Spatial spectrum in Hermite-Laguerre-Gauss modal space
Two best-known TEM families, Hermite- and Laguerre-Gauss modes, are complete eigen sets of the paraxial wave equation (PWE) in Cartesian and cylindrical coordinates, respectively, and their spatial wavefunctions at the waist plane (z=0) are given by8:
$${{\rm{HG}}}_{m,n}^{{\boldsymbol{\phi }}=0}(\mathop{r}\limits^{ \rightharpoonup})=\sqrt{\frac{{2}^{1-n-m}}{\pi n!m!}}\frac{1}{w}{{{\rm{e}}}^{-\frac{{x}^{2}+{y}^{2}}{{w}^{2}}}H}_{m}\left(\frac{\sqrt{2}x}{w}\right){H}_{n}\left(\frac{\sqrt{2}y}{w}\right)$$
(1)
and
$${\mathrm{LG}}_{{\mathcal{l}},p}(\mathop{r}\limits^{ \rightharpoonup })=\sqrt{\frac{2p!}{\pi \left(p+{\mathcal{l}}\right)!}}\frac{1}{w}{{{\rm{e}}}^{-\frac{{r}^{2}}{{w}^{2}}}\left(\frac{\sqrt{2}r}{w}\right)}^{\,\left|{\mathcal{l}}\right|}{{\rm{e}}}^{-i{\mathcal{l}}\varphi }{L}_{p}^{\left|{\mathcal{l}}\right|}\left(\frac{2{r}^{2}}{{w}^{2}}\right)$$
(2)
where \({H}_{m/n}\left(\cdot \right)\) and \({L}_{\left|{\mathcal{l}}\right|,p}\left(\cdot \right)\) are Hermite and Laguerre polynomials with bivariate modal indices \((m,n)\) and \(({\mathcal{l}},p)\), and \(w\) is the fundamental beam waist. Note that \(\phi \in \left[\mathrm{0,2}\pi \right]\) represents the rotational angle of Cartesian coordinates defined by longitude of a modal sphere and, in particular, \(\phi =0\) denotes \((x,y)\) coincides with horizontal and vertical directions. For a given modal order defined as \(N=2p+\left|{\mathcal{l}}\right|=m+n\), there are total \(N+1\) orthonormal HG and LG modes. Figure 1a shows all possible HG and LG modes with \(N\le 4\). Despite sharing the same fundamental mode TEM00, two families have their own special features in transverse patterns determined by corresponding modal indices. Specifically, indices \({\mathcal{l}}\) (an integer) and \(p\) (a positive integer) for LG modes determine numbers of azimuthal twisting (regarding OAM) and radial reversal in phase structures, respectively; while, for HG modes, positive integers \(m\) and \(n\) decide separately phase-reversal number along the \((x,y)\) coordinates predefined by \(\phi\). Except for the ground order (\(N=0\)), modes in each order can form \(\left(N+1\right)/2\) distinct Poincaré-like modal spheres29,30. For instance, Fig. 1b illustrates one and half such spheres for \(N=2\). On these higher-order SU(2) modal spheres, states on equator are rotating \({{\rm{HG}}}_{m,n}^{\phi }\), poles are occupied by \({{\rm{LG}}}_{\pm {\mathcal{l}},p}\) conjugates following the relation \({\mathcal{l}}=m-n\), and other areas represent intermediate Hermite-Laguerre-Gauss (HLG) states defined by the group structure. All possible states on the same sphere, uniquely determined by the longitude (\(\phi\)) and latitude (\(\theta\)) angles, can be connected reciprocally via astigmatic unitary transformation31,32.
a All possible single HG and LG patterns with orders no more than 4 at the waist plane. b Generalized HLG patterns on SU(2) modal spheres for N = 2. c 3D field profiles of two representative modes, \({{\rm{HG}}}_{\mathrm{7,0}}^{90^{\circ}}\) and \({{\rm{HG}}}_{\mathrm{4,4}}^{0^{\circ}}\), during resonance within a rotation-symmetry cavity, which have the same peak position and thus forming gain competition
Namely, (1) one can access the full spatial spectrum of TEM modes using astigmatic modal conversion (AMC) of either HG or LG complete set; and (2) the tunable range and fineness of a structured laser can thus be characterized by available modal orders \(N\) and indices \((m,n)\) or \(({\mathcal{l}},p)\), respectively.
