Control of spatially rotating structures in diffractive Kerr cavities
CControl of spatially rotating structures in di ff ractive Kerr cavities Alison M. Yao, Christopher J. Gibson and Gian-Luca Oppo SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, Scotland, E.U. ∗ Turing patterns in self-focussing nonlinear optical cavities pumped by beams carrying orbital angular mo-mentum (OAM) m are shown to rotate with an angular velocity ω = m / R on rings of radii R . We verify thisprediction in 1D models on a ring and for 2D Laguerre-Gaussian and top-hat pumps with OAM. Full controlover the angular velocity of the pattern in the range − m / R ≤ ω ≤ m / R is obtained by using cylindricalvector beam pumps that consist of orthogonally polarized eigenmodes with equal and opposite OAM. UsingPoincar´e beams that consist of orthogonally polarized eigenmodes with di ff erent magnitudes of OAM, the re-sultant angular velocity is ω = ( m L + m R ) / R , where m L , m R are the OAMs of the eigenmodes, assuming goodoverlap between the eigenmodes. If there is no, or very little, overlap between the modes then concentric Turingpattern rings, each with angular velocity ω = m L , R / R will result. This can lead to, for example, concentric,counter-rotating Turing patterns creating an ’optical peppermill’-type structure. Full control over the speedsof multiple rings has potential applications in particle manipulation and stretching, atom trapping, and circulartransport of cold atoms and BEC wavepackets. INTRODUCTION
Pattern formation is ubiquitous in nonlinear dynamical systems, the most famous example being Turing patterns in reaction-di ff usion systems [1]. In optical cavities, spontaneous spatial pattern formation results due to the interplay of a nonlinearity anda spatial coupling, such as di ff raction or dispersion. Such systems are very well described by the Lugiato-Lefever equation [2].In this paper we consider the e ff ect of pumping optical cavities containing a self-focussing Kerr medium with beams carryingoptical angular momentum (OAM) and show the formation of rotating Turing structures. We derive analytical expressionsthat fully describe these two-dimensional rotating Turing structures in single field (scalar) Kerr resonators and confirm ourpredictions numerically using pumps consisting of Laguerre-Gaussian modes or ‘’top-hats” carrying OAM. In particular, weshow that the angular velocity ω of the patterns is fully determined by the OAM m of the pump and the radius R of the ringstructure according to ω = m / R . Spatial structures rotating on a transverse ring including cavity solitons can be considered asslow light pulses with fully controllable speed and structure for use in optical quantum memories and delay lines. These studiescomplete early investigations that focused on optical parametric oscillators, semiconductor heterostructures and photorefractivematerials, respectively [3–5].Fully-structured light consisting of a vector superposition of two scalar OAM-carrying Laguerre-Gaussian eigenmodes withorthogonal circular polarizations [6–8], has attracted increasing attention for a number of applications [9–12]. The inclusionof a second field component in the light-matter interaction inside the cavity o ff ers further degrees of control in the shape andpolarization of the pump and the resultant nonlinear structures. In particular we show how the use of fully-structured lightto pump the cavity allows us full control over the angular velocity of the Turing structures. Using numerical simulations wedemonstrate how biasing cylindrical vector (CV) beam pumps - orthogonally polarized eigenmodes with equal and oppositeOAM ( m L = − m R ) - allows us to produce patterns with angular velocity − m / R ≤ ω ≤ m / R and that Poincar´e pumps- orthogonally polarized eigenmodes with di ff erent magnitudes of OAM - produce patterns with angular velocity ω = ( m L + m R ) / R . Applications of these rotating structures to particle manipulation, optical beam shaping and photonic devices will isdiscussed. Finally we give examples of fields with counter-rotating Turing patterns, the ‘’optical peppermill”, that may be ofparticular interest in trapping, manipulating and deforming biological specimens. THE LUGIATO-LEFEVER CASE
