Resonance-free Fabry-Pérot cavity via unrestricted orbital-angular-momentum ladder-up

Domain: physics
Published: 2025-11-24
Editorial outcome: major_revision

Abstract
Introducing elements into an optical cavity that modify the transverse spatial field structure can also impact the cavity spectral response. In particular, an intra-cavity spatial mode-converter is expected to induce modal runaway: unrestricted ladder-up in the modal order, concomitantly thwarting coherent field interference, thereby altogether suppressing the resonant response – a phenomenon that has yet to be observed in an optical cavity. Here we show that a single intra-cavity holographic phase mask placed in a compact free-standing planar Fabry-Pérot cavity renders the cavity spectral response resonance-free. By acting as a mode-converter on a basis of Laguerre-Gaussian (LG) modes, an incident broadband fundamental Gaussian mode exits the cavity in the form of a superposition of a large number of collinearly propagating broadband LG modes of fixed parity whose spectra coincide with that of the input. Crucially, the resonance-free spectral response is maintained while changing the cavity length by ~350%, raising the prospect of stable resonant optical sensors whose performance is impervious to length perturbations.

Introduction
Optical resonators are a key building block in optics and photonics, which have enabled the optical revolution that started with the invention of the laser1. Resonant field buildup resulting from intra-cavity light recycling (optical feedback via mirrors) is useful across a broad swath of applications: enhancing linear effects (e.g. coherent perfect absorption)2–5, nonlinear effects (e.g., harmonic generation and nonlinear wavelength conversion)6–10, sensing11, optical combs12 and clocks13, in quantum optics14, and even in gravitational wave detection15. Nevertheless, light recycling is achieved coherently only over a narrow linewidth at discrete resonant wavelengths separated by the free spectral range (FSR). Moreover, further enhancing the field buildup can be harnessed only over an ever-shrinking linewidth1. This fundamental complementarity between field buildup and resonant linewidth has spurred on efforts towards the realization of linewidth broadening without sacrificing the cavity quality factor. Examples along these lines include white-light cavities16,17 that incorporate an active medium endowed with anomalous dispersion (whether atomic species18,19 or nonlinear effects20), omni-resonant cavities21 in which angular dispersion is introduced into the externally incident field to couple a broad continuous spectrum into the cavity22–24, among other efforts25–28.

A different research enterprise has recently focused on modifying the transverse spatial structure of a laser field through introducing intra-cavity optical elements29; for example, amplitude stops or apertures30, spatial light modulators31,32, phase plates33, or metasurfaces34–36. These efforts have yielded lasers emitting prescribed orbital angular momentum (OAM) modes or other spatial field structures37. Only recently has it been recognized that intra-cavity modification of the spatial structure of the cavity field can exert a profound impact on its spectral structure38, thereby opening the possibility of spectrally broadening of the resonant bandwidth through spatial structuring of light. However, the long cavity lengths typically utilized for producing structured laser fields unfortunately mask any such effects by rendering the free spectral range (FSR) too small to be readily resolved, thus precluding the investigation of coupling between the spatial and spectral degrees-of-freedom.

This limitation can be lifted by exploiting on-chip waveguide-based cavities, which facilitates spectrally resolved measurements by reducing the cavity length (e.g., length ~150 μm with FSR ~1.5 nm at a wavelength of 1.5 μm in ref. 38). Intriguingly, the impact of an intra-cavity mode-converter on the cavity spectral response is sensitive to the modal basis selected. In the case of a waveguide geometry that supports Hermite-Gaussian-like (HG-like) modes in a Cartesian coordinate system, a single intra-cavity mode-converter toggles the cavity field between two waveguide modes after each cavity roundtrip, effectively doubling the cavity length and thus halving its FSR38. Further modification of the spectral structure in such a waveguide-based cavity requires introducing a sequence of spatial mode-converters – each responsible for converting between a particular pair of HG modes. In this case, a restricted ladder-up effect of the modal order can be produced (limited by the number of individual mode-converters implemented), which can be used to establish a folded back-and-forth light-recycling path with broadband operation as realized in ref. 39. However, a cavity structure was not established in 39, so that broadband operation is guaranteed in absence of the mode-converters, albeit with only a single pass through the waveguide. To rely on a single mode-converter to achieve this effect therefore requires changing the modal basis.

It is therefore intuitively expected that an intra-cavity mode-converter can, in principle, result in unrestricted modal ladder-up (or modal run-away), so that a single spatial mode externally incident on this cavity produces an output in the form of a superposition of a large number of collinearly propagating orthogonal modes. Furthermore, accompanying this complex spatial field response is a dramatic change in the spectral structure. Rather than the discrete spectral resonances, the orthogonality of the cavity transverse modes thwarts intra-cavity interference, thereby suppressing the resonant response. Such a cavity is expected to display a broadband resonance-free spectral response that is altogether independent of the cavity length.

Nevertheless, the use of a single intra-cavity element to enable modal ladder-up and a resonance-free spectral response has eluded observation to date in an optical cavity. Despite its conceptual intuitiveness, this goal represents a substantial technical challenge. First, the cavity need to remain compact after introducing the intra-cavity elements to facilitate spectrally resolving the resonances. Second, as mentioned above, waveguide HG-like modes are not conducive to modal ladder-up with a single intra-cavity element, and modes carrying OAM are preferred instead because a single phase element can implement a modal transformation in this basis for all mode orders. Third, high-efficiency modal conversion is required to eliminate any remnant resonant spectral response without deleteriously impacting the cavity finesse. Fourth, the orthogonality of the modes produced by the mode-converter is required to suppress intra-cavity interference of the cavity field. Fifth, large-area modes are necessary to reduce diffraction over the entire resonant path length in the cavity, to further maintain mode-orthogonality in all the roundtrips. These requirements place stringent constraints on the cavity design and construction to achieve a resonance-free spectral response.

Here we show that introducing a single, thin, intra-cavity phase element into a planar Fabry-Pérot (FP) cavity can radically change the cavity spectral response to yield a broadband resonance-free cavity. Upon incorporating the phase element, the conventional discrete resonant spectral structure is entirely eliminated, and the cavity spectral response is immediately rendered flat. By utilizing a free-standing yet compact FP cavity, we satisfy multiple desiderata simultaneously. First, any modal basis can be utilized, and we make use here of OAM-carrying Laguerre-Gaussian (LG) modes. The intra-cavity element is a holographic phase mask (HPM) written in a thin volume Bragg grating in transmission mode capable of changing the topological charge (OAM) of an LG mode traversing it by ±1 with high conversion efficiency (>85% for all modal orders), so that the cavity field increases the OAM by 2 after each roundtrip. Second, the FP cavity accommodates a large transverse beam profile whose associated Rayleigh range significantly exceeds the distance corresponding to the cavity photon lifetime (the product of the cavity finesse and roundtrip length), and is simultaneously compact to yield a large FSR. All the forward and backward propagating field components are mutually orthogonal, which thwarts coherent field interference and frustrates the emergence of resonances in the transmitted spectrum. We are therefore able to carefully correlate the elimination of the resonant spectral structure with the modification of the transverse spatial field structure. We have unambiguously verified our predictions experimentally using a variety of measurement approaches. We first verify the performance of the free-standing HPM as a mode-converter carrying out the OAM transformation LG0,m to LG0,m+1. After introducing the HPM into the cavity, we resolve the modal content produced by an incident broadband fundamental mode LG00 via an LG mode-sorter that reveals modes extending up to m=19 collinearly produced from the cavity. We next measure the spectral response of the cavity in absence and presence of the HPM: whereas the former displays the usual discrete resonant spectral lines, the latter has a flat spectral response devoid of any resonances, so that the output spectrum coincides with the incident spectrum. We have confirmed this resonance-free behavior while changing the cavity length by ≈350% (from 1.3 mm to 4.8 mm). Whereas the FSR of the cavity in absence of the HPM changes accordingly (from 274 pm to 100 pm) over this span of cavity lengths, the spectral response in the presence of the HPM remains invariant. These results may have profound implications for lasing in active micro-cavities in which the spatial field structure is modified, while offering opportunities for stable resonant optical sensors that are impervious to cavity-length perturbations, in addition to providing a platform for exploring the non-Hermitian dynamics of exceptional points40 along the dual synthetic dimensions of modal order and wavelength.