The complete three-dimensional spatial wavefunctions \({{\rm{HG}}}_{m,n}^{\phi }(\mathop{r}\limits^{ \rightharpoonup },z)\) and \({{\rm{LG}}}_{\pm {\mathcal{l}},p}(\mathop{r}\limits^{ \rightharpoonup },z)\) can be derived from the diffraction integral by employing Eqs. (1) and (2) as pupil functions. These beam profiles, see examples in Fig. 1c, exhibit propagation invariance, except for global scaling and the acquisition of N-dependent Gouy phases. This self-similar diffraction suggests that HG and LG maintain pattern invariance during resonance within optical cavities, forming the basis for higher-order mode generation through laser gain control capable of modal discrimination. Notably, a cylindrically symmetric cavity can resonate with a waist-matched set of LG modes; however, due to rotational symmetry, it also accommodates HG sets across all possible rotating Cartesian coordinates. The absence of restrictions on \(\phi\), i.e., rotational symmetry breaking, prevents the use of the off-axis pump technique to generate 2D (bivariate) HG modes with both non-zero m and n, as will be demonstrated subsequently.
Spatial modal gain by off-axis pump
The selection rule of HG cavity modes is governed by the local gain of the off-axis pump, facilitating the desired mode to achieve the lasing threshold prior to others. For a given HG cavity mode initially from the spontaneous radiation, the gross gain acquired from unit pump power is proportional to its spatial overlap with the pump, denoted as \(P(\mathop{r}\limits^{ \rightharpoonup })\), within the gain medium, given by:
$$G(m,n)=\kappa \int {{\rm{HG}}}_{m,n}^{\phi }(\mathop{r}\limits^{ \rightharpoonup })P(\mathop{r}\limits^{ \rightharpoonup}){dV}$$
(3)
where \(\kappa\) is the local gain factor describing the amplitude growth rate at the pump overlapping area, and the corresponding threshold pump power should be \({P}_{\text{th}}(m,n)\propto 1/G(m,n)\)6. According to the polynomial \({H}_{m/n}\left(\cdot \right)\) in Eq. (1), the maximum intensities of 1D and 2D HG modes occur at their respective outermost peaks, with 1D peaking at two ends and 2D modes at four corners. That is, an HG cavity mode with one of its peaks aligned with the Gaussian pump center achieves higher effective gain and a lower lasing threshold. However, this mechanism indicates that an inevitable gain competition exists between two specific yet representative modes in Fig. 1c—\(\,{{\rm{HG}}}_{\mathrm{7,0}}^{90^\circ }\) and \({{\rm{HG}}}_{\mathrm{4,4}}^{0^\circ }\)—because both exhibiting the same peak position at \(\triangle x=\triangle y=1.75w\). Beyond displacement-sensitive gross modal gain, the ‘self-healing’ characteristic of HG modes also plays a pivotal role making the off-axis pump technique valid. This characteristic can be attributed to the higher net modal gain obtained during each oscillation, given by:
$$N\left(m,n\right)=\left\langle {{\rm{HG}}}_{m,n}^{\phi }(\mathop{r}\limits^{ \rightharpoonup})|{{\rm{g}}}_{m,n}^{\phi }(\mathop{r}\limits^{ \rightharpoonup})\right\rangle G(m,n)$$
(4)
where \({{\rm{g}}}_{m,n}^{\phi }\left(\mathop{r}\limits^{ \rightharpoonup }\right)\) denotes spatial complex amplitude of the mode experienced the local amplitude gain by off-axis pump; see Supplementary Information for details.
The numerical analysis presented in Fig. 2a examines the modal gain and competition mechanism of the two competing HG modes. This analysis assumes that the outermost peak of each mode experiences the same local gain factor \(\kappa\), leading to around double net modal gain during each trip. Specifically, the simulation shows that \({{\rm{HG}}}_{\mathrm{7,0}}^{90^\circ }\) and \({{\rm{HG}}}_{\mathrm{4,4}}^{0^\circ }\) obtain 2.2 (12.9) and 1.7 (6.9) times net modal (local amplitude) gains as setting κ = 9 × 104, respectively. The findings indicate that the local gain offered by the off-axis pump does not substantially degrade modal purity, remaining 80.51% and 92.65% purity for the two modes, see spatial spectra displayed by density matrices in Fig. 2a right-hand, and thereby exhibiting diffraction self-healing, as shown in Fig. 2a middle. For that, the off-axis pump consistently provides effective net gain throughout cyclic oscillations. However, the inevitable gain competition between these two example modes also demonstrates that relying solely on off-axis pump gain control is impossible to generate 2D HG modes. More generally, when the maximum intensity patterns of 1D and 2D modes coincide, the latter is disadvantaged in gain competition due to lesser spatial overlap with the pump, thus restricting the tunable range always to the 1D HG modal space.
