## Abstract

Optical cavities in the near-concentric regime have near-degenerate transverse modes; the tight focusing transverse modes in this regime enable strong coupling with atoms. These features provide an interesting platform to explore multi-mode interaction between atoms and light. Here, we use a spatial light modulator (SLM) to shape the phase of an incoming light beam to match several Laguerre-Gaussian (LG) modes of a near-concentric optical cavity. We demonstrate coupling efficiency close to the theoretical prediction for single LG modes and well-defined combinations of them, limited mainly by imperfections in the cavity alignment.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Transverse modes of paraxial beams are a set of unique field patterns perpendicular to the propagation of electromagnetic waves. They have a wide range of applications, such as increasing the information-carrying capacity in free-space [1] and fiber [2,3] communications, creating smaller focal volumes to achieve superresolution imaging [4], utilizing orbital angular momentum (OAM) for quantum key distribution [5], and producing highly-entangled states [6]. In optical cavities, transverse modes have been used to track atomic position via the observed mode pattern [7–9], and to help enhancing the cooling process in atomic ensembles [10–12]. Optical cavities with near-degenerate transverse modes have also been used to engineer inter-mode coupling [13,14], and to study crystallization domains in Bose-Einstein condensates (BEC) [15–18]. Furthermore, transverse modes can be chosen as a degree of freedom for field quantization, along with wavelength and polarization, and can be utilized to explore atom-photon interaction as building blocks of a quantum network.

The near-degeneracy of transverse modes in an optical cavity arises in the region where the Gouy phase shifts of the cavity modes are fractions of $\pi$, notably in the confocal and concentric region [19,20]. Cavity modes in the near-confocal region have relatively large mode volume, which is suitable to explore multi-mode interaction in large atomic ensemble such as BEC [16,20]. On the other hand, cavity modes in the near-concentric region have small mode volumes with a beam waist on the order of the atomic cross section, and thus show potential for strong interaction between light and single atoms [21–24]. The spatial resolution of the transverse modes can also be utilized to trap and couple selectively to small ensemble of single atoms. In centimetre-sized near-concentric cavities, the frequency spacing of the transverse modes ranges between $\sim 0.01$ to $1\,\mathrm {GHz}$ – the lower limit is set by the last stable resonance from the critical point, which is less than half a wavelength away [25]. Unlike planar cavities, the frequency spacing is on the order of the hyperfine or the Zeeman level splitting of the atoms. This allows to explore single-quanta atomic nonlinearities with multiple optical modes coupled to different hyperfine or magnetic energy levels simultaneously, which has been previously demonstrated with two atomic transitions with $\sim 10\,\mathrm {THz}$ spacing using planar cavities [26].

The transverse modes of a cavity can be excited by modifying the wavefront of the incoming Gaussian beam in a TE$_{00}$ mode to match the transverse spatial profile of the modes. In this work, we use a liquid-crystal spatial light modulator (SLM) to perform mode conversion by modulating the spatial phase profile. This enables coupling of a SLM-converted beam to a specific mode or a superposition of transverse modes in a near-concentric cavity. Furthermore, we examine how close to the critical point the transverse modes are still supported. Previously, such phase SLM have been utilized to excite the transverse modes of multimode fibers [27], while excitation of cavity transverse modes in a near-confocal regime has been implemented with a digital micromirror device (DMD) – a type of binary-mask amplitude SLM [20]. Compared with amplitude SLMs, phase SLMs can ideally perform mode conversion and coupling with higher overall efficiency as it does not require parts of the beam to be attenuated or diverted away. While the near-concentric cavities exhibit some technical complexities specific to the highly diverging modes in approaching the critical point, an efficient mode conversion enables interfacing of atomic qubits with multiple near-degenerate photonic modes.

## 2. Theory

#### 2.1 Transverse modes of a cavity

The spatial modes of the near-concentric cavity we investigate here are still well described by the paraxial approximation up to the last stable resonance [25]. We briefly present the theoretical framework to express paraxial transverse modes in an optical cavity with a scalar field that forms a standing wave ( refer to Fig. 1) [19]. In a cylindrically symmetric cavity, the transverse mode profile can be described by a complex amplitude

Inside a cavity, the LG modes are bounded by the two spherical mirror surfaces of radii $R_1$ and $R_2$ spaced $L$ apart. The modes are geometrically stable when stability parameters $g_1 = 1 - L/R_1$ and $g_2 = 1 - L/R_2$ satisfy the confinement condition $0\leq g_1 g_2 \leq 1$ [28]. In symmetric cavities ($g_1=g_2=g$), the marginally stable concentric mode is obtained for a critical mirror separation of $L=2R$ and $g=-1$. Near-concentric modes depart from this point towards the stable region – the distance away from the critical mirror separation is characterized by the critical distance $d=2R-L$, with $g=-1+d/R$.