Results
Conventional FP cavity versus resonance-free FP cavity
A conventional FP cavity confines light between two mirrors, thereby leading to field-buildup in a standing wave at specific resonant frequencies (Fig. 1a,b)1,41. The transverse spatial structure of the resonant field is maintained after each cavity roundtrip, except for an overall amplitude factor (Fig. 1a). The transmission spectrum of an FP cavity comprises sharp peaks (Fig. 1b) separated by the free spectral range (FSR). For example, when a Gaussian beam is externally incident on this FP cavity, the beam reflects back and forth between the two mirrors, and multipass interference yields a Gaussian beam emerging on resonance, and the spectrum of the Gaussian beam is modulated by the cavity spectral response (Fig. 1b). We assume that the transverse size of the field is sufficiently large that diffractive effects can be neglected over the entire path length of the field within the cavity. In this case the spatial and spectral responses of the cavity here are separable.

Our goal here is to construct a non-standard FP cavity in which the formation of standing waves is suppressed (Fig. 1c), thereby eliminate the resonant response (Fig. 1d), without changing the number of roundtrips in the cavity. We refer to this cavity as a resonance-free cavity (RFC). This novel behavior is achieved by introducing a single intracavity purely phase element to act as a spatial mode-converter that increases the modal index m→m+1 (for all m) of a spatial mode belonging to an orthogonal basis, such that m→m+2 upon each cavity roundtrip (Fig. 1c). We select here the set of LG modes with radial index 0 and azimuthal (OAM) index m, and we contract the notation of these pure OAM modes to LGm. The mode-converter we utilize here is a holographic phase mask (HPM) that comprises an appropriate phase distribution imprinted within a volume Bragg grating to increase the OAM index by 1 after each pass (Methods).

Several requirements must be satisfied by the HPM for the RFC to exhibit resonance-free behavior. First, high diffraction efficiency (modal conversion) is required; finite modal conversion results in a remnant conventional resonant field component superimposed on the target resonant-free spectral response. Second, a thin component is desirable to minimize the cavity length and yield a large FSR to render the resonance suppression spectrally unambiguous. Third, a small diffraction angle (6.3° here) between the incident and diffracted fields is beneficial in optimizing the angular and spectral acceptance of the HPM, and to further ensure a minimal cavity length42,43, while still eliminating non-diffracted light. Fourth, a large HPM area is desirable to accommodate a large beam profile (and, therefore, a large Rayleigh range). All of these requirements are satisfied by the HPM that has a device thickness of 1 mm, an active area of diameter 3 mm (in a volume Bragg grating of area 12×12 mm2), and high diffraction efficiency for all modes (as confirmed below).

The intracavity HPM modifies the phase front upon each pass. Traversing from left to right yields the transformation LGm → LGm+1, and from right to left LGm → LGm−1 35,44. The orthogonality of the LGm modes disrupts constructive multipass interference, yielding a broadband and flat spectral transmission response (Fig. 1d), devoid of the sharp resonant peaks typical of a conventional FP cavity. Therefore, when a Gaussian beam is incident externally on the RFC, the spatial profile of the transmitted beam comprises a superposition of odd-order LG modes (m≥1), and the spectrum of each mode in the output field coincides with the input spectrum.

Mode conversion in an RFC
We first highlight a key feature of the OAM modes in the LG basis in contrast to other modal bases such as those of the HG modes. To convert HG0 to HG1 in one transverse dimension, a π-phase step is required (Fig. 2a). Traversing from left to right results in HG0 → HG1, and from right to left HG1 → HG0. Note that reflection from a mirror does not impact the modal order of an HG mode. By introducing such a phase pattern in an FP cavity, the field toggles between HG0 and HG1, no modal ladder-up or run-away occurs, and the resonant spectral structure survives, except for a potential modification of the FSR due to a change in the effective cavity length, as demonstrated recently in ref. 38. Accessing higher-order HG modes requires utilizing a sequence of individualized mode-converters for each conversion HGm → HGm+1, which was realized in ref. 39 (up to m=6) to fold the propagation path in a multimode waveguide and increase the propagation distance. No cavity was established in ref. 39, and thus no resonant response follows when the mode-converters are removed.

A feature common to the work reported in 38 and 39 is field confinement to an on-chip multimode waveguide that supports HG modes rather than LG modes. Relying instead on LG modes opens the new possibilities illustrated in Fig. 2b. Normal incidence from the left produces a diffracted field on the right in which the OAM index increases by 1 (LGm → LGm+1), while retracing the path from the right to the left decreases the OAM index by 1 (LGm → LGm−1), for all values of m; see Supplementary Fig. 1 for experimental verification of all the configurations in Fig. 2b. Coupled with the change in the OAM sign upon reflection from a mirror m→−m, this mode-converter leads to a ladder-up effect in which the modulus of the OAM order progressively increases with each roundtrip (in principle without bound), rather than the modal toggling in the HG basis. Critically, maintaining the orthogonality of these modes further requires complete propagation-collinearity and minimal diffraction over multiple cavity roundtrips.

The evolution of the cavity field is illustrated in the distribution matrix in Fig. 2c. The fields on the right and left sides of the HPM in the RFC (Fig. 1c) are represented in Fig. 2c by green and orange columns, respectively. A fundamental Gaussian beam LG0 entering the cavity diffracts at the HPM to LG1, a percentage of which exits the RFC at M2. Reflection from M2 produces LG−1 propagating back to the HPM, whereupon it is converted to LG−2, and then to LG2 upon reflection from M1. Each roundtrip therefore increases the modal index by 2, thereby generating a sequence of LG modes with increasing order, with the forward and backward propagating modes having opposite OAM signs. As shown in Fig. 1c, only odd LG modes with positive orders are transmitted through the RFC at M2, while negatively signed even LG modes exit from M1. The effective number of mode conversions is limited by the amplitude reduction upon each roundtrip (i.e., by the cavity quality factor), which is determined by the HPM efficiency, the inherent divergence of the converted LG modes, and the intrinsic cavity roundtrip losses. The modal evolution is further elucidated by the directed graph in Fig. 2d that depicts the ordered sequence of modal conversion over two cavity roundtrips. The plot in Fig. 2e of the modal order with roundtrip number then illustrates the unrestricted modal-order ladder-up that is expected in the RFC.

Theory of RFC
Consider a conventional, symmetric, planar FP cavity with roundtrip phase χ, and field reflection and transmission coefficients for the mirrors of r and t, respectively, with corresponding reflectance and transmittance R=|r|2 and T=|t|2. When a fundamental Gaussian mode ψ00(r,φ;ω) is incident, the transmitted field is:

ψFP(r,φ;ω) = t2eiχ/2 [1 + r2eiχ + r4ei2χ + ...] ψ00(r,φ;ω) (1)

We have ignored diffraction effects over the course of the photon lifetime in the cavity. The spatially averaged transmitted intensity is:

IFP(ω) = 1 / (1 + 4R sin2(χ/2) / (1 − R)2) (2)

which is the usual spectral transfer function of a planar FP cavity.

In presence of an ideal HPM in the cavity (unity diffraction efficiency for all mode orders), the transmitted field takes on a different form:

ψRFC(r,φ;ω) = t2eiχ/2 [ψ01(r,φ;ω) + r2eiχψ03(r,φ;ω) + r4ei2χψ05(r,φ;ω) + ...] (3)

where ψ0ℓ(r,φ;ω) is the modal profile of a LG mode with radial index 0 and azimuthal (OAM) index ℓ. Assuming that the mode profiles form an orthonormal basis, ∫ rdrdφ ψ*0m(r,φ;ω) ψ0n(r,φ;ω) = δmn, then a simple calculation of the spatially averaged transmission yields IRFC(ω) = 1; that is, the spectral transmission is flat and devoid of resonances. The requirements for this result are: (1) the modal profiles are orthonormal; (2) diffraction is minimal across the photon lifetime; and (3) perfectly performing HPM. Note that introduction of radial modes ψpm(r,φ;ω) indexed by p into the field via the HMP does not impact the operation of the RFC as long as the modes remain orthogonal; i.e., ∫ rdrdφ ψ*pm(r,φ;ω) ψqn(r,φ;ω) = δmn δpq. In other words, even in presence of radial modes, Eq. (3) still holds. We thus confine our analysis to the azimuthal modes without loss of generality.

