Aberration-driven tilted emission in degenerate cavities

The compensation of chromatic dispersion opened new avenues and extended the level of control upon pattern formation in the \textit{temporal domain}. In this manuscript, we propose the use of a nearly-degenerate laser cavity as a general framework allowing for the exploration of higher contributions to diffraction in the \textit{spatial} domain. Our approach leverages the interplay between optical aberrations and the proximity to the self-imaging condition which allows to cancel or reverse paraxial diffraction. As an example, we show how spherical aberrations materialize into a transverse bilaplacian operator and, thereby, explain the stabilization of temporal solitons travelling off-axis in an unstable mode-locked broad-area surface-emitting laser. We disclose an analogy between these regimes and the dynamics of a quantum particle in a double well potential.

Dispersion compensation consists in combining elements with opposed chromatic properties to achieve an overall partial or total cancellation of the second order dispersion.This simple yet powerful idea permitted exploring the influence of higher order contributions.While optical temporal localized structures often result from the balance between self-phase modulation and anomalous second-order dispersion [15][16][17], it was recently proven that third and fourth-order dispersion lead to unforeseen effects such as the stabilization of solitons and frequency combs [18][19][20], symmetry breaking [21], the control of modulational instabilities [22,23] or the realization of purely quartic solitons [24,25] as predicted in [26].
The paraxial diffraction emerging as a beam propagate is mathematically equivalent to that of secondorder chromatic dispersion.Optical cavities in which the path of light is folded onto itself may contain a transverse plane that is its own image after a round-trip [27,28].This so-called stigmatic condition is equivalent to an effective cancellation of paraxial diffraction.Imposing additionally the nullification of the round-trip wavefront curvature does not only achieve the self-imaging condition (SIC) for the field intensity, but also for its amplitude.Self-imaging cavities have attracted a great deal of attentions for their rich spatio-temporal dynamics [5,10,[29][30][31][32] and for their application in specklefree imaging [33,34], frequency comb multiplexing [35], controllable and reconfigurable multimode fields [36,37], see [38] for a review.The proximity of the SIC was used to manipulate the spatial coherence of the field [34,36,[39][40][41], create a perfect coherent absorber [42], realising topological band structures [43], or to form propa-gation invariant beams [44].Aberrations become crucial as the SIC is approached [31,45,46], yet, their effect on spatio-temporal laser dynamics received comparatively less attention.Photonic crystals also allow for dispersion control [47,48] and lead to the zero-diffraction regime [49].Higher order spatial operators occur in fiber lasers [50] and in optical cavities either at the onset of bistability [51] or containing a photorefractive or semiconductor medium [2][3][4]52].Diffraction control was also proposed using metamaterials [53][54][55][56] or atomic resonances [57].In this paper, we study how aberrations appear as leading effects in nearly-degenerate broad-area surface-emitting lasers.We show that these effects can crucially modify the spatio-temporal mode-locking dynamics and give rise to either spatially tilted pulses or spatio-temporal solitons.Our results represent a step further towards the understanding of spatio-temporal mode-locking [12][13][14], the realization of fully confined three-dimensional light bullets [10,58,59] and may lead to new ideas and applications for beam steering, tweezing [60] and tailored optical energy potential landscapes [61,62].
The dynamics of the transverse profile of the field close to the SIC and in presence of spherical aberrations is equivalent to the dynamics of a quantum harmonic oscillator featuring a fourth-order derivative where c, b and s are real coefficients that correspond to the residual wavefront curvature, diffraction, and spherical aberration, respectively.We will focus our attention on spherical aberration as it is often the dominant form of aberration.A detailed derivation is provided in the Appendix I. Notice that Eq. ( 1) was studied already albeit in a different context in [63].
It is well known that the potential V (x) = c x 2 in Eq. ( 1) supports localized solutions for bc < 0 [28].Setting A(x, θ) = A s (x)e −iωθ with ω denoting the frequency of the eigenmode and imposing boundedness of the the solution defines a singular Sturm-Liouville problem (SLP) that allows determining ω: For bc < 0 and s = 0 the solutions of Eq. ( 2) are the so-called Hermite-Gauss (HG) modes H n ( x σ ) with σ = −b/c and frequencies ω n = b σ 2 (2n+1) while n ∈ N is the modal index.We will see later that this situation corresponds to the case of a stable cavity devoid of aberration.