a Simulated single-pass gross (G) and net (N) gain for two competing modes (lefthand), \({{\rm{HG}}}_{\mathrm{7,0}}^{90^{\circ}}\) and \({{\rm{HG}}}_{\mathrm{4,4}}^{0^{\circ}}\), within a cylindrically symmetric cavity, with the pump spot size configured to \(0.7w\), their pattern evolution from the origin (\({z}_{0}\)) to the Fourier (\({z}_{\infty }\)) planes after the local amplitude gain (middle), as well as associated density matrices (righthand). b Theoretical demonstration of modal blockade through intracavity astigmatism, where the continuous pattern variation of the undesired \({{\rm{HG}}}_{\mathrm{7,0}}^{90^{\circ}}\) results in its failure to compete in net gain with the \({{\rm{HG}}}_{\mathrm{4,4}}^{0^{\circ}}\). c Theoretical steady-state wavefunctions of cavity outputs, both in the absence and presence of astigmatic detuning for nonzero-\(\phi\) modal blockade, where the spatial wavefunctions were simulated using Fox-Li iteration, and the corresponding density matrices were obtained through numerical projection on the \({{\rm{HG}}}_{m,n}^{0^{\circ}}\) space
Nonzero \({\boldsymbol{\phi }}\) mode blockade by astigmatic detuning
How to circumvent the limitation, or more specifically, to blockade modal gain for undesired competitors?
A natural consideration is seeking to reduce somehow the average pump spatial overlap for HG modes with nonzero \(\phi\) throughout the oscillation process. Although this task appears challenging, the astigmatic unitary transformation mentioned previously offers a straightforward and viable solution, with which we do not even have to consider any complex active intracavity control. As demonstrated in Fig. 2b, the introduction of a subtle astigmatic detuning of \({\rm{\pi }}/q\) along the Cartesian coordinates with ϕ = 0 within a cavity that supports q cycles induces distinct diffraction behaviors in the competing mode pair \({{\rm{HG}}}_{\mathrm{7,0}}^{90^\circ }\) and \({{\rm{HG}}}_{\mathrm{4,4}}^{0^\circ }\). Due to the axially separable Gouy phase characteristic of HG modes, their beam profiles remain axially self-similar within the same Cartesian coordinates30,31. That is, the target 2D mode \({{\rm{HG}}}_{\mathrm{4,4}}^{0^\circ }\) consistently exhibits ‘pattern revival’ with each passage through the gain medium33, analogous to the transmission of a linear polarization state along a birefringence axis, thereby maintaining stable net modal gain. In contrast, owing to the intracavity astigmatic detuning, the pattern of the undesired \({{\rm{HG}}}_{\mathrm{7,0}}^{90^\circ }\) gradually transforms into that of the orthogonal one \({{\rm{HG}}}_{\mathrm{7,0}}^{-90^\circ }\) via the \({{\rm{LG}}}_{\mathrm{7,0}}\) mode during oscillations. This results in a significant loss in pump overlap and associated net gain, facilitating a modal blockade for nonzero ϕ HG spectra.
Consequently, the HG modal set in the astigmatic cavity is constrained to rectangular symmetry as defined by Cartesian coordinates with \(\phi =0\), unlocking full spectrum tunability in HG space. In addition to the primary 1D competitor, it is noteworthy that the intracavity astigmatic detuning also blockade all possible undesired modes, potentially enhancing the output modal purity (or spatial linewidth). The theoretical steady-state wavefunctions of cavity output depicted in Fig. 2c, simulated using the Fox-Li iteration method with a pattern correlation threshold 99% for adjacent cycles, corroborate this inference. Specifically, the modal purity of the \({{\rm{HG}}}_{\mathrm{7,0}}^{90^\circ }\) from a cylindrically symmetric cavity without the modal blockade for nonzero-ϕ modes is approximately 94.3%, which is actually closer to the Ince-Gaussian mode \({{\rm{IG}}}_{\mathrm{7,7}}^{90^\circ }\) with an ellipticity of 11 (99.7% purity), whereas the desired \({{\rm{HG}}}_{\mathrm{4,4}}^{0^\circ }\) from the astigmatic detuned cavity exhibits a near-perfect purity of 99.6%. Here, the modal purity, for both the theory and following experiments, was characterized by the inner product of cavity outputs and their nearest eigen modes.