We demonstrate below the orthogonality of the LG modes produced via the HPM, and the large transverse area of the cavity and HPM minimize the impact of diffraction, leaving the finite HPM diffraction efficiency. If the HPM has diffraction efficiency η (independent of the modal index), and we first ignore the non-diffracted fraction of the power, then the spectral transmission remains flat, but has a reduced amplitude: IRFC(ω) = (1 − (1 − η) / (1 + ηR2)). Finally, assuming a finite diffraction efficiency η for the HPM but retaining the non-diffracted field components in the cavity yields a weak contribution from a resonant spectrum atop a dominant flat spectral transmission. We model this scenario numerically while making use of the measured values of the HPM diffraction efficiency using the model described in Supplementary Section 6. The measurements reported below are compared to calculations based on this full model.

Modal characterization of the HPM
The fabricated HPM consists of a volume Bragg grating superimposed with a first-order LG-mode phase profile (Methods). Two distinct measurements verify that the intended HPM functionality is realized: (1) verifying the transverse phase profile imparted by the HPM and the orthogonality of the cavity LG modes, and (2) assessing the diffraction efficiency of the HPM for different incident LG orders.

First, we verify that a precise azimuthal 2π phase change is imparted by the HPM to the wavefront of an incident optical field via holographic reconstruction45,46. Any deviation from this 2π phase change compromises the orthogonality of cascading LG modes47,48, resulting in the response of the RFC deviating from the target resonance-free behavior. As illustrated in Fig. 3a, a Gaussian beam LG0 is superposed with the converted mode LG1 produced by the HPM from LG0 (Fig. 3b), and the phase profile is retrieved from their interference pattern using off-axis holography (Fig. 3c)46, which shows a phase change of ≈(0.96 ± 0.025) × 2π rad per pass (this phase is averaged over the FWHM bandwidth of ~5 nm of the laser used). This provides near-orthogonal modal conversions after each pass through the HPM. This measurement is repeated for the LG1 and LG2 modes as reported in Supplementary Fig. 2, where the orthogonality of the collinearly propagating modes is confirmed.

Second, we estimate the diffraction efficiency of the HPM across the spectral bandwidth of interest for 7 LG modes. We prepare the modes LG0 to LG6 (Methods), which are directed to the HPM to undergo diffraction into a higher-order mode LGm → LGm+1 (Fig. 4b, inset in top row). The spatial intensity distribution of prepared LG modes (LG0 through LG6) and the converted modes produced by the HPM (LG1 through LG7) are shown in Fig. 4a and Fig. 4b (left panels), respectively. In both cases, the order of the LG field is determined using the astigmatic transform technique49–51, which involves analyzing the intensity pattern of the LG beam at the Fourier plane of a plano-convex cylindrical lens (Fig. 4a,b, right panels). In this technique, the OAM number is equal to n−1, where n is the number of lobes observed in the Fourier plane of a plano-convex cylindrical lens with a focal length of 150 mm. The diffraction efficiency is estimated from the ratio of the diffracted power at the optimal Bragg angle of the HPM to the input power, both measured using a calibrated silicon photodetector (Gentec-EO PH100-SI-HA-D0). The HPM used is not provided with anti-reflection coatings, so the measured diffraction efficiency includes a reduction induced by Fresnel reflections. The diffraction efficiency of the HPM for converting the first seven LG modes to their next higher-order counterparts are provided in Fig. 4b (at the bottom of the left panels), whose average overall modes exceeds ≈87.5%.

Modal characterization of the RFC
The base FP cavity consists of two high-reflectivity flat mirrors (R≈94% at a wavelength ~1096 nm), which we illuminate with a continuous-wave Ytterbium fiber laser (YLR-100; IPG Photonics) having a spectral width of 5 nm (full width at 1/e2 maximum) centered at 1095.6 nm. The randomly polarized laser is rendered vertically polarized via a polarizing beamsplitter and a zero-order half-wave plate. The RFC is produced by inserting between the two mirrors an HPM of dimensions 12.5×12.5×1 mm3 diffracting at an angle of 6.3° at the central wavelength, with spectral and angular bandwidths of ≈32.5 nm and ≈4.33 mrad, respectively (Fig. 5a). The phase pattern constituting the HPM occupies a circular area of diameter ≈3 mm in the volume Bragg grating in which it is embedded, which implements the LG mode transformation. Laterally transposing the HPM area to intercept the beam path in the FP cavity allows us to convert the FP cavity into an RFC, and vice versa, by moving the HPM area out of the cavity beam path, all without changing the cavity length. The mirror M2 is normal to the diffracted beam from the HPM, and is thus tilted with respect to M1. Consequently, the separation between the two mirror edges are different, as shown in the photographs of the RFC from the ‘long’ and ‘short’ sides in Fig. 5b (see also Supplementary Fig. 3a).

We depict in Fig. 5a the measurement modalities employed at the cavity exit (see Supplementary Fig. 3b-d): (1) spatial beam profiling of the cavity output; (2) analysis of the LG modal content; and (3) measuring the transmitted spectrum. Measurement results for three different cavity configurations are shown in Fig. 5c-e (left column): a conventional FP cavity with the HPM area displaced out of the beam path (Fig. 5c); the RFC incorporating the HPM (Fig. 5d); and the RFC with the out-coupling mirror M2 slightly tilted to angularly displace the field after each roundtrip (Fig. 5e). We focus on the transmitted field because the reflected field is dominated by the reflected LG0 mode. In all three cases, the cavity is illuminated with a Gaussian beam LG0. To record the spatial profile, the cavity is followed by a spherical lens (focal length 150 mm) and the intensity profile is captured by a camera (BladeCam S-BC-XHR) positioned at a distance 120 mm after the lens.

To analyze the LG modal content of the cavity output, we make use of an LG mode-sorter that separates spatially overlapping LG modes (in a polar coordinate system) into displaced parallel lines (in a Cartesian coordinate system)52. The mode-sorter comprises two axially separated phase plates that implement a log-polar coordinate transformation53,54, after which the azimuthally varying phase distributions of the circularly symmetric LG modes are converted to spatially overlapping, rectangular strips endowed with linearly varying phases, which are spatially delineated using a spherical lens52,55,56 (Supplementary Section 4). The lateral position of the LG modes after the mode-sorter is mλf/d, where m is the mode order, λ is the wavelength, f is focal length of the lens (f=400 mm here), and d is the length of the transformed beam at the mode-sorter exit55. This results in a transverse separation of ≈84 μm between adjacent LG-mode orders. A portion of the field at the FP cavity exit (enclosed in a white dashed square in the second column of Fig. 5c–e) is directed to the mode-sorter entrance, and the output from the mode-sorter (third column in Fig. 5c–e) where each mode appears as a vertical strip. In the integrated intensity at the mode-sorter output (fourth column of Fig. 5c–e), each mode appears as a peak. In the case of the conventional FP cavity (Fig. 5c), where the HPM area is displaced out of the cavity, the output field is still Gaussian and matches the input field. After directing the output field to the mode-sorter, only a single vertical strip is visible after the mode-sorter since the cavity transmits only the LG0 mode.

By moving the HPM area into the FP cavity, a profound change is observed at the RFC output (Fig. 5d). Rather than a Gaussian mode, we observe a doughnut mode, and upon directing this field to the mode-sorter, we observe a cascade of LG modes reaching to the order m=19 (the odd modes from LG1 to LG19), each diminishing by ≈14.5% with respect to the preceding order. This drop, and the presence of a remnant LG0 mode, is attributed to the finite reflectivity of the cavity mirrors, the efficiency of the HPM, and the deviation from the ideal 2π phase change per pass.

Tilting the out-coupling mirror M2 in presence of the HPM in the RFC (Fig. 5e) introduces a small angular displacement upon each cavity pass, so that the lens after the cavity spatially displaces the field from each roundtrip, resulting in spatial separation of the transmitted LG modes. This allows each mode to be spatially isolated and directed to the mode-sorter separately. An example is shown in Fig. 5e, where the LG1 mode is sent alone to the mode-sorter, which is verified by the appearance of a single line at the mode-sorter output at the expected transverse location.

Spectral characterization
Finally, we examine the spectrum of the transmitted field. For an ideal HPM, we expect a flat transmission spectral response (limited by the bandwidth of the underlying VBG), so that the output spectrum from the RFC coincides with the input source spectrum. The spectrum of the transmitted field is measured for various cavity lengths L by coupling the cavity output to a multi-mode fiber connected to an optical spectrum analyzer (OSA; Yokogawa AQ6370, spectral resolution ~0.02 nm). We plot in Fig. 6 the measured spectra from the conventional FP cavity (left column) and the RFC (right column) for cavity lengths L=1.3, 2.3, and 3.3 mm, corresponding to FSR values of 274, 205, and 150 pm, respectively (measurements for other cavity lengths are provided in Supplementary Fig. 6). The cavity length L is the average of the ‘long’ and ‘short’ sides of the cavity. For example, the L=1.3 mm cavity corresponds to a ‘short’-side length of 1.1 mm (the edges of the cavity mirrors are nearly in contact) and a ‘long’-side length of 1.5 mm (Fig. 5b).