We start with the description of Kerr media in optical cavities through the well-known Lugiato-Lefever equation (LLE) in twotransverse dimensions [2]: ∂ t E = P − (1 + i θ ) E + i β | E | E + i ∇ E (1)where E is the intracavity field, P is the amplitude of the input pump, θ is the detuning between the input pump and the closestcavity resonance, β is proportional to the Kerr coe ffi cient of the nonlinear material, and the term with the transverse Laplacian ∇ describes di ff raction and can be written in either Cartesian or polar coordinates. The time scale has been normalised by τ p the mean lifetime of photons in the cavity given by 2 L / cT for a unidirectional ring cavity and by 4 L / cT for a Fabry-Perot cavity,with L being the cavity length, T the (intensity) transmission coe ffi cient of the cavity mirrors and c the speed of light in vacuum. a r X i v : . [ n li n . PS ] A ug The transverse spatial scale ( x , y ) has been normalised by a = (cid:18) c τ p K (cid:19) / = (cid:32) c λτ p π (cid:33) / = (cid:18) LkT (cid:19) / (2)where k and λ are the wavevector and wavelength of the input light, respectively.In order to derive some analytical results, we note that LG modes with m > R m normalised via (2). We therefore express the transverse Laplacian in polar coordinates ( R , ϕ ) and, as R can be considered aconstant, we can write the LLE Eq. (1) in one angular transverse dimension: ∂ t E = P − (1 + i θ ) E + i β | E | E + iR ∂ E ∂ϕ . (3)As the focus of this work is the e ff ect of pumping the ring with light carrying orbital angular momentum (OAM), we considerpumps of the form: P = P m e im ϕ (4)where P m is a complex amplitude independent of ϕ , and m is an integer corresponding to the topological charge of the opticalvortex. In this case we consider solutions of the form: E ( ϕ, t ) = F ( ϕ, t ) e im ϕ (5)that satisfy the equation: ∂ F ∂ t = P m − (cid:34) + i (cid:32) θ + m R (cid:33)(cid:35) F + i β | F | F − mR ∂ F ∂ϕ + iR ∂ F ∂ϕ . (6)One e ff ect of the OAM-dependent solution (5) is that the detuning is modified by an amount m / R . We note that this phaseshift is independent of the sign of OAM (i.e. left- or right-hand phase circulation) and the overall e ff ect is to increase the cavityo ff -tuning for positive θ and to (partially) compensate the detuning in the case of negative θ . Moreover, this OAM-dependentdetuning increases when the radius of the ring decreases. Homogeneous Stationary States
For any value of m , the homogenenous stationary solutions F s are obtained from: P m = F s (cid:34) + i (cid:32) θ + m R − β I s (cid:33)(cid:35) (7)where I s is the intensity of the stationary solution I s = | F s | = | E s | . Once the stationary intensity I s is selected, the amplitudeand phase of the pump field are obtained from (7) implicitly. We highlight the radial dependent detuning term that comes fromthe OAM associated with the helical phase of the stationary solution (5) and note that for m (cid:44)
0, homogeneous stationary statesof Eq. (6) correspond to stationary states for the field E that are not homogeneous in the phase ϕ . Turing instabilities on the ring: m = For m = F = E and Eq. (6) is equivalent to Eq. (3). Both Eqs. (1) and (3) are well known to display a Turing instabilityof the homogeneous stationary state. In order to analyze the stability of the solutions we introduce a small perturbation withwavevector k : E = E s e λ ( k ) δ E and neglect terms nonlinear in δ E , δ E ∗ . Performing a linear stability analysis (LSA) we find thatabove the Turing instability, both Eqs. (1) and (3) with m = k given by [2, 13] : λ ( k ) = − ± (cid:113) ∆ β I s − β I s − ∆ . (8)where ∆ = θ + k . It is clear from Eq. (8) that if the square root is imaginary, there are no instabilities