Experimental setup
Figure 3 depicts the experimental configuration designed to achieve a tunable single-transverse-mode laser throughout the HLG space. This setup incorporates an astigmatic oscillator operating at 1064 nm, which is capable of generating arbitrary \({{\rm{HG}}}_{m,n}^{0^\circ }\) modes, and an extra-cavity AMC for further unitary transformations of the modes as required.
The left-bottom inset illustrates the theoretical lateral displacement of the pump center as a function for selecting modal indices \(m\) and \(n\), which is normalized to the fundamental waist of cavity modes. The upper-right inset illustrates the origin of astigmatic detuning and the corresponding retardance accumulation upon oscillations; further details are provided in “Methods”
The astigmatic oscillator is built using a V-shaped cavity arrangement with a folding angle of \(\delta =\mathrm{10}^\circ\). The crucial astigmatic retardance originates from the use of a common curved mirror, CM, rather than an off-axis parabolic mirror avoiding astigmatism, to fold the cavity, resulting in different focal powers on the tangential (\({f}_{T}=(R\cos \delta )/2\)) and sagittal planes (\({f}_{S}=R/(2\cos \delta )\)). This precise oscillator design (see “Methods”), achieved without the need for additional astigmatic elements, can accumulate a total \(\pi\) astigmatic retardance throughout the oscillation process, as shown in the upper-right inset. This intrinsic astigmatic detuning breaks the rotational (or cylindrical) symmetry of intracavity diffraction, thereby permitting the excitation of arbitrary 2D modes \({{\rm{HG}}}_{m,n}^{0^\circ }\) (here \(\phi =0^\circ\) aligns with the table plane) through lateral displacement of the pump beam, as previously discussed.
The extra-cavity AMC comprises a Dove prism and a pair of cylindrical lenses, each having an identical focal length of \(f=\mathrm{100}{\rm{mm}}\), serving as \(\lambda /2(\phi )\) and \(\lambda /4(\theta )\) astigmatic retarders, respectively. Additionally, a relay lens with a focal length of \(f=\mathrm{100}{\rm{mm}}\) is utilized for waist matching (not depicted). The sequential arrangement of the \(\lambda /2\) and \(\lambda /4\) retarders facilitate the unitary rotation of the generated mode \({{\rm{HG}}}_{m,n}^{0^\circ }\), allowing access to all possible SU(2) coherent states on the modal sphere, i.e., generalized HLG modes29,30,31.
Experimental results
The tunability of the oscillator within the spatial degrees of freedom defined by HG space, characterized by the accessible range of bivariate modal indices \(m\) and \(n\), was initially validated. In experiment, the pump center was initially aligned with the cavity axis, allowing the oscillator to operate in the fundamental mode \({{\rm{HG}}}_{\mathrm{0,0}}^{0^\circ }\). Using this position as a reference, higher-order modes \({{\rm{HG}}}_{m,n}^{0^\circ }\) were achieved by laterally displacing the pump center. As demonstrated in [Video-1], the modal indices \(m\) and \(n\) increase continuously with lateral displacements in the horizontal (\(\varDelta x\)) and vertical (\(\varDelta y\)) dimensions, respectively, while displacements in both dimensions yield a 2D modal output as anticipated.
In both dimensions, the maximum displacements of the pump center were constrained to approximately 1.9 mm by the crystal aperture, corresponding to a normalized displacement exceeding 14 times the fundamental cavity waist of 136 μm. According to the relationship depicted in the left-bottom inset of Fig. 3, the accessible modal order for both \(m\) and n are exceeding 200. That is to say, by simply adjusting the pump position, it is possible to generate over 40,000 distinct HG modes from the oscillator on demand. Experimentally, the highest mode order achieved was \(N=4\mathrm{30}\) when \(\Delta x=\Delta y=1.9{\rm{mm}}\) under a 2 W pump power, although this mode \({{\rm{HG}}}_{\mathrm{214,216}}^{0^\circ }\) exhibited unequal \(m\) and \(n\). This observation accords with the fact that the cavity beam waist has a slightly elliptical profile.