The measured spectra for the conventional FP cavity exhibit a clear resonant response with the expected FSR. In contrast, the transmission spectrum for the RFC exhibits an early smooth spectrum that closely matches the input source spectrum (Fig. 6a, inset), thereby confirming the resonance-free spectral response from the RFC. A residual resonant response is observed atop the resonance-free spectrum, which is attributed to the imperfections of the HPM: its finite conversion efficiency and deviation from the ideal 2π phase change per pass. In addition, diffractive spreading with propagation, along with any cavity misalignment result in mode-mixing and cross-talk, and are thus expected to yield a remnant resonant response superposed with the resonance-free response57–59. The spectral measurements are compared to predictions based on a theoretical model that is described in Supplementary Section 6.

The spectra plotted in Fig. 6a-c correspond to the total spatial field distributions emerging from the FP and the RFC. In the case of the RFC, this comprises the large number of LG modes seen at the mode-sorter output in Fig. 5d. In addition, we plot in Fig. 6d the spectra for the first three odd LG modes measured separately after being spatially isolated at the mode-sorter output (for L=1.3 mm). The measurements are compared to simulations based on the above-described theoretical model. Each mode undergirding the output field shows no spectral resonant structure. Because the HPM imperfections are expected to exacerbate the higher-order LG modes, we note that the resonant spectral components atop the resonance-free response is stronger for LG5 compared to LG1 (Fig. 6d).

Methods
Holographic Phase Mask (HPM)
HPMs represent an advanced wavefront modification technology that offers significant advantages over significant advantages conventional phase masks42,69–73. Crucially, because the phase in conventional elements is encoded typically in the optical path length (e.g., through geometric surface variations), they are thus limited to specific wavelengths, and are susceptible to surface contamination42,72,73. In contrast, HPMs encode phase information volumetrically within transmissive Bragg gratings (TBGs)42,69–73, which enables HPMs to maintain intricate wavefront modifications across a broad bandwidth by varying the angle of incidence to meet the Bragg condition at each wavelength, thereby providing greater versatility and broadening their range of applications42,72,73.

HPM material
We record the HPM in a photosensitive medium. Here we make use of photo-thermo-refractive (PTR) glass74,75, which is an ideal medium due to its low absorption in the visible and near-infrared (≈10−4 cm−1) and high laser-damage threshold in both pulsed and continuous-wave modalities73,74, and multiple HPMs can be multiplexed within the same volume of PTR glass with minimal crosstalk42. Current PTR-glass technology allows up to 1000 ppm of refractive-index change, which cannot be bleached by laser radiation, while it possesses minimal absorption and scattering from 300 nm in the near ultraviolet to 2000 nm in the near infrared70. These specific properties of PTR glass enable the fabrication of HPMs with near-unity diffraction efficiency, and an extended degree of tolerance to high average- or peak-power laser beams, mechanical shocks, and elevated temperatures.

HPM recording
The HPM is produced by embedding the desired phase information onto a TBG, and holographically recorded into a thick PTR glass69 using a two-beam interference setup, where a phase element is introduced into one of the beams42. The resulting interference pattern in the material contains both the grating structure and the desired phase information. As light travels through the HPM, it undergoes diffraction and simultaneously acquires the encoded phase profile, which remains consistent across a wider range of wavelengths when the Bragg condition is met42,70–73. Only the diffracted beam incurs the embedded phase distribution, while uncoupled light in the transmitted beam observes minimal to no phase modulation. By designing the TBG with a broad acceptance bandwidth, a complete phase reconstruction can be achieved for all operating frequencies within the allowed spectrum42,71. The phase profile of the first LG mode76 is recorded in the volume of a PTR glass using a holographic interference setup. LG modes possess OAM, which manifests as a spiral-shaped phase front, and exhibit an intensity profile resembling a donut or ring76,77. Moreover, they satisfy the paraxial wave equation in cylindrical coordinates and constitute a complete orthogonal basis set76. The holographic recording setup for a typical HPM follows a standard Mach-Zehnder interferometer setup. A volume of PTR glass is exposed to the standing-wave pattern produced by the interference of the signal and reference UV beams in the interferometer. Once the TBG’s Bragg period is determined, the half-angle separation θrec between the two interfering beams is adjusted to produce a periodic pattern with a matching modulation period. Upon further thermal development, a permanent change in the local refractive-index is induced, forming the volume Bragg grating72. To construct an HPM, a master phase-plate encoding the desired phase distribution is inserted into one of the interferometric arms72. The phase element’s spatial profile, incurred by the signal beam, is transferred to the photosensitive recording medium, superimposing it on to the plane phase-front of a periodic pattern originated from the interference between the signal and reference beams.

Preparing LG modes to characterize the HPM
We prepared the LG modes (from LG0 to LG6, shown in Fig. 4a) used in testing the performance of the HPM and assessing its diffraction efficiency by passing the fundamental Gaussian mode LG0 through an appropriate surface fork diffractive grating (FDG). We produced 6 transmission FDGs with 150 μm period and the proper singularity using electron-beam lithography. The Gaussian laser beam at a central wavelength of 1095.6 nm impinges upon the surface grating, and the first diffraction order beam is selected using a circular aperture.