since both eigenvalues havenegative real part. For a real eigenvalue to be positive, the quantity in the square root has to be larger than one. The instabilityboundary where the square root in (8) is exactly equal to one, provides a relation between the detuning ∆ (which contains thewavevector k ) and the stationary intensity I s : ∆ = β I s ± (cid:113) β I s − . (9)This shows that there is an instability threshold in the stationary intensity given by I cs = /β . For a given I s > I cs = /β the mostunstable wavevector is obtained by finding the maximum of the square root in (8) when changing ∆ : k c = (cid:112) β I s − θ . (10)Above threshold, N peaks appear along the ring separated by a distance given by, or close to, the wavelength of the Turingstructure Λ c = π/ k c . Note that for a ring of circumference 2 π R , the number of peaks is N = π R / Λ c = R (cid:112) β I s − θ , and exactly N peaks fit inside a ring of radius R to satisfy the periodic boundary conditions. Rotating solutions: m (cid:44) We now consider the case of pumps carrying OAM, i.e. m (cid:44)
0. Above the instability threshold these are seen numerically toform patterns that move around the ring at a constant angular velocity. We start by rearranging Eq. (6) such that the first orderderivatives are on the l.h.s.: ∂ F ∂ t + mR ∂ F ∂ϕ = P m − (cid:34) + i (cid:32) θ + m R (cid:33)(cid:35) F + i β | F | F + iR ∂ F ∂ϕ . (11)Note that this is the generalization of the analysis of a tilted wave front [14] to polar coordinates on a ring. We then considertravelling wave solutions to Eq. (11) of the form F ( q ) that depend on the variables ϕ and t through q = ϕ − ω t , (12)where ω is the angular velocity. In this case we can write the l.h.s. of Eq. (11) as ∂ F ∂ t + mR ∂ F ∂ϕ = ∂ F ( q ) ∂ q (cid:32) − ω + mR (cid:33) . (13)Clearly this equals zero when ω = mR , (14)and thus there exist rotating solutions F ( q ) with angular velocity ω = m / R that can be determined via P m = (cid:34) + i (cid:32) θ + m R (cid:33)(cid:35) F − i β | F | F − iR ∂ F ∂ q . (15)Apart from a renormalization of the detuning, Eq. (15) is equivalent to the stationary solutions of Eqs. (6) and (3) for m = m = m (cid:44) τ = t − ϕ/ω , as described in Appendix A.Among these travelling waves solutions we can identify Turing patterns for m (cid:44) m = ∆ = θ + m R + k (16)and critical wavevector: k c = (cid:114) β I s − θ − m R . (17)The wavelength of the Turing structure Λ c = π/ k c and the number of peaks N = π R / Λ c = R (cid:114) β I s − θ − m R (18)now both depend on the OAM and the radius of the ring. Note that for detuning θ di ff erent from 2 β I s and for m / R small,e.g. for small magnitudes of OAM and large radii, the critical wavevectors from (10) and (17), and hence the number of peaks,are approximately the same. Historically, expressions for the angular velocity similar to (14) had been obtained and applied torotating domain walls in optical parametric oscillators [3] and used as numerical ansatz for self-trapped necklace-ring beams ina self-focusing nonlinear Schr¨odinger equation [15].The present analysis is confined to Turing patterns close to the threshold of instability of the homogeneous stationary state. InAppendix B we show that it is possible to obtain equations that describe rotating Turing patterns for generic values of the pump P m , detuning and OAM. NUMERICAL SIMULATIONS
Althought the analysis in the previous section assumed a quasi-1D geometry (rings of fixed radius), all of our numericalsimulations are performed in 2D.
Laguerre-Gaussian Pumps
We start by numerically modelling equation 1 using a Laguerre-Gaussian pump with radial index p = LG m ( r , φ ) = (cid:114) π | m | ! 1 w r √ w | m | exp − r w e im ϕ = P m e im ϕ , (19)where m is the OAM and w is the beam waist. In Fig. (1) we show the time evolution of the field from an LG mode-like ring toa number of bright peaks equally spaced around a ring of maximum intensity.For our given parameters, I s = . , θ = , β = / , w = . R = . ± .