To assess the quality of the single-transverse-mode output across the tunable range, we sampled several modes from \({{\rm{HG}}}_{\mathrm{0,0}}^{0^\circ }\) to \({{\rm{HG}}}_{\mathrm{20}\mathrm{3,20}3}^{0^\circ }\) using spatial complex amplitude measurements34. Figure 4a presents the observations, which display nearly perfect complex amplitude patterns. Beyond subjective qualitative visual assessment, these patterns provide comprehensive information about the laser fields and can be used for further quantitative modal characterization. Specifically, the modal purity indicated in the table in Fig. 4d was obtained via numerical projection measurement, while the beam quality \({M}^{2}\) values next to purities were calculated using numerical diffraction methods, further details on the characterization can be founded in Supplementary Information. Both characterizations confirmed the high modal quality of the oscillator output.
Typical spatial complex amplitudes of a HG and b LG modes recorded in the experiment; corresponding intensity patterns are provided in additional data in Supplementary Information. c Generalized HLG modal extensions with \({{\rm{HG}}}_{\mathrm{8,4}}^{0^{\circ}}\) and \({{\rm{HG}}}_{\mathrm{6,6}}^{0^{\circ}}\) as inputs. d Modal purities and beam quality factors obtained from corresponding spatial complex amplitudes
Subsequently, by employing the Dove prism to rotate the oscillator modes to \(\phi =90^\circ\), the final output was transformed into \({{\rm{LG}}}_{{\mathcal{l}},p}\) after passing through the \(\lambda /4\) astigmatic retarder, see [Video-2] demonstrating the continuous tunability in LG space. Figure 4b illustrates the measured spatial complex amplitudes, corresponding modal purity and beam quality \({M}^{2}\) are provided in Fig. 4d. In comparison to their HG counterparts, each LG mode exhibits a consistent Gouy phase and a corresponding \({M}^{2}\) value within the radial dimension. Furthermore, the conversion from 1D HG modes results in pure azimuthal modes with \({\mathcal{l}}=m\) or \(n\); whereas, a 2D mode input with equal \(m\) and \(n\) generates pure radial modes that carry no net OAM. Beyond these point-to-point pairs of HG and LG outputs, by rotating both the \(\lambda /2\) and \(\lambda /4\) astigmatic retarders, each original oscillator mode can be further transformed into numerous ‘elliptical’ HLG modes, which can full cover the surface of their higher-order SU(2) modal sphere. Figure 4c presents two examples of such SU(2) modal extension with \({{\rm{HG}}}_{\mathrm{8,4}}^{0^\circ }\) and \({{\rm{HG}}}_{\mathrm{6,6}}^{0^\circ }\) as inputs, respectively. It is shown that, on the basis of a high-quality HG oscillator, new extending HLG modes via the SU(2) rotation also exhibit high beam quality.
In a manner similar to the power-frequency dependence observed in conventional tunable lasers, the tunable structured laser presented in this study also exhibits a power dependence on the operating spatial mode. This phenomenon primarily arises from the fact that the pump overlap in Eq. (3) and the associated effective gain achieved in each oscillation diminish with increasing modal order. Figure 5a illustrates the measured mode-threshold dependence, which aligns well with theoretical predictions. Experimentally, the threshold was observed to initiate at 36 mW for the fundamental mode and progressively increased with pump displacement to generate higher-order modes. For instance, the thresholds for \({{\rm{HG}}}_{\mathrm{102,104}}^{0^\circ }\) and \({{\rm{HG}}}_{2\mathrm{14,2}\mathrm{16}}^{0^\circ }\) outputs reached 1.13 W and 1.67 W, respectively.
a Laser threshold of typical 2D \({{\rm{HG}}}_{m,n}\) modes excited by off-axis pumping, the numbers in the bracket are the mode indices obtained in the experiment. b Laser output power of typical 2D \({{\rm{HG}}}_{m,n}\) modes as a function of input pump power
Figure 5b depicts the measured output power curve as a function of the incident pump power. The slope efficiency of the output modes gradually decreased from 41.9% to 22.3% and 13.7% as the modal order increased from \({{\rm{HG}}}_{\mathrm{0,0}}\) to \({{\rm{HG}}}_{8\mathrm{6,89}}\) and \({{\rm{HG}}}_{2\mathrm{14,216}}\), respectively. The output power of the highest order \({{\rm{HG}}}_{2\mathrm{14,216}}\) achieved 78 mW under an incident pump power of 2.3 W. Notably, both theoretical and experimental results mentioned above were obtained with a constant pump radius of 100 μm. Therefore, employing a variable pump radius to optimize the pump overlap and effective gain of higher-order modes can, in principle, mitigate the power and modal purity decline associated with increasing modal order. Besides, employing multi-edge pumps may be another feasible path to over such declines.