The resonance frequencies of the cavity depend on the transverse mode numbers $p$ and $l$,

where $q$ is the longitudinal mode number of the cavity, $v_F=c/2L$ is the cavity free spectral range, and $\Delta \zeta =\zeta (z_{M2})-\zeta (z_{M1})$ is the Gouy phase difference between the two cavity mirrors. In near-concentric symmetrical cavities, the frequency spacing between two consecutive transverse modes is given by#### 2.2 Atom-light coupling in near-concentric cavities

The strength of atom-light interaction is characterized by the coupling constant $g_{ac}\propto d_a/ \sqrt {V_m}$, which depends on the atomic dipole moment $d_a$ and the effective mode volume $V_m = \pi w_0^2 L/2$ [29]. Small mode volumes can be achieved either with short cavity length $L$ or small waist radius $w_0$. Due to the small $w_0$ in approaching the critical point, near-concentric cavities exhibit strong atom-light coupling strength $g_{ac}$, comparable to $\mathrm {\mu m}$-length cavities or fiber cavities [23].

In addition, all the radial transverse modes (LG modes with $l=0$) at a particular critical distance $d$ have identical effective mode volumes $V_m$. Imposing the normalization condition $\int dV \, |U_{p,0}(\rho ,\phi ,z)|^2 = 1$ with the prefactor $A_{p,0}=1/\sqrt {V_m}$ from Eq. (1), we obtain $V_{m}=\pi w_0^2 L/2$ for all values of $p$ (radial mode number), by applying the relation $\int _0^{\infty } du\, e^{-u} \mathcal {L}^0_p (u)^2 = 1$ where $u=2\rho ^2/w^2$. This relation also implies that even though higher order radial modes appear to be “larger”, their intensity cross-section areas remain the same. This allows coupling between an atom and cavity modes with equal strength across all radial transverse modes. A more thorough calculation of the atom-cavity coupling constant for radial transverse modes is provided in Supplement 1.

#### 2.3 Mode-matching to a cavity

We briefly describe the method to measure the mode matching efficiency in a cavity with realistic losses, following the cavity characterization technique in Ref. [30]. The power transmission through a cavity with mirrors of the same reflectivity is given by

The mode matching efficiency $\eta$ can be obtained from Eq. (5) and Eq. (6) on the cavity resonance ($\omega =\omega _0$),

where $\alpha = \kappa _m/(2\kappa _l +\kappa _m)$ is determined by the cavity decay rates, and thus is a physical property of the cavity mirrors – for mirrors with no scattering or absorption losses, $\alpha =1$. The parameter $\alpha$ can be estimated from the measurement of the cavity transmission and reflection at resonance: which represents the effectiveness of the cavity transmission. The cavity decay rates can be obtained as $\kappa _m= 2\kappa \alpha /(1+\alpha )$ and $\kappa _l= \kappa (1-\alpha )/(1+\alpha )$ from measured values of $\kappa$ and $\alpha$.#### 2.4 Beam shaping with SLM

To prepare LG beams and couple to the transverse modes of the near-concentric cavity, we use a liquid-crystal phase SLM to perform mode conversion from a collimated single mode fiber ouput (approximating a Gaussian beam). Such a transformation can be performed with a spatial filter which modulates both the amplitude and the phase of the incoming mode, and described by a generalized filter function $T(\mathbf {x}) = M(\mathbf {x}) \exp (i \,\Phi (\mathbf {x}))$. However, a liquid-crystal SLM only modulates the phase of the incoming beam and hence only provides the transformation $T(\mathbf {x}) = \exp (i \,\Phi (\mathbf {x}))$.

There are several methods to perform both amplitude and phase modulation using only a phase SLM. In one method, the SLM can be operated in a phase-grating configuration – this produces both the carrier and first-order diffraction beams, where phase and amplitude can be varied using the modulation angle and the modulation depth, respectively [31,32]. This method typically requires a high-resolution SLM to encode the phase and amplitude information sufficiently precise with the phase grating. However, recent works explored encoding techniques with different sets of amplitude modulation bases which allow the usage of a low-resolution phase SLM [33–36]. Another method relies on using two SLMs with a polarizer to modulate the amplitude and phase of the incoming beam independently [37–39].

