Photonic crystal (PhC) structures with subwavelength periods are widely used for diffractive optics, including high reflectivity membranes with nanoscale thickness. Here, we report on a design procedure for 2D PhC silicon nitride membrane mirrors providing optimal crystal geometry using simulation results obtained with rigorous coupled-wave analysis. The Downhill Simplex algorithm, a robust numerical approach to finding local extrema of a function of multiple variables, is used to optimize the period and hole radius of PhCs with both hexagonal and square lattices, as the membrane thickness is varied. Following these design principles, nanofabricated PhC membranes made from silicon nitride have been used as input couplers for an optical cavity, resulting in a maximum cavity finesse of 33 000, corresponding to a reflectivity of 0.999 82. The role played by the spot size of the cavity mode on the PhC was investigated, demonstrating the existence of an optimal spot size that agrees well with predictions. We find that, compared to the square lattice, the hexagonal lattice exhibits a spectrally wider reflective range, less sensitivity to fabrication tolerances, and higher reflectivity for membranes thinner than 200 nm, which may be advantageous in cavity optomechanical experiments. Finally, we find that all of the cavities that we have constructed exhibit well-resolved polarization mode splitting, which we expect is due primarily to a small amount of anisotropic stress in the silicon nitride and PhC asymmetry arising during fabrication.

## I. INTRODUCTION

Diffraction from periodic grating structures with subwavelength periods has long been recognized as a tool that can enable a rich variety of optical functionalities. Fundamentally, this arises from the fact that when the period *a* of the grating structure is smaller than that of the wavelength *λ* of the incident light, only the zero-order diffracted component is allowed, and the astute choice of the physical and geometric parameters of the grating can result in useful constructive and destructive interference phenomena in the forward and backward directions. Such “high-contrast gratings” have been used for tunable lasers,^{1} second harmonic generation,^{2} optomechanics,^{3,4} and as building blocks for optical computation;^{5} a comprehensive summary^{6} describes both the physics and applications of these devices. The one-dimensional structure of such gratings is simple enough that an exact analytic solution to the scattering problem is known,^{6} although it is not simple.

The analogous situation of scattering from a two-dimensional (2D) photonic crystal (PhC) membrane has also been studied extensively. The principal attractions of 2D structures are as follows: first, they have much less polarization sensitivity, and second, in the case of suspended structures where the dielectric contrast is provided by air, they are much more mechanically robust. Indeed, one of the most important applications for such structures is in the field of optomechanics, where the goal is to fabricate a mechanically compliant structure having a very high reflectivity and mechanical quality factor *Q* and simultaneously a very low mass. Early work demonstrated experimentally that the reflectivity of a silicon nitride membrane with a thickness of *t* = 50 nm could be enhanced from *R* ≈ 0.15 to *R* ≈ 0.57 by patterning the membrane with a square lattice of air holes^{7} and that of a 66 nm thick silicon nitride membrane^{8} to *R* ≈ 0.68. More recently, such devices have been incorporated into trampoline^{9} and phononic crystal^{10} structures to reduce the mechanical dissipation, and the reflectivity of silicon nitride membranes has reached values of *R* = 0.999 47 using a square lattice structure^{11} and *R* = 0.999 835 using a hexagonal lattice.^{12} Other dielectrics have been used as well, including silicon^{13} and LiNbO_{3}.^{14}

Unlike the case of the 1D grating, however, analytic solutions to the 2D photonic crystal membrane are not available. Considerable insight into the physical mechanisms for the high reflectivity has been found in studies of the band structure, Fano resonances, and symmetries of both square^{15,16} and hexagonal^{17} lattices. Here, we present a design procedure and supporting experimental results for producing 2D photonic crystal membranes with the highest reflectivity reported to date. The applications for these devices inevitably involve compromises, involving, for example, the material thickness that can be tolerated while allowing an acceptable mechanical quality factor^{18} and the overall size that allows an acceptably low mass. We emphasize the importance of considering the off-normal wavevectors in a plane-wave decomposition of an incident Gaussian beam in ascertaining the maximum possible reflectivity.

Thin-film silicon nitride is used here for operation at wavelengths near 1550 nm. The most common method for depositing silicon nitride thin films for the fabrication of membranes with nanoscale thickness is low-pressure chemical vapor deposition (LPCVD). The mechanical and optical properties of these films can be adjusted by altering the gas flow ratio between dichlorosilane (SiH_{2}Cl_{2}) and ammonia (NH_{3}). Cavity optomechanics research often favors stoichiometric silicon nitride (Si_{3}N_{4}) due to its high tensile stress on silicon when deposited using LPCVD, which yields mechanical resonators with a high *f* × *Q* product resulting from dissipation dilution.^{19} Conversely, Si-rich silicon nitride (SiN) has substantially reduced film stress when deposited by LPCVD, but the refractive index is slightly higher than that of Si_{3}N_{4} films and greater thicknesses can be achieved without cracking. We show how the optimal lattice parameters can be found for a given material thickness and how the limiting reflectivity is affected by the membrane thickness, choice of (square or hexagonal) lattice structure, optical spot size (beam collimation), and index of refraction. The role of material absorption is explored, as well as the loss due to the finite size of the PhC structure.