5, as shown in the top panel of Fig. (2). This is in accordance with the closest integer value from ourpredicted value, using Eq. (18), of 10 . ± . counter-clockwise at a constant angular velocity ω = . ± . R , as shown in thebottom panel of Fig. (2), where the diagonal red lines correspond to the peaks of intensity. Angular velocity ω = ∆ ϕ/ ∆ t =∆ s / ( R ∆ t ) where ∆ s is the distance a peak travels around the circumference of the circle in a time ∆ t . Numerically we find ω = . ± . m = −
1, and found that the now peaks rotated clockwise at the same speed, ω = . ± . m = , , ω = . ± . R for each OAM.For any LG mode the radius of maximum amplitude is r max = w √| m | /
2. Substituting this into the angular velocity we find ω = m / r = ± / w . We therefore expect the angular velocityof LG modes to be independent of m but inversely proportional tothe beam waist, w .To confirm this we numerically integrated Eq. (1) using LG pumps with m = . , . , .
0, whichformed rings of bright spots at radii 19 . , . , .
5, and with a beam waist of 25 .
0, which formed two rings of bright spots atradii 39 . , .
0. Fig. (4) shows that we have very good agreement between the angular velocity of the rings measured numerically(blue line) and the predicted values from Eq. (14) using the measured radii (red line).These measurements simultaneously confirm the direct proportionality of the angular velocity to the OAM index m and inverseproportionality to the square of the radius of the input ring (see Eq. (14). We note that in previous studies of the e ff ect of LGpumps on rotating cavity solitons in semiconductor microresonators [4] and on rotating patterns in photorefractive media insingle mirror feedback configurations [5], the pump radius and the OAM index where changed simultaneously leading to weakdependencies of the angular velocity on the OAM index (see Fig. (3). We believe that both these investigations provide supportto the universality of Eq. (14) when one includes the changing radius of the input pump of LG modes with di ff erent OAM. FIG. 1: Density plot of intensity (left) and phase (right) during evolution to pattern formation for m =
1. Parameters are: I s = . , θ = , β = / , w = . t = , , R at t = R from t = t = I s = . , θ = , β = / , w = . , m = FIG. 3: Angular velocity ω vs OAM index m for LG input pumps. Parameters are: I s = . , θ = , β = / , w = . , m = −
5. The blueline corresponds to numerical simulations of Eq. (1), the red line to the analytical result Eq. (14).FIG. 4: Angular velocity ω vs the radius of LG input pumps with m =
1. Parameters are: I s = . , θ = , β = / , w = . − . , m = ‘’Top-hat Pumps carrying OAM” We next consider the case where the input pump has a top-hat shape amplitude multiplied by an azimuthal phase: P = P m − tanh ( S ( r − r t ))] e im ϕ . (20)Here P m is a spatially independent complex amplitude, S and r t control the steepness of the sides and the radius of the top-hat,respectively, and m is an integer corresponding to the topological charge of the optical vortex, as before.Di ff raction due to the finite size of the pump induces concentric rings, whose amplitude decreases from the outer ring inwardsfor m =
0, as shown in Fig. (5 (a)). The amplitude of the outermost ring increases with the steepness of the sides of the pumpand this can allow its intensity to trigger the Turing instability, see the red line in Fig. (6) in a comparison with I cs = /β , and anazimuthal pattern forms on the ring. Once the pattern has formed on the outer ring (left panel Fig. (7)), we observe a sequenceof azimuthal instabilities from the outer to the inner ring. Each patterned ring forms a number of peaks separated by the criticalwavelength corresponding to the radius of the particular di ff raction ring. The final patterns for m = ff raction rings, however, are similar to the well-known daisy or sunflower patterns as observed, for example, inVCSELs with an electronic pump with a steep oxide confinement [17–21]. When the pump carries OAM, i.e. m (cid:44)
0, the phaseat the centre of the pump is undefined and hence the field at the origin has to be zero, as is typical for Laguerre-Gaussian modes[22]. The physical e ff ect of this on-axis vortex is the induction of di ff ractive rings close to the centre of the beam, see the dashedblue line in Fig. (6). In this case the intensity of the inner ring may exceed I cs and undergo an azimuthal Turing instability. Thedistance between the concentric rings is close to the critical wavelength Λ c along the radial direction. Once the Turing pattern FIG. 5: Time evolution of the intensity of a top-hat pump with m = m = ff raction rings at t = t = ,