Here, we use a much simpler technique that does not require parts of the beam to be diverted away or attenuated, because LG modes with relatively high purity can be created by spatially modulating the incoming Gaussian beam with only the phase component of the desired LG modes [40–42]. The cavity then acts as a filter to attenuate the remaining off-resonant LG mode components, while transmitting the desired LG mode. The SLM phase function for this transformation is given by

The mode overlap is defined as $\int (d\sigma ) U_1^*(\rho ,\phi ) U_2(\rho ,\phi )$, evaluated over the cross section at $z=0$, where $U_1(\rho ,\phi ) = A_0 \exp \left (-\rho ^2/w_0^2\right ) \exp \left [i \Phi (\mathbf {\rho ,\phi }) \right ]$ is the SLM-modulated output of the incoming Gaussian mode, and $U_2(\rho ,\phi )$ is the targeted LG mode, while the normalization coefficients $A_0$ and $A_{p,l}$ are chosen such that the modes are normalized, i.e. $\int (d\sigma ) U_i^*(\rho ,\phi ) U_i(\rho ,\phi )=1$. Thus, the modulus square of the mode overlap is equivalent to the mode matching efficiency $\eta$ as defined in Section 2.3. The ratio between the targeted LG mode waist and the incoming Gaussian mode waist $w/w_0$ can also be varied to maximize the mode overlap. For relatively small mode indices $p$ and $l$, the mode matching efficiencies of the same LG modes are relatively high, with low mode matching efficiencies to different LG modes (see Table 1). Due to the simplicity of the phase function, this technique can also be implemented using physical phase plates [43,44].

## 3. Experiment

#### 3.1 Experimental setup

The design and construction of the near-concentric cavity was described previously [22,25]. The cavity is formed by two lens-mirrors with mirror radius of curvature of 5.5 mm and aperture diameter of 4.07 mm – the anaclastic design allows highly divergent modes of the near-concentric cavity to be transformed into collimated modes with a single element. This simplifies the requirement of the optical components to generate and measure collimated LG beams on the input and output of the cavity (see Fig. 2).

### 3.1.1 Mode conversion with SLM

We use a liquid-crystal SLM (Meadowlark HV 512 DVI) with an active area of 12.8 mm $\times$ 12.8 mm and resolution of 512x512 pixels. As this SLM only modulates light with a particular linear polarization, a sequence of a polarizing beam-splitter (PBS) and a half-wave plate (HWP) prepares the correct polarization to match the SLM polarization axis. We minimize the pixelation artifact by using a significant portion of the SLM area. To achieve this, we prepare a slightly divergent beam with beam diameter ($1/e^2$ width) ranging from 3 to 7 mm, measured at the SLM.

The phase modulation applied on the SLM consists of three components: the LG mode-generating phase pattern as described in Eq. (9), the correction phase pattern provided by the manufacturer, and a quadratic phase pattern which effectively acts as a Fresnel lens with variable focal length. This SLM-generated Fresnel lens helps in supressing the unmodulated light on the SLM output (more commonly done with a blazed grating pattern [42]). In addition, the combination of the Fresnel lens with a telescope of variable length and magnification creates a collimated LG beam with tunable beam size. The appropriate values for the Fresnel lens and telescope parameters are obtained with ray-tracing simulations.

The overall diffraction efficiency of the SLM due to the aforementioned phase modulation is measured to be around $60\%$, similar throughout different LG mode-generating phase patterns, which can potentially improve with better SLM designs. Thus, the overall fiber-to-cavity coupling efficiency is only limited by the SLM diffraction efficiency, losses due to on-path optical elements, and the mode matching efficiency as measured in Section 3.1.3. This is much better than using the DMD-based devices which have much higher loss due to the amplitude mask [45,46].

### 3.1.2 Cavity alignment

In the cavity design [25], one cavity mirror is placed on 3D piezo translation stage (Fig. 2) to allow for both the longitudinal (z direction) and transverse alignment (x and y directions). The longitudinal alignment changes the cavity length to be resonant to a particular optical frequency, while the transverse alignment is performed to establish cylindrical symmetry of the system. Small tip-tilt misalignment can also be corrected by the transverse alignment, if the mirrors are perfectly spherical. However, such a correction misaligns the two anaclastic lens-mirror axes from the cavity axis, resulting in slightly asymmetric collimated output modes.