Data availability
Data supporting the findings of this study are available within the article and its supplementary information files. All the corresponding spectral data obtained in this study, including conventional cavity and RFC are publicly available on Zenodo at https://doi.org/10.5281/zenodo.17139354. Any correspondence and material requests should be addressed to the corresponding author.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This paper presents an innovative approach to realizing a resonance-free Fabry-Pérot (FP) cavity by incorporating a single intra-cavity holographic phase mask (HPM). The concept of achieving a broadband and flat spectral response through unrestricted orbital-angular-momentum (OAM) ladder-up is novel and holds significant potential for applications in lasers, sensing, and non-Hermitian optics. The paper is well-organized, the experimental design is rigorous, and the results are convincing. However, I still have several concerns need to be fixed before recommedation for publication.\n1. While a theoretical model is presented, further exploration of the underlying physical mechanisms, particularly related to non-Hermitian and topological optical effects, could strengthen the paper.\n2. Discussing potential applications of the resonance-free FP cavity, such as in lasers, sensors, or non-Hermitian optical devices, would highlight the practical significance of the research.\n3. Providing more details on key experimental parameters, such as the fabrication precision of the thin HPM and the reflectivity of the cavity mirrors, would aid readers in replicating the experiments.\n4. Some figures (e.g., Figures 2 and 3) could benefit from clearer labeling and explanations to facilitate rapid understanding.", "reviewer_2": "The authors report on a simple but interesting implementation of structured light cavity. It fits timely in a rising trend decoupling spectral and spatial properties of structured light resonators. The work is overall well presented, though some points require clarification, as outlined below.\nMajor comments:\n- In my view this is not the first realized system hosting LG modal ladder-up. Also in [35] the cavity comprised an OAM converter and likely should have led to a modal ladder-up. This aspect should be analyzed based on the comment on [35] published as \"Reconsidering metasurface lasers\" by Dandan Wen, Jasper J. Cadusch, Zhiqiang Fang & Kenneth B. Crozier in Nature Photonics volume 15, p.337–338 (2021). The authors should comment on the difference between the present work and [35] and why the ladder-up was not appearing in the previous work, but instead it was claimed that the roundtrip condition could be experimentally observed. In essence, there's a fundamental disagreement between the conclusion of [35] and the present work. In my view the analysis of the present work is correct but should be well discussed within the literature.\n- I'd like to have further analysis on why the HG modes cannot lead to modal ladder-up, also in view of the fact that HG can be obtained from LG modes. What if the cavity instead of being FP is a ring cavity, what would be the effect on HG modes?\n- Alternatives to the HPM written in volume Bragg gratings could be discussed in more details. Metasurfaces are just mentioned en passant in the Discussion. And what about SPPs? A proper discussion of pros and cons of the different technologies could increase the impact of this work.\n- The HPM is phase only. Therefore at every pass there'll be a superposition of p-modes (e.g. Revealing the radial modes in vortex beams, Bereneice Sephton, Angela Dudley, and Andrew Forbes). This aspect is ignored by the authors as it seems they assume the LG_0,m modes are only populated in the resonator (e.g. line 133 and others). Without a proper discussion (and ideally simulations) of the role of p-modes the system analysis is not accurate.\n- line 200: the precise 2pi azimuthal phase could only be achieved strictly at 1 wavelength. The authors should discuss what happens in case a broadband pulse is used in the resonator.\n- line 346: The possibility of adding a gain medium is mentioned too briefly. It's not clear how stimulated emission would work in such a system with modal ladder-up. The authors should clarify this, otherwise it sounds like speculation.\nMinor comments:\n- The term \"light recycling\" is quite unusual in my experience (I see it also in [39]), if really considered appropriate then it should be better introduced and defined\n- line 71: typo \"ceonceptual\"\n- line 74: typo \"orbaital\"\n- line 142: the diffraction angle of the HPM could be specified already here for clarity", "reviewer_3": "The manuscript prepared by the authors proposes a single intra-cavity holographic phase mask (HPM) embedded within a compact Fabry–Pérot cavity, achieving a broadband, resonance-free spectral response. Furthermore, the authors have conducted thorough experimental validation and demonstrated that this resonance-free behavior is maintained even under a cavity length variation of approximately 350%. Both the manuscript and the Supplementary Material are well-structured and clearly written.\nHowever, a pioneering work on this topic has already been reported in Nat. Commun. 14, 495 (2023). The present manuscript appears to be more of a technical development based on that prior work, rather than a fundamentally novel conceptual advancement. The principle that the spatial structure of the cavity field can profoundly affect its spectral characteristics is only implicitly mentioned, and deserves a more rigorous theoretical discussion.\nIn addition to this primary concern, the following issues also raise questions that warrant further clarification by the authors:\nThe manuscript lacks a rigorous explanation for how the superposition of spatial modes leads to the elimination of frequency-domain resonances. The authors should elaborate on how the spatial structure of the intra-cavity field governs its spectral behavior.\nThe reported diffraction efficiency of the HPM for converting the first seven Laguerre–Gaussian (LG) modes to their next higher-order counterparts exceeds ~87.5%, as shown in Fig. 4b. However, this implies a ~12.5% energy loss per step in OAM index, which would lead to an exponential decay in power for higher-order modes. This calls into question the claim of unrestricted orbital angular momentum (OAM) ladder-up capability.\nIn Cartesian coordinates, Hermite–Gaussian-like modes (HG-like) cannot achieve all order ladder-up through a single optical element. In contrast, within the OAM basis, a single phase element can theoretically enable such all-order mode conversion. The manuscript lacks a clear definition of the OAM-carrying modes employed. It should be clarified whether the authors refer to modes carrying a single well-defined OAM value (as opposed to superposition states like HG modes), and if so, whether Bessel–Gaussian modes would also be a viable alternative.\nThe authors propose an all-order mode conversion mechanism based on LG modes but only discuss the azimuthal index ℓ (i.e., the OAM quantum number), without addressing the influence of the radial index p. The role of p in achieving mode ladder-up, as well as its impact on resonance suppression and linewidth broadening, should be clarified theoretically.\nIn Fig. 5, the authors compare the output profiles with and without tilting mirror M2. It appears that the unrestricted ladder-up feature is preserved only when M2 is not tilted. This suggests that mirror alignment may be critical to the effect. The authors should discuss this dependence in detail.\nIn summary, the authors present a technological advance in resonator linewidth engineering. However, it should be noted that this work appears to be a further development of the earlier concept introduced in previous work (Nat. Commun. 14, 495 2023), rather than a fundamentally novel contribution. For this reason, following NC criteria, the manuscript is not recommended for publication in NC."}, "author_response": "Response to reviewers:\nReviewer 1:\nThis paper presents an innovative approach to realizing a resonance-free Fabry-Pérot (FP)\ncavity by incorporating a single intra-cavity holographic phase mask (HPM). The concept of\nachieving a broadband and flat spectral response through unrestricted orbital-angular-\nmomentum (OAM) ladder-up is novel and holds significant potential for applications in\nlasers, sensing, and non-Hermitian optics. The paper is well-organized, the experimental\ndesign is rigorous, and the results are convincing.\nAuthors’ response:\nWe thank the Reviewer for this high praise indeed.\nHowever, I still have several concerns need to be fixed before recommendation for\npublication.\n1. While a theoretical model is presented, further exploration of the underlying physical\nmechanisms, particularly related to non-Hermitian and topological optical effects, could\nstrengthen the paper.\nAuthors’ response:\nWe appreciate the reviewer’s comment. While our primary focus was on demonstrating the\nresonance-free nature of the nFP cavity through an unrestricted orbital-angular-momentum (OAM)\nladder-up process, we agree that highlighting non-Hermitian effects can broaden the context of our\nfindings.\nTo clarify, our HPM inserted in the FP cavity functions as a mode converter, incrementally\nincreasing (decreasing) OAM order by one in each single pass from left to right (right to left).\nThese dynamics can be represented by the following transfer matrices in the modal basis ℓ\nassociated with LG modes:\n0ℓ\n∣ ⟩\n𝑇𝑇L→R = 𝜅𝜅𝑛𝑛 |2𝑛𝑛+ 1⟩⟨2𝑛𝑛−1|,\n𝑛𝑛\n𝑇𝑇R→L = 𝜅𝜅′𝑛𝑛 |−2𝑛𝑛− 2⟩⟨−2𝑛𝑛|,\nwhere and are modal coupling coefficients. These operators are represented by upper-\n𝑛𝑛\ndiagonal matrices that correspond to Jordan normal forms. In the context of tight-binding models,\nthis cla𝜅𝜅ss 𝑛𝑛 of ma𝜅𝜅tr′𝑛𝑛 ices are known to describe non-Hermitian models that can host non-Hermitian\ndegeneracies known as exceptional points (EPs). Similar operators are also known to arise in non-\nHermitian topological lattices such as the Hatano-Nelson model. Our current study shows that the\nLG modes of various OAM numbers in our platform can indeed exhibit dynamics that are governed\nby these non-Hermitian operators. In our future studies, we intend to explore such non-Hermitian\noperators more closely, and to experimentally demonstrate their characteristic signatures such as\neigenmode coalescence associated with EPs as well as non-Hermitian topological effects such as\nthe non-Hermitian skin effect (NHSE).