The HG modes are invariant under Fourier transform since the multiplicative and differential terms in Eq. ( 1) are exchanged upon performing the latter.However, the role played by the fourth order derivative in Eq. (1) becomes more clear in Fourier space, where the SLP becomes Here, we defined Âs as the Fourier transform of A s .Hence, the presence of a fourth order derivative in Eq.Eq. ( 1) converts the situation to that of a particle in single or a double-well potential.In case of b < 0 and c > 0 the potentials V (x) or V (k) = −b k 2 + sk 4 both feature a minimum at the origin as shown in Fig. 1 (a).The bounded solutions of Eq. (1) remain practically unaffected by small values of s, since the solutions remain concentrated at low values of k for which bk 2 ≫ sk 4 .Let us know consider the situation where b changes its sign, which usually signals the transition from a stable to an unstable cavity.In that case, the potential V (k) develops a negative curvature around k = 0 for b > 0. If there were no aberrations present in the system, i.e. s = 0, the laser would simply turn off as there is no transverse mode to support emission.Mathematically speaking, this means that the bounded solutions of the SLP cease to exists.
However, the situation can be remedied by the fourth order derivative as depicted in Fig. 1(b): For s > 0, two new minima emerge symmetrically at k 0 = ± b/2s which allows bounded solutions to continue to exist in the otherwise unstable domain.The ground state solution Âs (k) in Fourier space can be approximated by a superposition of two bell-shaped functions localized around ±k 0 which, in real space, amounts to a strongly modulated eigenmode with wavelength λ ⊥ = 2π/k 0 , akin to the interference between Bose-Einstein condensates in a double-well potential [64,65].This situation is depicted in Fig. 2 where we solved for the ground state of Eq. (1) numerically using complex time evolution.Note, that the waist of the mode decreases monotonically upon approaching b = 0, but remains finite at b = 0.This is due to the fact that the higher order derivative start to play a more dominant role.In the unstable range b > 0 we notice that the second order moment increases, albeit at at a faster rate.
While we could not obtain closed form analytical solutions of Eq. ( 2) for s ̸ = 0, the following change of variable permits cancelling the first-order derivative and leads to Slowly varying envelope solutions, where the characteristic length scale for the envelope A s is much larger than 2π/k 0 correspond to the inequality  this condition is fulfilled, one may neglect the third and fourth order derivative in Eq. ( 5), leading to By comparing Eq. ( 1) with Eq. ( 2) we note that the latter is again a Hermite-Gauss equation where the parameter b has been replaced by −2b, which fully explains the increased diverging rate of the second moment in Fig. 2. We note that the frequencies ω are shifted due to the additional term b 2 / (4s).
In summary, we disclose the surprising result that an unstable cavity does not necessarily turn off upon crossing the SIC, and lasing eigenmodes can still be supported by spherical aberrations.The modes can be approxi-mated by Since k 0 ∈ R, it is convenient to define two families of eigenfunctions Γ n (x, θ) and Ψ n (x, θ) as: The latter are depicted in Fig. 3 and Fig. 4, respectively.We observe the effect of the parameter s on the ground states Γ 0 and Ψ 0 in Fig. 3.In order to stabilize a bounded eigenmode, the decrease of s must be compensated by additional oscillations and, therefore, an increase in k 0 ∼ 1/ √ s.In Fig. 4 we also depict how the intensity of the modes Γ n (x, θ) and Ψ n (x, θ) alternate between even and odd when varying the modal index n.