000 for m = t = ,
000 for m =
1. Whenthe pump carries OAM (bottom) the di ff raction rings have maxima on both outer and inner rings, there is a vortex in the centre, and the Turingpatterns are arranged in concentric rings. Parameters are: I s = . , θ = , β = / , S = . m = m = k c . Green line is pump intensity threshold for Turing instability. Parameters are: I s = . , θ = , β = / , S = . , t = has formed on the inner ring we observe a sequence of azimuthal instabilities taking place from the inner to the outer ring. Asbefore, azimuthal peaks are separated by the critical wavelength corresponding to the radial length of the particular di ff ractionring, modified by the factor m / R as shown in Eq. (17). Note: The e ff ect of the factor m / R on the wavelength is strongertowards the centre of the beam where the radius is smaller. Moreover, the size of the central vortex, and hence radius of the firstdi ff raction ring, increases with increasing OAM, m . These e ff ects alter the radial wavevector in the vicinity of the vortex and canprevent regular patterns from forming on the innermost rings.As mentioned above, the steepness of the top-hat pump, determined by S in Eq. (20), and also of the optical vortex, a ff ectsthe relative intensity of the di ff raction rings. By careful choice of steepness, and / or the OAM of the pump we can control if theTuring instability first occurs on the inner or the outer ring, or even on both simultaneously. This is demonstrated in Fig. (7)for a top-hat pump with OAM m = S = . , . , . S = . S = . S = . m (cid:44) R equal to the radius of the specific di ff raction ring. It is interesting tocompare the final phase and final far-field distributions for the m = m = m (cid:44)
0, the central singularity and the line of phase discontinuity
FIG. 7: Formation of pattern for top-hat pump with OAM m = S = . , . , . S = . S = . S = . I s = . , θ = , β = / m = m =
1, respectively. Parameters are: I s = . , θ = , β = / , S = . corresponding to the OAM of the input pump are clearly visible. Note that there are no further phase discontinuities in spiteof the presence of rotating rings. For m = m (cid:44) Λ c but do not display any specific 2D geometry since they are located on rotating rings (see Fig. (8)(d)). Each ring is decoupledfrom the rest of the structure, meaning that they behave as independent 1D azimuthal structures although embedded in a fully2D field.In Fig. (9) we plot the angular velocity of each ring that forms versus its radius for top-hat pumps with m = / cyan), m = / magenta) and m = / orange). The numerical results (solid lines) show excellent agreement with the analyticalresults (dashed line) calculated using (14) provided with the measured radii of the rotating rings. This confirms that scalar pumpscarrying OAM m form independent Turing patterns on concentric rings of radius R each rotating with constant angular velocity ω = m / R . For completion, we note that the dynamics leading to the asymptotic ring rotation for m (cid:44) m = FULLY-STRUCTURED PUMPS
Vector, or fully structured light (FSL), beams [6–8] have attracted increasing attention for a number of applications. Thesebeams consist of a vector superposition of two scalar orbital angular momentum (OAM) carrying Laguerre-Gaussian (LG)eigenmodes with orthogonal circular polarizations: (cid:126) E ( r , φ ) = cos( γ ) LG m L ( r , φ ) (cid:126) e l + e i α sin( γ ) LG m L ( r , φ ) (cid:126) e r , (21)where γ and α give the relative amplitudes and phase, respectively, of the two modes. We assume throughout that each of thespatial modes takes the form of a Laguerre-Gaussian beam with radial index p = FIG. 9: Angular velocity vs ring radius for top-hat pumps with m = / cyan), m = / magenta) and m = / orange). Solidlines are numerical results, dashed lines are calculated using (14) with measured radii of rotating rings. Parameters are: I s = . , θ = , β = / , S = . uniform spatial intensity, phase and polarization distributions. To investigate the e ff ect of using an FSL pump we use coupled Lugiato-Lefever equations [23]: ∂ t E L , R = P L , R − (1 + i θ ) E L , R + i ∇ E L , R + i β (cid:16) | E L , R | + | E R , L | (cid:17) E L , R . (22)Note that if either E L or E R is zero, then the resultant beam is a scalar LG mode with spatially uniform right- or left-handedcircular polarisation, respectively, and Eq. (22) reduces to the scalar LLE (1) that we have considered so far.