\nlast paragraph of the Discussion section:\n“The evolution of the LG modes in our nFP structure upon propagating in the cavity and passing\nthrough the HPM element can be described by non-Hermitian operators defined by upper-\ndiagonal matrices of Jordan normal form.”\nWe have also added the above mathematical description in a new section in the Supplementary\ndocument (Supplementary Section 7).\n2. Discussing potential applications of the resonance-free FP cavity, such as in lasers, sensors,\nor non-Hermitian optical devices, would highlight the practical significance of the research.\nAuthors’ response:\nWe thank the Reviewer for bringing up the potential for practical applications. However, since our\nproof-of-principle demonstration is still in an early stage of development, we chose not to state\nany specific concrete applications beyond what we listed in the Introduction and Conclusion\nsections. These will emerge with further research on the topic.\n3. Providing more details on key experimental parameters, such as the fabrication precision\nof the thin HPM and the reflectivity of the cavity mirrors, would aid readers in replicating\nthe experiments.\nAuthors’ response:\nWe have revised the Supplementary Information as much as possible to include experimental\ndetails. The mirror reflectivities are already stated in the main text and the Supplementary\ndocument (95% for M and 94% for M at the operating wavelength of 1095.6 nm).\n1 2\nThe precision of the recorded HPM is primarily governed by the interference pattern formed by\nthe UV laser beams in the grating-writing procedure, which we estimate to correspond to a total\nphase error below 0.05π over the 3 mm active area. Additional details on the HPM fabrication\nprocess and cavity alignment protocol are provided in the Supplementary Information (Sections\nS and S ) and in Reference [42].\n1 3\n4. Some figures (e.g., Figures 2 and 3) could benefit from clearer labeling and explanations\nto facilitate rapid understanding.\nAuthors’ response:\nWe have updated the labeling in Figure 2 for clarity according to the Reviewer’s suggestion.\nThe authors report on a simple but interesting implementation of structured light cavity. It\nfits timely in a rising trend decoupling spectral and spatial properties of structured light\nresonators. The work is overall well presented, though some points require clarification, as\noutlined below.\nAuthors’ response:\nWe thank the Reviewer for appreciating our work reported here.\n-In my view this is not the first realized system hosting LG modal ladder-up.\nAuthors’ response:\nThere have been only a few reports of observing LG modal ladder-up in a cavity; for example,\nRef. [66], which was cited in the original manuscript. However, because such cavities are\nstructured typically by including a spatial light modulator, the cavity length was too long to observe\nthe spectral structure, and hence no spectral-free response has been reported to date. Indeed, the\ncentral finding of our paper is not simply the observation of modal ladder-up, but that this\nphenomenon is intrinsically responsible for broad spectral operation as a consequence of\nresonance-free behavior.\nAlso in [35] the cavity comprised an OAM converter and likely should have led to a modal\nladder-up. This aspect should be analyzed based on the comment on [35] published as\n\"Reconsidering metasurface lasers\" by Dandan Wen, Jasper J. Cadusch, Zhiqiang Fang &\nKenneth B. Crozier in Nature Photonics volume 15, p.337–338 (2021).\nAuthors’ response:\nWe agree in principle with the Reviewer. Indeed, Ref. [35] did not observe modal ladder-up as\nshould be expected and pointed out in the comment by Crozier et al. (which we have cited as Ref.\n[59] in the original manuscript, now Ref. [44] in the revised manuscript). Furthermore, no\nresonance-free behavior was observed in [35], although to be fair to the authors of [35], they did\nnot search for that signature behavior. Therefore, despite the relevance of Ref. [35] and the\ncomment by Crozier et al., our observations are quite distinct, where both modal ladder-up and\nresonance-free behavior were observed. We have opted not to engage significantly with the results\nin [35] since they are not under our control and probably many unknowns remain that led to their\nreported results.\nThe authors should comment on the difference between the present work and [35] and why\nthe ladder-up was not appearing in the previous work, but instead it was claimed that the\nroundtrip condition could be experimentally observed. In essence, there's a fundamental\ndisagreement between the conclusion of [35] and the present work. In my view the analysis\nof the present work is correct but should be well discussed within the literature.:\nAuthors’ response:\n– in agreement with the Reviewer that Ref. [35] should have observed modal ladder-up.\nNevertheless, we feel that it is not our place to speculate why the authors of [35] did not observe\nthis phenomenon. Consequently, there is indeed a disagreement between our results and those in\n[35]. However, our cavity is passive whereas that in [35] is active, which makes the comparison\ndifficult.\n- I'd like to have further analysis on why the HG modes cannot lead to modal ladder-up, also\nin view of the fact that HG can be obtained from LG modes. What if the cavity instead of\nbeing FP is a ring cavity, what would be the effect on HG modes?\nAuthors’ response:\nFigure 2a describes the modal transformations that occur for HG modes upon traversing a pi-phase\nstep. Such a transformation leads to toggling between HG and HG rather than a modal ladder-\n0 1\nup. The central point is that there does not exist a purely phase distribution capable of\nimplementing the transformation HG to HG for all m even approximately. Indeed, each of\nm m+1\nthese transformations requires a different number of pi-phase steps. Rather, a complex (that is,\nphase and amplitude) transformation is required for that purpose (inter-converting any pair of HG\nmodes). Generalized transformations of HG modes require volumetric devices (e.g., MPLC’s),\nwhich would occupy a large space and hinder the observation of resonance-free behavior. To date,\nHG transformations in a cavity have either relied on the HG to HG toggling as in Ref. [38], or\n0 1\nhave made use of a series of phase modulations each specialized to a single modal order m as in\nRef. [39]. In contrast, LG OAM modes have the unique characteristic that a single phase\ndistribution can implement a modal-order increase for any OAM mode; for example, we have\nimplemented an HPM that adds 1 to the OAM order regardless of its initial value.\nAlthough HG modes can indeed be written as a superposition of LG modes, this superposition\nchanges with the HG modal order (every HG mode has a distinct decomposition into LG modes).\nHence, the phase distribution that modifies the LG modes will not have the same impact on the\nHG modes.\n- Alternatives to the HPM written in volume Bragg gratings could be discussed in more detail.\nMetasurfaces are just mentioned en passant in the Discussion. And what about SPPs? A\nproper discussion of pros and cons of the different technologies could increase the impact of\nthis work.\nAuthors’ response:\nThe Reviewer is correct in pointing out that a multiplicity of platforms could be utilized in lieu of\nthe HPM to implement the target OAM transformation. We have focused on the HPM and\nmetasurfaces because they are the most suitable for low-loss operation in free-standing cavities. A\nfull discussion of the pros and cons of the various nanophotonic platforms is perhaps out of the\nscope of our paper. We have nevertheless added mention of other realizations in the Discussion\nsection (highlighted in blue).\n4\n- The HPM is phase only. Therefore, at every pass there'll be a superposition of p-modes (e.g.\nRevealing the radial modes in vortex beams, Bereneice Sephton, Angela Dudley, and Andrew\nForbes). This aspect is ignored by the authors as it seems they assume the LG modes are\n0,m\nonly populated in the resonator (e.g. line 133 and others). Without a proper discussion (and\nideally simulations) of the role of p-modes the system analysis is not accurate.\nAuthors’ response:\nThe Reviewer is correct that the HPM does not purely transform the azimuthal index, but that some\nimpact on the radial index is also expected. However, the change in the radial modal content is\nsmall, which we have confirmed experimentally in comparison with commercially available OAM\nphase plates (where the results are in agreement with the Forbes paper cited by the Reviewer). This\ncan be seen first in Fig. 4b by analyzing the beam after the HPM using a cylindrical lens. Any\nchange in the radial index will directly impact the field structure after the lens [Opt. Lett. 42, 4387\n(2017)], which we do not observe. Instead, the observed field structure in our measurements is\nconsistent with almost complete conversion with zero-order radial index. This result can be\nunderstood by the fact that the phase imprinted in the HPM is not uniform everywhere. Rather the\nphase is confined to the volume resulting from the interference of the LG and LG modes\n00 01\ninvolved in the holographic writing. This improves the conversion into zero-order radial modes.\nSuch a configuration is not covered by the analysis and experiment in the abovementioned Forbes\npaper.\nCrucially, any conversion into non-zero radial index does not detract from the resonance-free\nresponse. As shown in the theoretical analysis added to the revised manuscript, the only\nrequirement is that all the mode sin the cavity belon to an orthogonal basis, which is the case for\nLG modes whether the radial or azimuthal indices are varied.\n- line 200: the precise 2pi azimuthal phase could only be achieved strictly at 1 wavelength.\nThe authors should discuss what happens in case a broadband pulse is used in the resonator.\nAuthors’ response:\nThe Reviewer brings up an excellent point. Indeed, for a conventional phase plate, the 2pi phase\nwould strictly apply to a single wavelength, with the phase changing over a broad bandwidth.