We now turn our attention to the two-dimensional (2D) transverse profile whose evolution is governed by Here, we assume that the cavity possesses two orthogonal axes denoted r ⊥ = (x, y) and, for the sake of generality, we introduced dichroism in Eq. 12, i.e. b x ̸ = b y and c x ̸ = c y .The effect of spherical aberration in 2D translates into a bilaplacian operator y .We note that ∇ 4 ⊥ is the only fourth order operator that preserves the rotational symmetry both in real and in Fourier space, which is consistent with the idea of spherical.The equivalent SLP for When s = 0 and the two conditions b x c x < 0 and b y c y < 0 are simultaneously verified, the SLP admits separable solutions that are simply the product of the HG modes in the x and y directions discussed previously [28].We now consider the effect of aberration close to the SIC when the cavity becomes unstable.Due to the inherent dichroism present in any realistic experimental system, one can assume that the cavity becomes unstable first in one direction, say the x-direction.Applying the Fourier transform to Eq. 13 leads to where we defined the potential in Fourier space A representation of V (k ⊥ ) for three different values of b x is given in Fig. 5.We see that the transition from a stable to an unstable cavity is obtained when b x changes its sign from negative to positive which gives rise to the appearance of two new minima as observed in the 1D case in Fig. 1.Their position is obtained by simply setting As already discussed, this corresponds to off-axis emission on the x-axis and stable on-axis emission on the y-axis.The curvature of V (k ⊥ ) around the two minima in k ± ⊥ upon back-transforming to direct space corresponds to the effective diffraction experienced by a wave-packet centered around this tilted wavevector k ± ⊥ .We obtain The curvature in the x-direction changed from −b x → 2b x , similarly to the 1D case.However, the perpendicular direction is also affected by the off-axis emission, leading to the substitution b y → b y − b x .In the case of a cavity that changes behaviour from stable b x < 0 to unstable b x > 0, the off-axis emission along the x-direction that re-stabilizes the emission renders the perpendicular direction "more diffractive" since b The results of Fig. 5 entice us to seek again a modulated profile with k 0 = b x / (2s).Truncating to second order we get the approximate SLP problem where the modification of the diffraction coefficients in Eq. 18 is fully consistent with the discussion of the curvature of V given in Eq. 16.The eigenmodes of such an unstable cavity are the product of a modulated HG mode in the unstable direction and of a regular HG mode in the y-direction.As such, we define the eigenmode family with two modal indices (n, m) The situation considered in Eq. ( 1) materializes for instance in the study of passively mode-locked integrated external-cavity surface-emitting laser (MIXSELs) in a near-degenerate cavity [29] as depicted in Fig. 6.Here, the gain (G) and saturable absorber (SA) media are enclosed in a single micro-cavity and the external mirror is assumed to be ideal while the collimating lens is not.Our modeling approach is based on a Haus master equation model for passive mode-locking (PML) adapted to the experimentally relevant long cavity regime regime [59,[66][67][68], where the PML pulses become individually addressable temporal localized states (TLSs).The Haus equation relates the slow evolution of the three-dimensional intra-cavity field E (r ⊥ , t, θ) to the dynamics of the population inversion in the gain N 1 (r ⊥ , t) and the saturable absorber N 2 (r ⊥ , t) as In this formalism, the variable t ∈ [0, τ ] represents the round-trip time in the external cavity and θ is a second dimensionless time scale normalized by τ .The latter corresponds to the slow evolution of the pulse under the combined effect of gain, absorption and spatiotemporal filtering.The two transverse dimensions are denoted r ⊥ = (x, y) and allow for pattern formation in the plane perpendicular to the propagation direction of the pulse, cf.Fig. 6.Note that in the long cavity regime, the spatio-temporal distributions of the carriers N j (r ⊥ , t) are slaved to the evolution of the optical field [66,68] and do not depend explicitly on the slow timescale θ.