The transmission and reflection spectrum of the cavity are obtained by measuring the light intensity with a photodetector while varying the cavity length linearly over time. The detuning from the cavity resonance is expressed in corresponding units of light frequency – the conversion factor is determined by measuring the spacing of the frequency sideband generated with an electro-optical modulator.

### 3.1.3 Measurement of the mode-matching efficiency

The mode matching efficiency $\eta$ (Eq. (7)) quantifies how well the input mode couples to the cavity mode. It only depends on the resonant power transmission at resonance $T(\omega _0)$ and the effective transmission coefficient $\alpha$ (Eq. (8)). We characterize the value of $\alpha$ by coupling a Gaussian beam (from a collimated single mode fiber output mode) into the cavity without the SLM. The transmission and reflection spectrum were recorded. From the fitting, we obtain $T(\omega _0)=19.5(1)\%$, $R(\omega _0)=33.6(2)\%$, and $\kappa = 2\pi \times 24.8(8)\,\mathrm {MHz}$, which corresponds to a cavity finesse of $\mathcal {F}=275(9)$. From these parameters, we estimate $\alpha =0.294(2)$, which results in a mode matching efficiency of $\eta =94(1)\%$ for Gaussian beam, and cavity decay rates of $\kappa _m= 2\pi \times 11.3(4)\,\mathrm {MHz}$ and $\kappa _l= 2\pi \times 13.5(4)\,\mathrm {MHz}$.

To estimate the mode matching efficiencies for SLM-generated LG modes, we obtain the cavity transmission spectrum $T(\omega )$ and multiply it with $(1+\alpha )^2/(2\alpha )^2$ (the prefactor in Eq. (7)) to obtain the mode transmission spectrum $\eta (\omega )$. We fit this spectrum with a Lorentzian profile, and estimate the mode matching efficiency $\eta =\eta (\omega _0)$ from the fit amplitude. The parameters from the ray-tracing simulation helps to start the coupling procedure, and we fine-tune these values further to maximize the mode matching efficiency.

#### 3.2 Mode-matching to single LG modes

We generate a single LG mode using the SLM and couple it to the near concentric cavity. The cavity is located at a critical distance of $d = 4.8(2) \mathrm {\mu m}$ with $g = -0.99912(4)$, corresponding to a measured transverse mode spacing of $\Delta v_{tr} = v_F (1-\Delta \zeta / \pi ) = 182(5)\,\mathrm {MHz}$ between adjacent LG modes. The cavity spectra and the camera-captured output modes are depicted in Fig. 3 for LG modes with no angular momentum ($l=0$), and in Fig. 4 for LG modes with angular momentum ($l\neq 0$). The measured mode matching efficiencies are close to the simulated values (see Table 2), although they decrease with higher mode numbers. We attribute this to limited SLM pixel resolution, axial mismatch between the cavity and the anaclastic lens axis due to tip-tilt misalignment, and a mirror surface deviation from a perfect spherical profile. These factors also contribute to some irregularities on the output mode observed by the mode camera.

#### 3.3 Mode-matching to a superposition of LG modes

Superpositions of transverse modes in a cavity provide an interesting avenue to explore multi-photon interaction with atomic medium [26]. We demonstrate the coupling of the SLM-generated beam to an arbitrary superposition of LG modes. We use the method described in Section 3.1.1 by considering the resultant mode as a superposition of individual LG modes,

where $A_{p,l}$ is the amplitude of each constituting LG mode and $\xi _{p,l}$ is the relative phase of the LG mode.Figure 5 (left) shows the mode matching efficiency in coupling the SLM-generated beam to the cavity superposition mode $U_{\{00,10\}}=\left (LG_{00} + e^{i\xi } LG_{10} \right )/\sqrt {2}$ with a varying relative phase angle $\xi$. To obtain a balanced distribution of LG$_{00}$ and LG$_{10}$, we introduce a mode amplitude $A_{10}$ to the SLM spatial phase pattern,

Figure 5 (right) shows the transmission spectra of a superposition of three modes. Modes LG$_{00}$, LG$_{10}$, and LG$_{20}$ are superposed with a relative phase difference of $2\pi /3$ to distribute the phases evenly on the complex plane. The corresponding SLM spatial pattern is given by

#### 3.4 Mode-matching at different critical distances

Small critical distances provide strong field focusing and a small mode volume. In addition, the frequency spacing of the transverse modes decreases with smaller critical distances, leading to the mode degeneracy at the critical point [25]. We study how the mode matching of a single LG mode performs at different critical distances. We use the SLM to couple to LG$_{00}$, LG$_{10}$, and LG$_{20}$ modes of the cavity, and obtain the cavity transmission spectra. We find that the linewidth of the cavity spectra increases for smaller critical distances, while the mode transmission amplitude decreases. This is likely due to diffraction losses as the cavity approaches the critical point.