\nBecause our cavity is now resonance-free and thus can operate in principle over a large bandwidth,\nthis issue is critical. In our experiments, the operating bandwidth was ~ 5 nm at a central\nwavelength of ~ 1 micron, so the expected change is negligible; indeed, the measured phase and\nthe estimated error was averaged over the 5 nm bandwidth of the laser used (now mentioned in the\nrevised manuscript). Nevertheless, the HPM technology in principle would alleviate the problem\nif a larger bandwidth were used. The reason is that the phase is not simply introduced through a\nsurface modification and is rather incorporated into the volume. Consequently, as long as the\nincident wavelength satisfies the Bragg condition, the 2pi phase shift is imparted. Our spectral\nacceptance for the HPM was 32 nm; any wavelength within this bandwidth would experience the\nsame phase shift – in contrast to a surface phase plate. In principle, the HPM writing procedure\ncan be modified to further increase the acceptance bandwidth, all of which would experience the\nsame phase distribution.\n5\n- line 346: The possibility of adding a gain medium is mentioned too briefly. It's not clear\nhow stimulated emission would work in such a system with modal ladder-up. The authors\nshould clarify this, otherwise it sounds like speculation.\nAuthors’ response:\nThe Reviewer is correct: lasing after the addition of a gain medium is at this point a speculation.\nAs mentioned by the Reviewer earlier, the results in Ref. [35] are indeed in disagreement with\nours, and perhaps they should have observed modal ladder-up. The question of lasing in the\npresence of modal ladder-up will be best tackled experimentally. We are currently carrying out\nthese experiments and have thus limited what we have said in the current manuscript on this topic.\nMinor comments:\n- The term \"light recycling\" is quite unusual in my experience (I see it also in [39]), if really\nconsidered appropriate then it should be better introduced and defined\nAuthors’ response:\nThe term ‘light recycling’ is quite common in our experience and we have thus preferred not to\nbelabor it with a definition.\n- line 71: typo \"ceonceptual\"\n- line 74: typo \"orbaital\"\nAuthors’ response:\nWe thank the Reviewer for catching these typos, which have been corrected in the revised\nmanuscript.\n- line 142: the diffraction angle of the HPM could be specified already here for clarity\nAuthors’ response:\nWe have mentioned the Bragg angle at the location suggested by the Reviewer.\nThe manuscript prepared by the authors proposes a single intra-cavity holographic phase\nmask (HPM) embedded within a compact Fabry–Pérot cavity, achieving a broadband,\nresonance-free spectral response. Furthermore, the authors have conducted thorough\nexperimental validation and demonstrated that this resonance-free behavior is maintained\neven under a cavity length variation of approximately 350%. Both the manuscript and the\nSupplementary Material are well-structured and clearly written.\nAuthors’ response:\nWe thank the Reviewer for this succinct summary of our results and for praising the manuscript.\nHowever, a pioneering work on this topic has already been reported in Nat. Commun. 14,\n495 (2023). The present manuscript appears to be more of a technical development based on\nthat prior work, rather than a fundamentally novel conceptual advancement.\nAuthors’ response:\nWe disagree with the Reviewer. While we think highly of the paper cited here, it is not the first\npaper on the topic. Indeed, the concept of modifying the spatial mode structure of a cavity has a\nlong history predating this reference by at least three decades, some of which we cited in our\nmanuscript. Indeed, this reference is not even the first paper on the topic from the Capasso group\nitself, which has published an earlier version (Ref. [35]) that led to some controversy. Moreover,\nthe paper in question is intimately related to Ref. [39] from the Lipson group, which predates it.\nThe principle that the spatial structure of the cavity field can profoundly affect its spectral\ncharacteristics is only implicitly mentioned and deserves a more rigorous theoretical\ndiscussion.\nAuthors’ response:\nWe are confused by this statement from the Reviewer (unless this statement is directed at the\nreference the Reviewer cited). Our entire manuscript is directed towards investigating the impact\nof the spatial structuring on the spectral response, culminating in the data presented in Figure 6\nthat compares the spectral response of our structured cavity to a conventional counterpart.\nHowever, to address the Reviewer’s concern, we have now added a theoretical description that\ncontrasts the spectral response of a conventional FP cavity and one modified by inserting an HPM.\nThis analysis shows clearly introducing modal ladder-up converts the conventional discrete\nspectral response to a flat resonance-free spectrum.\nIn addition to this primary concern, the following issues also raise questions that warrant\nfurther clarification by the authors:\nThe manuscript lacks a rigorous explanation for how the superposition of spatial modes leads\nto the elimination of frequency-domain resonances. The authors should elaborate on how the\nspatial structure of the intra-cavity field governs its spectral behavior.\nAuthors’ response:\nbehavior quite obvious – and perhaps the first two Reviewers also share this opinion – it is worth\nelaborating more rigorously in this in the main text. We have now revised the manuscript to include\na Section that describes analytically why this phenomenon arises as mentioned above. In the new\nSection entitled ‘Emergence of the resonance-free response’ we provide two equations that\ndescribe the spatio-spectral transmission through an FP cavity and an nFP cavity. Whereas the\nspatio-spectral transmission is separable in the first case, it is non-separable in the second.\nThe orthogonality of the spatial modes (which we confirmed in Fig. 2) wipes out the resonant spectral\nresponse after tracing over the spatial coordinates.\nWe note that remnant resonant structure can appear at the output in the following cases: (1) if the\nHPM leaks part of the original mode (not 100% diffraction efficiency); and (2) if the mode\nconversion is not perfect 2pi azimuthal phase (the modes are not completely orthogonal).\nThe reported diffraction efficiency of the HPM for converting the first seven Laguerre–\nGaussian (LG) modes to their next higher-order counterparts exceeds ~87.5%, as shown in\nFig. 4b. However, this implies a ~12.5% energy loss per step in OAM index, which would\nlead to an exponential decay in power for higher-order modes. This calls into question the\nclaim of unrestricted orbital angular momentum (OAM) ladder-up capability.\nAuthors’ response:\nThe Reviewer strictly speaking is correct; no system has 100% efficiency. However, this does not\ndetract from our result. In principle the efficiency can be increased significantly by adding anti-\nreflection coating. Even with the 12.5% drop, we observe modal ladder-up from 1 to 19, which\nhas never been observed before. The resonance-free behavior is minimally affected by the finite\ndiffraction efficiency of the HPM, as is clear in the data reported in Fig. 6. Indeed, the theoretical\nsection added in the revised manuscript shows that a finite conversion efficiency does not detract\nfrom the resonance-free behavior. We chose to use the term “unrestricted” because it conveys the\ncentral concept relevant to the physics of the problem.\nIn Cartesian coordinates, Hermite–Gaussian-like modes (HG-like) cannot achieve all order\nladder-up through a single optical element. In contrast, within the OAM basis, a single phase\nelement can theoretically enable such all-order mode conversion. The manuscript lacks a\nclear definition of the OAM-carrying modes employed. It should be clarified whether the\nauthors refer to modes carrying a single well-defined OAM value (as opposed to\nsuperposition states like HG modes), and if so, whether Bessel–Gaussian modes would also\nbe a viable alternative.\nAuthors’ response:\nThe OAM modes used here are the LG modes as mentioned explicitly in the main text. To avoid\nany ambiguity, we have now written out the equation for the modes in the Supplementary.\nThe authors propose an all-order mode conversion mechanism based on LG modes but only\ndiscuss the azimuthal index ℓ (i.e., the OAM quantum number), without addressing the\n8\ninfluence of the radial index p. The role of p in achieving mode ladder-up, as well as its impact\non resonance suppression and linewidth broadening, should be clarified theoretically.\nAuthors’ response:\nAs mentioned explicitly in the main text, we hold the radial index of the modes at 0 and only\nmodulate the azimuthal OAM index. In practice, we expect that some portion of the energy will\nbe converted into non-zero-radial index modes. However, the theoretical analysis we have added\nto the revised manuscript reveals that maintaining a zero radial index is not a requirement fro the\nresonance-free response. Rather, the sole requirement is that all the modes generated belong to a\northogonal basis. Therefore, the radial field component does not impact the results of resonance-\nfree behavior. It is an open question whether exploiting the radial index can be of benefit in this\ncontext.\nIn Fig. 5, the authors compare the output profiles with and without tilting mirror M2. It\nappears that the unrestricted ladder-up feature is preserved only when M2 is not tilted. This\nsuggests that mirror alignment may be critical to the effect. The authors should discuss this\ndependence in detail.\nAuthors’ response:\nUnfortunately, the Reviewer is confused about the content of Figure 5. Modal ladder-up occurs in\nboth cases of mirror M2 aligned (Fig. 5d) and misaligned (Fig. 5e). We have used these two\ndifferent configurations to highlight the modal ladder-up in two different ways. When the cavity\nis perfectly aligned, all the modes are co-aligned, so that observing the spatial structure directly\ndoes not reveal the modal content. In this case, we direct the output field to the modal sorter, which\nreveals the amplitudes of the various underlying modes. In the case of tilted mirror M2, the modal\nladder-up can be immediately observed in the output field structure because the underlying modes\nare no longer co-propagating and are instead spatially displaced. The confusion seems to arise\nfrom the fact that the modal sorter output in this case shows only a single peak. But that is a\nconsequence of us selecting only a single output mode to be directed to the modal sorter. This is\nthe portion of the output field enclosed in the white square in the second column of Fig. 5e, which\nthe Reviewer perhaps missed.\nIn summary, the authors present a technological advance in resonator linewidth engineering.\nAuthors’ response:\nWe disagree with the Reviewer here on both counts.