In this regime, one can safely assume a full recovery of the carrier between pulses and a total loss of memory from one round-trip toward the next one.We define in Eqs. ( 20)-( 22) κ as the round-trip cavity losses, and α j are the linewidth enhancement factors of the two active media that relax with timescales γ −1 j toward the equilibrium values J j .The ratio of the saturation fluence of the gain and of the SA is denoted by ŝ.A standard speudospectral split-step method was used for the numerical simulations [59].The effective cavity spatio-temporal linear operator L, accounts for the finite gain bandwidth and chromatic dispersion, but also for non-perfect imaging conditions, diffraction, parabolic wavefront curvature and mirror losses due to finite aperture.L is given by where d g is the temporal diffusion coefficient representing the gain bandwidth.The Fresnel transform [69] permits calculating the transverse effects occurring at each round-trip analytically from the round-trip (ABCD) matrix in the paraxial approximation [28].In presence of aberrations, its calculation is achieved by expanding the exact (non-paraxial) operator corresponding to the lens as a sum of an ideal element plus a deviation.The latter potentially contains all the wavefront curvature contributions beyond the parabolic approximation.Assuming a large ratio between the focal length of the mirror and that of the lens permits expressing the spherical aberrations as a bilaplacian operator, see the Appendix I for details.The transverse round-trip operator L ⊥ is given by The finite size of lenses and the numerical aperture of the whole optical system is modeled by as a soft aperture and a real transverse diffusion parameter d f .Finally, b is the normalized paraxial diffraction parameter, c is the normalized parabolic wavefront curvature that are the off-diagonal elements of the round-trip ABCD matrix [28] and s is the spherical aberration parameter.We note that moving the lens in Fig. 6 modifies c while moving the mirror modifies both b and c.A qualitative model for the transverse profile of the TLSs such as the one derived in [10,59] can be obtained by essentially adapting New's method for PML [70] to the situation at hand.This method exploits the scale separation occurring between the pulse evolution, the socalled fast stage in which stimulated emission is dominant, and the slow stage that is controlled by the gain recovery processes.Assuming, as in [10,59], that the four-dimensional spatio-temporal profile E (x, t, θ) can be factored as E (x, t, θ) = A (x, θ) p (t) with p (t) a normalized TLS profile, one obtains a Rosanov equation [71] for the slow evolution of the TLS transverse profile A (x, θ) as The normalized coefficients b, c and s represent the residual paraxial diffraction, parabolic wavefront curvature and aberration coefficients close to SIC.Their expression and their derivation are given in the Appendix I.
We define the effective nonlinearity as with J 1,2 and α 1,2 denoting the bias and the line-width enhancement factors of the gain and absorber, ŝ the ratio of their saturation energies and κ the round-trip losses.
The nonlinear response of the active material to a pulse is given by g (P ) = 1 − e −P /P [10,59,70].The equations (25,26) provide a unified framework which allow bridging our results for spatio-temporal dynamics with the former results of [71,72] for the case of static auto-solitons in bistable interferometers.There, the function g (P ) should be replaced by the saturated lineshape transition ∼ (1 + P ) −1 .As such, our discussion of the effect of aberrations close to SIC is equally valid for temporal solitons, regular mode-locking and CW beams.
In one-dimension, Eq. ( 1) is recovered for the empty cavity, i.e., f = 0. Hence, we expect the emergence of a stable family of Γ n and Ψ n tilted HG modes (10,11) in the unstable cavity, where the condition bc < 0 is violated.We performed a bifurcation analysis of Eq. ( 25) in one transverse dimension using path-continuation framework pde2path [73].The results are summarized in Fig. 7 in the unstable cavity regime, where the peak powers for the fundamental tilted modes Γ 0 (red) and Ψ 0 (blue) are shown in the panel (a) as a function of the gain bias J 1 normalized to the threshold value J th .Furthermore, two exemplary profiles of Γ 0 and Ψ 0 at the same fixed gain value (gray dashed line) are depicted in Fig. 7 (b,c), respectively.For both modes, we observe the typical subcritical transition that leads to bistable TLSs as detailed in [17,59]; The high intensity branch is stable while the lower branch is unstable and creates a separatrix with the stable off solution.For the parameters chosen, the leftmost limiting fold bifurcations F 1 and F 2 are almost identical for Γ 0 and Ψ 0 .A small region of the Andronov-Hopf (AH) instability for Γ 0 exists between points H 22 and H 23 .In this region, a small amplitude oscillation is visible in time simulations.Both modes are limited by the AH bifurcations H 11 and H 24 , respectively, for high gain values.We stress that these nonlinear modulated HG solutions that are solutions of Eqs.(25,26), correspond to a train of TLSs whose profile is a tilted beam supported by spherical aberrations in an unstable cavity.