The critical distance can be estimated from the transverse mode spacing. By changing the cavity length and keeping the laser frequency fixed, we obtain neighbouring cavity spectra spaced $\Delta d=\lambda /2$ apart. Figure 6 shows the cavity transmission amplitudes and the cavity linewidths for various critical distances. Without diffraction loss, the mode transmission amplitude is equivalent to the mode-matching efficiency $\eta$. However, as the diffraction loss increases, the effective transmission coefficient $\alpha$ also changes. Hence, the mode transmission amplitude describes the mode-matching efficiency weighted by a factor associated with the diffraction loss. In the high diffraction loss regime, it becomes hard to couple to a particular lossy eigenmode, and characterize its linewidth to obtain $\alpha$, as different transverse modes start to overlap in frequency. Figure 7 shows the spatial profile of the cavity transmission, captured with the mode camera. Diffraction rings become visible at the critical distance where the linewidth increases.

The near-concentric cavity can support several LG modes reasonably close ($\sim$ a few $\mu$m) to the critical point. However, higher order LG modes start to exhibit diffraction losses at larger critical distances, due to larger LG beam sizes. The performance of the cavity mirrors can be characterized with an effective aperture – for every round trip, the cavity mode is clipped by a circular aperture with diameter $a$ on the mirror, effectively blocking some outer parts of the beam. As a first order approximation, we assume the LG modes to be unperturbed after subsequent round trips. To estimate the onset of the diffraction loss, we choose an aperture size to block $\sim 1\%$ of the mode (the diffraction loss is $2\kappa _{ap} \sim 2\pi \times 20\,\mathrm {MHz}$), which is on the same order as the mirror transmission and scattering losses. From Fig. 6 (right), the effective aperture diameter is estimated to be $a_{\mathrm {exp}} = 1.40(6) \,\mathrm {mm}$ with the onset of the diffraction loss at critical distances of $0.46(8) \,\mathrm {\mu m}$ for LG$_{00}$, $1.8(3) \,\mathrm {\mu m}$ for LG$_{10}$, and $3.8(6) \,\mathrm {\mu m}$ for LG$_{20}$.

The estimated effective aperture $a_{\mathrm {exp}} = 1.40(6) \,\mathrm {mm}$ is comparatively lower than the nominal aperture of the anaclastic lens-mirror design $a_{\mathrm {nom}} = 4.07 \,\mathrm {mm}$. We suspect this to be due to a combination of: (1) local aberrations of the mirror surface due to mechanical stresses induced by the temperature change and the clamping process [47,48], (2) angle-dependent variation on the wavefront due to the multi-layered coating [49], and (3) the validity of the paraxial approximation for strongly diverging modes [50], particularly for higher orders. By slightly modifying the mirror shape or the coating layers, it might be possible to increase the effective aperture of the cavity and obtain stable LG modes even closer to the critical point.

## 4. Conclusion

In summary, we presented a mode-matching procedure to excite several transverse modes of a near-concentric cavity with a relatively high conversion efficiency. We use an SLM to engineer the spatial phase of an input Gaussian beam to selectively match a specific LG mode, and observe experimental mode matching efficiencies close to theoretical predictions for several low-order LG modes, despite the imperfections in the cavity alignment and mirror surface, and the limited resolution of the SLM. We demonstrated that a superposition of cavity modes can be generated with a high fidelity, and showed that a near-concentric cavity can support several LG modes up to critical distances of a few $\mathrm {\mu m}$ before the diffraction loss dominates.

The near-concentric regime of an optical cavity supports transverse modes which are spaced close to one another, on the same order of the magnetic level or hyperfine splitting of the atoms. Exciting the transverse modes in such a regime is a step towards exploring interaction between atoms and strongly focused near-degenerate spatial modes. The nonlinearity arising from multiple photons interacting with single atoms can therefore provide a building block for scalable quantum networks.

## Funding

National Research Foundation Singapore (RCE programme); Ministry of Education - Singapore (RCE programme).

## Disclosures

The authors declare no conflicts of interest.

## Supplemental document

See Supplement 1 for supporting content.

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