\nFirst, the reported experiment is not merely a technical improvement over previous studies. As we\nlisted explicitly in the Introduction, almost all previous studies on spatially structured cavities have\nfailed to observe the associated spectral content. The merit of the paper repeatedly cited by the\nReviewer [Nat. Commun. 14, 495 (2023)] is that the cavity was short enough to resolve the\nlongitudinal modes by adopting an on-chip platform. The reduced cavity length is only one out of\nat least five necessary conditions required to observe the ladder-up phenomenon. Because of the\ninherent difficulty of combining all five constraints in a single cavity, the modal ladder-up has\ngone mostly unreported to date.\nexperiment is specifically designed to not impact the intrinsic resonance linewidth. The data in\nFig. 6 shows this clearly. In Fig. 6 left column, the spectral structure of the longitudinal cavity\nmodes is clear. In transitioning to Fig. 6 right column, we only transport the spiral-phase in the\nHPM into the beam path. However, the Bragg grating structure of the HPM remains inside the\ncavity in both cases. Consequently, the cavity length and quality factor are not changed going from\nFig. 6 left column to Fig. 6 right column. The dramatic change in the observed spectrum is not the\nresult of added losses or any change in the intrinsic resonant linewidth and is solely a consequence\nof the spatial structuring of the field; namely, the unrestricted modal ladder-up. We regret that the\nReviewer has missed the central point of the paper.\nHowever, it should be noted that this work appears to be a further development of the earlier\nconcept introduced in previous work (Nat. Commun. 14, 495 2023), rather than a\nfundamentally novel contribution.\nAuthors’ response:\nAs described before, we would like to point out that our unrestricted ladder-up and resonant free\nFabry-Perot cavity are fundamentally new phenomena that have not been demonstrated until now.\nThe paper cited by the Reviewer reports a change in the cavity FSR, but the discrete spectral\nresponse of a conventional FP cavity remains. We hope that our responses here have helped clarify\nthis issue for the Reviewer and thus have changed his/her mind.\n10"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors already answered all my questions and I think this manuscript can be accepted.", "reviewer_2": "I am satisfied with the replies the authors gave to the points I have raised but I believe the authors could expand on the modifications in the revised manuscript related to these points. In essence, apart from the section \"Theory of RFC\", the authors have changed just 3 sentences in the main text and two sections in the supplemental. I encourage the authors to expand further on the manuscript clarifications given the discussions with the reviewers.\nFor instance:\n- line 346: The possibility of adding a gain medium is mentioned too briefly. It's not clear\nhow stimulated emission would work in such a system with modal ladder-up. The authors\nshould clarify this, otherwise it sounds like speculation.\n--> the authors agree with this comment but I don't see a change\n- Alternatives to the HPM written in volume Bragg gratings could be discussed in more detail.\nMetasurfaces are just mentioned en passant in the Discussion. And what about SPPs? A\nproper discussion of pros and cons of the different technologies could increase the impact of\nthis work.\n--> I am not satisfied with the brief generic sentence added. I'm still wondering what would happen with metasurfaces and\nSPPs and I believe it's important to address this\n- The term \"light recycling\" is quite unusual in my experience (I see it also in [39]), if really\nconsidered appropriate then it should be better introduced and defined\n--> I'm still of the opinion that this term is quite unusual and for a broad audience like that of Nat Comm a brief definition\nshould be given\n- The HPM is phase only. Therefore, at every pass there'll be a superposition of p-modes (e.g.\nRevealing the radial modes in vortex beams, Bereneice Sephton, Angela Dudley, and Andrew\nForbes). This aspect is ignored by the authors as it seems they assume the LG0,m modes are\nonly populated in the resonator (e.g. line 133 and others). Without a proper discussion (and\nideally simulations) of the role of p-modes the system analysis is not accurate.\n--> I could not find the discussion of the role of p-modes in the revision", "reviewer_3": "The authors have addressed all my concerns and I would like to recommend the publication of this manuscript."}, "author_response": "Response to reviewers:\nFirst reviewer:\nThe authors already answered all my questions, and I think this manuscript can be accepted.\nAuthors’ response:\nWe sincerely thank the reviewer for their positive feedback and recommendation for acceptance.\nWe appreciate the time and effort invested in reviewing our manuscript.\n---------------------------------------------------------------------------------------------------------------------\nSecond reviewer:\nI am satisfied with the replies the authors gave to the points I have raised but I believe the\nauthors could expand on the modifications in the revised manuscript related to these points.\nIn essence, apart from the section \"Theory of RFC\", the authors have changed just 3\nsentences in the main text and two sections in the supplemental. I encourage the authors to\nexpand further on the manuscript clarifications given the discussions with the reviewers.\nAuthors’ response:\nWe sincerely thank the reviewer for their constructive comments. We greatly appreciate the time\nand effort they dedicated to reviewing our manuscript, as well as their positive assessment of our\nwork.\nIn response, we have now added additional explanatory content in the main text to better reflect\nthe clarifications provided during the review discussions. We are grateful for the reviewer’s\nthoughtful engagement, which has helped us significantly strengthen the final version of the paper.\nFor instance:\n- line 346: The possibility of adding a gain medium is mentioned too briefly. It's not clear\nhow stimulated emission would work in such a system with modal ladder-up. The authors\nshould clarify this, otherwise it sounds like speculation.\n--> the authors agree with this comment, but I don't see a change.\nAuthors’ response:\nWe agree with the reviewer, we are speculating and now we have specified our speculation in the\nmain text and added the sentence “It is expected that modal ladder-up will frustrate lasing in such\na cavity. However, we speculate that mode-dependent cavity losses could clamp the ladder-up\neffect, thereby leading to lasing in one or a small subset of modes.” to the Discussion section.\n1\n- Alternatives to the HPM written in volume Bragg gratings could be discussed in more detail.\nMetasurfaces are just mentioned an passant in the Discussion. And what about SPPs? A\nproper discussion of pros and cons of the different technologies could increase the impact of\nthis work.\n--> I am not satisfied with the brief generic sentence added. I'm still wondering what would\nhappen with metasurfaces and SPPs and I believe it's important to address this.\nAuthors’ response:\nWe appreciate the reviewer’s interest in a comparative discussion of alternative implementations.\nAs noted in the main text, any compact optical element that introduces controllable desired phase\ntransformation is sufficient for the resonance-free cavity and hence, we found it unnecessary to\nadd significant details about alternative platforms. Nevertheless, we have added the following\nsentence to the main text to acknowledge the broader range of possibilities: “In general, any of the\nvariety of currently available nanphotonics platforms can be exploited in this context, including\nsurface plasmon polariton structures [62]. All that is needed is for the optical element introduced\ninto the cavity to implement the requisite phase transformation while satisfying the desiderata\ndiscussed above.”\nAlso - The term \"light recycling\" is quite unusual in my experience (I see it also in [39]), if\nreally considered appropriate then it should be better introduced and defined.\n--> I'm still of the opinion that this term is quite unusual and for a broad audience like that\nof Nat Comm a brief definition should be given.\nAuthors’ response:\nWe appreciate the reviewer’s observation, and we agree with the reviewer. To address this, we have\nnow included a brief definition in the main text, clarifying it as “optical feedback via mirrors.”\n- The HPM is phase only. Therefore, at every pass there'll be a superposition of p-modes (e.g.\nRevealing the radial modes in vortex beams, Bereneice Sephton, Angela Dudley, and Andrew\nForbes). This aspect is ignored by the authors as it seems they assume the LG0,m modes are\nonly populated in the resonator (e.g. line 133 and others). Without a proper discussion (and\nideally simulations) of the role of p-modes the system analysis is not accurate.\n--> I could not find the discussion of the role of p-modes in the revision.\nAuthors’ response:\nWe thank the reviewer for pointing this out and apologize for the oversight in the previous version.\nWe now address the role of radial modes in the revised manuscript, discussing their influence on\nthe operation principle of the resonance-free cavity. We added “Note that introduction of radial\n2\nmodes ψ (r, φ;ω) indexed by p into the field via the HMP does not impact the operation of the\npm\nRFC as long as the modes remain orthogonal, i.e., rdrdφψ* (r, φ;ω) ψ (r, φ;ω)= δ δ . In\npm qn mn pq\nother words, even in presence of radial modes, Eq. 3 still holds. We thus confine our analysis to\nthe azimuthal modes without loss of generality.” in ∬the main text.\n---------------------------------------------------------------------------------------------------------------------\nThird reviewer:\nThe authors have addressed all my concerns, and I would like to recommend the publication\nof this manuscript.\nAuthors’ response:\nWe thank the Reviewer for this high praise indeed and for recommending the publication of our\nmanuscript. We appreciate his support and constructive input throughout the review process.\n3"}, {"round": 3, "reviewer_comments": {"reviewer_2": "I am satisfied with the latest round of revision and the final clarifications added in the manuscript, which is now in my view suitable for publication."}, "author_response": ""}]

Decision letter