In two spatial dimensions, we assume the system to be weakly astigmatic and, as discussed previously, that the self-imaging condition is not reached simultaneously for both transverse dimensions.For a cavity that is stable in the vertical and unstable in the horizontal direction we observed the tilted localized patterns shown in Fig. 8 (ac) by solving Eqs.(25,26) numerically.The typical evolution for large values of b x (i.e., entering more deeply into the unstable region) is presented together with the corresponding far-field power spectrum | Â| 2 .The panels (d-f) shows the two peaks in the far field related to the value of the transverse horizontal wave-number ±k 0,x .Clearly, an increase of b x leads to an increase of k 0 , which corresponds to a higher frequency of the mode oscillations in the near-field.These results would correspond closely to the experimental situation described either in [74] for a mode-locked vertical surface-emitting external-cavity semiconductor laser using a saturable absorber or in [32] for a CW broad area laser.
In conclusion, we have investigated the effect of wavefront aberrations in a degenerated cavity.We found that the interplay between spherical aberrations and the proximity to the SIC may lead to modulated beams that can support either CW or temporal solitons in a mode-locked broad-area VECSELs.These modulated beams are analogous to the eigenmodes of a quantum particle in a double well potential.They can be analytically approximated by spatially modulated Hermite-Gauss modes.We linked the wavelength of their modulation to the parameters of the cavity.We note that the transition from a stable towards an unstable cavity around SIC can be obtained in two ways: Either for c > 0 and changing the sign of b from negative to positive, or for c < 0 and chang- ing the sign of b correspondingly.These two situations are not identical with respect to the effect of spherical aberrations since the off-axis wavevector k 0 = b/2s requires b and s to have the same sign.As such, the sign of s dictates the situation in which one can observe these modulated Hermite-Gauss beams.While we focused on the influence of spherical aberrations, we proposed a general framework that, in principle, permits calculating the effect of the other Seidel aberrations such as coma, distortion or field curvature in a cavity close to SIC.We believe that exhibiting the link between these wavefront curvature defects and their equivalent representations as spatial operators provides a new framework that could lead to a range of interesting new research avenues in photonics.Furthermore, the condition of a large ratio between the focal distances of the aberrated lens and the non-aberrated other elements could be relaxed.In this situation, spherical aberrations translate into a non-local spatial operators that may lead to rich pattern formation scenarios, as observed in other fields [75][76][77].
In this appendix we derive how the effect of spherical aberration can be recast into the form of a bilaplacian differential operator in the paraxial equation governing the field evolution after each round-trip.We will use the formalism of wave optics provided by generalized Huygens-Fresnel transform (HFT) for ABCD systems [28].We note that the HFT allows for the composition of parabolic operators, i.e. quadratic wavefront profiles induced by parabolic lenses as well as paraxial diffraction, since the latter also corresponds to a parabolic operator in Fourier space.We characterize our system by an ABCD roundtrip matrix with det (W ) = AD − BC = 1.The generalized HFT for the passage of light through a first-order optical system [69] composed of parabolic elements is given in one transverse dimension by where O (x, 0) is the incoming field passing through the system characterized by the ABCD matrix W and k = 2π/λ is the wave-vector of light.We note that B plays the role of a propagation distance while C is an inverse of a distance and represents wavefront curvature.Further, A, D are dimensionless quantities that correspond to spatial and angular magnification, respectively.Close to the self-imaging condition we consider the limit B → 0 which renders Eq. ( 28) singular.This difficulty can be avoided by factoring the quadratic form as follows: Using that D − 1 A /B = C/A, the Huygens-Fresnel integral given in Eq. (28) becomes The equation ( 30) will be the form of the Huygens-Fresnel integral used in the rest of the Appendix section.