Cylindrical vector beam pumps
If the two modes have equal but opposite
OAM the resultant beam is know as a cylindrical vector (CV) beam [6, 8] (cid:126) E ( r , φ ) = cos( γ ) LG − m ( r , φ ) (cid:126) e l + e i α sin( γ ) LG + m ( r , φ ) (cid:126) e r . (23)If the two modes have equal amplitude ( γ = π/ linear polarization,as shown in Fig. (10) for eigenmodes with | m | =
3. In this case we find that a pattern of equally-spaced bright spots appearsaround the ring, as in the scalar case, but this time there is no rotation as the net OAM is zero. By changing γ in (23) we FIG. 10: Cylindrical vector beam with OAM ± I s = . , θ = , β = / , w = . , γ = π/ , α = . change the relative amount of the two eigenmodes, i.e. the bias between the eigenmodes. For γ = π/
2) the pump is a scalarleft (right) circularly polarized beam with − m ( + m ) and we find exactly the same behaviour as earlier; in particular, the outputTuring pattern rotates at ω = ∓ m / R . For γ = π/ < γ < π/ m < π/ < γ < π/ m > FIG. 11: Angular velocity (normalised) vs bias parameter γ for cylindrical vector (CV) beams with | m | = | m | = | m | = I s = . , θ = , β = / , w = . , α = . In Fig. (11) we plot the angular velocity (normalised to 2 m / R ) versus the bias parameter γ for cylindrical vector (CV) beamswith | m | = | m | = | m | = γ ) between the two eigenmodes of the CV beam,we can fine tune the angular velocity of the output field from − mr ≤ ω ≤ mr . (24)(Recall that LG modes all have the same angular velocity for any given beam waist w .) Poincar´e pumps
If the two modes have di ff erent magnitudes of OAM, the resultant beam is know as a Poincar´e beam [7]: (cid:126) E ( r , φ ) = cos( γ ) LG m L ( r , φ ) (cid:126) e l + e i α sin( γ ) LG m R ( r , φ ) (cid:126) e r . (25)This carries a net OAM and the polarization can cover all polarization states on the Poincar´e sphere. In Fig. (12)we plot the numerically measured values of ω R against net OAM for di ff erent Poincar´e modes ( m L , m R ) = ( − , , (0 , , ( − , , (1 , , ( − , , (1 , , ( − , , (2 ,
3) (red circles). We keep γ = π/ , α =
0. We can see that there is verygood agreement between our numerical results and the blue line for m L + m R , suggesting that for Poincar´e beams, the angularvelocity of the output field depends on the net OAM according to: ω = m L + m R R . (26) Optical ‘’peppermill”
Up until now we have only considered vector beams with some degree of spatial overlap. By considering eigenmodes withsignificantly di ff erent transverse radii such that there is very little interaction between them we can create, for example, counter-rotating rings of spots such as the “optical peppermill” shown in Fig. (13). In this case the pump consists of a left -circularlypolarized mode with m L = − m R =
8. The output field has two rings of peaks, with theouter ring rotating counter-clockwise and the inner clockwise with the same angular velocity ω = . ± . FIG. 12: Numerically measured values of ω R against net OAM for di ff erent Poincar´e modes ( m L , m R ) = ( − , , (0 , , ( − , , (1 , , ( − , , (1 , , ( − , , (2 ,
3) (red circles). Blue line is m L + m R . Parameters are: I s = . , θ = , β = / , w = . , γ = π/ , α = . m L = − m R =
