (2016–2019) on electromagneticresonance-based communication and intelligence of biomaterials. 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