As mentioned in the main text, we consider the simplest case of the self-imaging cavity [29] consisting of one lens of focal length f and one mirror or radius of curvature R, separated by distances d 1 and d 2 , see Fig. 6.The resulting round-trip propagation matrix reads with The SIC for which one finds W = Id is encountered for the following values of of d 1,2 Expanding to first order in δd j = d j − d * j one obtains We conclude that, close to the SIC, small displacements from the lens induce a B coefficient, whereas moving the mirror modifies both the values of B and C. Finally, we can safely assume that A = D = 1 at first order in O (δd j ).In addition to small deviations from the SIC, we consider that the focal length of the lens depends on its radial position and we model the spherical aberration as f (x) = f 0 + σx 2 .Denoting the field profile before and after the lens as E i and E o , respectively, we have Denoting the operator corresponding to the effect of the lens by L, we separate the contribution from the unperturbed lens with focal length f 0 and matrix L 0 as Using that we write Eq. 38 as which allows to express the action of the aberration operator δL : Finally, we will employ the proximity of the SIC to simplify the representation of the round-trip operator in the presence of the aberrations.The full operator W (cf. Eq. ( 31)) is given by Using Eq. ( 39) we can expand W as where we defined the unperturbed round-trip operator as The expression of δW reads after simplification and to the first order in O (δL) ) Further, we can simplify the dependence on δL in Eq. ( 44) using the exact self-imaging condition which amounts to setting W 0 = Id in Eq. (44).Indeed, since δL is already a small quantity, the error incurred in setting W 0 = Id will be second order.We are left calculating the two contributions of aberrations to the round-trip field evolution.We have W = F 1 + F 2 with In both cases, we can express F j as a double integral involving the HFT that involve the spherical aberration of the lens.We combine the two steps corresponding to γ into a single Fresnel transform.Further, we employ the SIC to find As a last approximation, we consider the limit of a long cavity for which the focal length of the mirror is large in comparison with the focal length of the collimator lens.
As such, we can define ε = f /R ≪ 1.This allows to approximate the operators γ j as Writing the action of F 1 using three integral transforms F 1 : E 0 → E 3 leads to The action of F 1 can be expressed by the following kernel where the kernel K 1 is defined as For small aberration, we can expand K 1 at first order in σ which leads to Using t = 2πy/ (λf 0 ) we find where we used the fact that Here, we defined δ (n) as the n th derivative of the Dirac delta which associates with the local value of the n th derivative of a function.Since δ (n) is a well-localized function, we can set z = x and obtain Performing the convolution with δ (n) is identical to taking the fourth derivative in direct space which yields the fourth derivative contribution.The action of F 1 to the first order is The action of F 2 can be found in a similar fashion using three integral transforms F 2 : E 0 → E 3 and reads Similarly, the action of F 2 can be expressed as a integral where we defined the kernel K 2 as Expanding analogously the aberration contribution to first order in σ and using Eq. ( 54), we obtain the desired kernel We conclude that, to the first order in σ, the action of F 2 is identical to that of F 1 In summary, the total effect due to spherical aberration close to self-imaging in the one-dimensional case reads The generalization of these calculations to twodimensions proceeds without difficulties and yields

FIG. 1 .
FIG. 1. Schematic of the fundamental Hermite-Gauss (HG) mode As(x) (blue) and the corresponding potential V (x) (orange) of Eq. (1) in the real (left) and Fourier space (right) for (a) b < 0, c > 0 and s = 1 (b) b > 0, c > 0 and s = 1.The resulting double-well potential in the Fourier space with the minima located at k0 shown in (b) correspond to a tilted HG mode in the real space.

FIG. 2 .
FIG. 2. Numerical solution of Eq. (1) for different values of b and fixed values of s = 1 and c = 4.97 × 10 −4 .White line indicates the second moment.

FIG. 6 .
FIG.6.A schematic of the MIXSEL, where both the gain (green) and the saturable absorption (pink) are contained in the same micro-cavity.It is coupled face-to-face to a distant external mirror by an imperfect self-imaging system.