8. The inner ring rotates clockwisewhile the outer ring rotates counter-clockwise at the same angular velocity. Parameters are: I s = . , θ = , β = / , w = . , γ = π/ , α = . we can produce even more complex superpositions of modes as shown: (cid:126) E ( r , φ ) = cos( γ ) E L ( r , φ ) (cid:126) e l + e i α sin( γ ) E R ( r , φ ) (cid:126) e r ; E L ( r , φ ) = n L (cid:88) i = A i LG m i (cid:113)(cid:80) n L i = A i ; E R ( r , φ ) = n R (cid:88) j = B j LG m j (cid:113)(cid:80) n R j = B j , (27)where A i and B j are the contributions of left- and right-circularly polarized LG modes with OAM m i , m j , respectively. Thiscan, for example, allow us to produce the peppermill-type beam shown in Fig. (13) but with full and individual control over thespeeds of each of the rings simply by biasing each with modes of opposite OAM and orthogonal polarisation, as in Fig. (11). CONCLUSION
We have demonstrated formation and rotation of spatio-temporal patterns in self-focussing nonlinear optical cavities pumpedby beams carrying orbital angular momentum, m . For scalar pumps we see the formation of a ring, or concentric rings, aroundan optical vortex that rotate at angular velocity ω . Using a 1D Lugiato-Lefever model we find that ω = mR , where R is the radiusof each ring. For a 1D azimuthal model this formula is exact but we confirm numerically that these angular velocities extend tothe 2D case and demonstrate this using input pumps that are Laguerre-Gaussian modes and ‘’top-hat” shaped pumps with OAM.2We note that the radius of maximum intensity of an LG mode scales with the OAM s.t. LG beams with the same beam waist w have the same angular velocity, ω = ± / w . This means that we can control the angular velocity of the patterns by the choice of: • OAM, m • beam waist of LG pump, w • radius of top-hat pump, R .Note that the numer of independent concentric rings that can form inside the top-hat depends on its diameter and the Turingpattern wavelength. Our analysis confirms earlier results on rotating domain walls in optical parametric oscillators and self-trapped necklace-ring beams in a self-focusing nonlinear Schr´’odinger equation.Further control over the angular velocity of the pattern can be achieved using vector pumps with orthogonally polarizationeigenmodes with good spatial overlap. • Using cylindrical vector beams, that have eigenmodes with equal and opposite OAM m , controlling the relative weightingsof the eigenmodes, the bias, allows the angular velocity to range from − mR ≤ ω ≤ mR . • Using Poincar´e beams, that have eigenmodes with di ff erent magnitudes of OAM m L , m R , the resultant angular velocity is ω = ( m L + m R ) / R .If there is no, or very little, overlap between the modes then concentric Turing pattern rings, each with angular velocity ω = m L , R / R will result. This can lead to concentric, counter-rotating Turing patterns creating, for example, an ’opticalpeppermill’-type structure with full and individual control over the speeds of each counter-rotating ring of pattern. This haspotential applications in particle manipulation by using the rotating peak intensities to dipole trap atoms, molecules and smallparticles. The di ff erential rotation of concentric rings can also be applied to stretching and breaking of cells in a way analogousto optical stretchers [25]. Finally, rotating Turing patterns can be used to induce circular transport of cold atoms and BECwavepackets using opto-mechanic nonlinearities instead of Kerr [26]. Funding
We acknowledge support from the Leverhulme Trust Research Project Grant No. RPG-2017-048, European Training NetworkColOpt, which is funded by the European Union (EU) Horizon 2020 programme under the Marie SkÅodowska-Curie action,grant agreement 721465, and Engineering and Physical Sciences Research Council DTA Grant No. EP / M506643 / Disclosures
The authors declare that there are no conflicts of interest related to this article.
APPENDIX A. RETARDED TIME TRANSFORMATION
Equation (11) is the LL equation on a ring for the field F ( ϕ, t ). By using the angular velocity (14) it is possible to introduceretarded time transformations ζ = ϕ ; τ = t − ϕω . (28)s.t we can write ∂∂ϕ = ∂∂ζ − ω ∂∂τ ; ∂∂ t = ∂∂τ . (29)We can then write the l.h.s of Eq. (11) in the retarded time variables: ∂ F ∂ t + ω ∂ F ∂ϕ = ∂ F ∂τ + ω (cid:32) ∂ F ∂ζ − ω ∂ F ∂τ (cid:33) = ω ∂ F ∂ζ ω ∂ F ∂ζ = P m − (cid:34) + i (cid:32) θ + m R (cid:33)(cid:35) F + i β | F | F + iR (cid:32) ∂ ∂ζ + ω ∂ ∂τ − ω ∂∂ζ ∂∂τ (cid:33) F . (30)Steady states in the retarded variable ζ , obtained by imposing ∂ F /∂ζ = APPENDIX B. ROTATING TURING PATTERNS AWAY FROM THRESHOLD
Here we investigate azimuthal Turing patterns on a ring due to pump fields carrying OAM well above threshold. From theanalysis close to threshold, Turing patterns are spatially modulated structures with wavelength Λ c = π/ k c where k c is the criticalwavevector given by Eq. (17). We consider spatially modulated solutions of the ring LLE (3) of the form E ( ϕ, t ) = F ( ϕ, t ) e im ϕ = A [ Q ( ϕ, t )] exp { i Φ [ Q ( ϕ, t )] + i ψ } e im ϕ (31)where ψ is a constant phase and A and Φ are amplitude and phase functions that are periodic in the variable q = ϕ − ω t andspatially normalised for Turing patterns of wavevectors k c , given by: Q ( ϕ, t ) = k c R q = k c R ( ϕ − ω t ) (32)where ω is the angular frequency. By replacing (31) in the ring LLE (6) one obtains: − k c R ω (cid:32) ∂ A ∂ Q + iA ∂ Φ ∂ Q (cid:33) exp ( i Φ + i ψ ) = + P m + (cid:40) − (cid:34) + i (cid:32) θ + m R (cid:33)(cid:35) A + i β A (33) + k c i ∂ A ∂ Q − A ∂ Φ ∂ Q − ∂ A ∂ Q ∂ Φ ∂ Q − iA (cid:32) ∂ Φ ∂ Q (cid:33) − mk c RR (cid:32) ∂ A ∂ Q + iA ∂ Φ ∂ Q (cid:33)(cid:41) exp ( i Φ + i ψ ) . This demonstrates that above threshold, Turing patterns with an amplitude and a phase that are spatially modulated at the criticalwavevector k c are solutions of Eq. (6) provided that they rotate at an angular velocity ω = m / R and that they satisfy P m exp ( − i ψ ) = (cid:40)(cid:34) + i (cid:32) θ + m R (cid:33)(cid:35) A − i β A (34) − k c i ∂ A ∂ Q − A ∂ Φ ∂ Q − ∂ A ∂ Q ∂ Φ ∂ Q − iA (cid:32) ∂ Φ ∂ Q (cid:33) exp ( i Φ )We have verified condition (34) by integrating Eq. (3) well above the threshold of pattern formation and for a variety of OAMindices m . In all of these tests, the numerically found rotating Turing patterns are of the form (31) and satisfy condition (34)with an error smaller that 2% up to stationary intensities almost twice the pattern formation threshold. ∗ Electronic address: [email protected], [email protected][1] A. M. Turing,
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