Following the design procedure, we present several experimental results with Si-rich SiN membranes that both validate and complement it. We demonstrate square and hexagonal PhC devices, such as those shown in Fig. 1 with reflectivity *R* = 0.999 82(2), as determined from the finesse of Fabry–Perot cavities using these PhC mirrors as input couplers. We show that the hexagonal structure imparts a larger spectral zone of high reflectivity than does the square lattice. By varying the cavity length, we force the spot size of the confined mode on the PhC to vary and study the associated “clipping” loss caused by light falling outside of the perimeter of the PhC. We show that varying the radii of the holes in the PhC lattice is an effective way to adjust the lattice to a desired wavelength and find that the hexagonal lattice is less sensitive to variations in the hole radii than the square lattice. Finally, we show a phenomenon that is not captured in our simulations: All the Fabry–Perot cavities that we have constructed exhibit polarization mode splitting with well-resolved modes, attributable to a combination of nonuniform stress in the SiN film and slight asymmetry in the nanofabricated structures.

## II. DESIGN OPTIMIZATION PROCEDURE

### A. RCWA analysis with plane-wave decomposition of a Gaussian beam

Various numerical approaches are available to solve Maxwell’s equations for photonic crystal membranes, including the finite difference time domain (FDTD) method, finite element analysis (FEA), and rigorous coupled wave analysis^{20} (RCWA). We use RCWA in this work on the basis of computational speed and the availability of free software. Unlike FDTD and FEA, RCWA is limited to plane-wave excitation of infinite periodic structures, whereas FDTD and FEA can handle more general optical excitation and structures of finite extent. Of these limitations, the restriction to plane-wave excitation is the less serious one. The excitation of interest is typically a Gaussian laser beam, which can be decomposed onto a plane-wave basis by means of Fourier analysis; the individual plane waves can then be treated by RCWA and the results summed to obtain the response of the structure. Techniques to handle the case of a finite-sized structure have been developed for 1D gratings.^{21} Here, we take a simpler approach that is in excellent agreement with our experimental data; we defer further discussion to Sec. IV B.

Figure 2(a) shows a cross section of a Gaussian laser beam^{22} in the vicinity of its waist. The spot size at the waist is denoted by *ω*_{0}; this is the distance from the axis at which the intensity has fallen off by a factor of 1/*e*^{2}. As the beam propagates, it spreads due to diffraction, with a far-field diffraction angle given by *θ* = *λ*/(*πω*_{0}). A Fourier decomposition of the Gaussian beam consists of a distribution of plane waves with different angles of incidence, whose intensity drops off from the axial intensity by 1/*e*^{2} at the angle *θ*. We start by considering the response of the photonic crystal membrane to a normally incident plane wave.

Figure 2(b) shows a color plot of the simulated transmission of a PhC membrane with a thickness of *t* = 200 nm patterned with a hexagonal lattice having period *a* of round holes with radius *r*. The computation was performed using RCWA with *S*^{4}, the Stanford Stratified Structure Solver.^{23} The index of refraction of the membrane is taken to be *n* = 2.14, corresponding to a value^{24} characteristic of Si-rich SiN. We start with an idealized situation where material losses are ignored, so that transmission *T* and reflection *R* are related by *T* + *R* = 1. Plane-wave illumination at normal incidence is assumed, with a wavelength of *λ* = 1.56 *μ*m. A low-transmission zone is evident, containing the points indicated by the crosses labeled “A,” “B,” “C,” and “D.” At each of these points, the transmission vanishes identically, as it does on the entire curve from which these points are selected. In the absence of absorption, this locus of points describes the possible geometries of a perfect reflector (reflectivity *R* = 1) at *λ* = 1.56 *μ*m for normally incident light. Figure 2(c) shows the transmission spectrum corresponding to the choice of point “B” shown in Fig. 2(a); indeed, the transmission vanishes at the wavelength of *λ* = 1.56 *μ*m. A larger wavelength sweep would show this to be one of many minima associated with Fano resonances, as discussed theoretically elsewhere.^{15}

We now consider the corresponding spectra for plane waves incident at *θ*_{i} = 15 mrad with S-polarization; *θ*_{i} = 15 mrad is the far-field diffraction angle of a Gaussian beam with *λ* = 1.56 *μ*m and a spot size at the waist of *ω*_{0} = 33 *μ*m. Despite the small angles involved, such off-normal components play a crucial role in determining the overall reflectivity.^{8} Figure 2(d) shows the calculated reflection spectra for geometries characterized by the periods and radii for the points labeled by “A,” “B,” and “C” shown in Fig. 2(b). Narrower resonances appear atop the minimum associated with the Fano resonance shown in Fig. 2(c), and the tails of the resonances labeled “A” and “C” raise the transmission at *λ* = 1.56 *μ*m to *T* = 1.1 × 10^{−2} and *T* = 3.2 × 10^{−3}, respectively, limiting the attainable reflectivity. The situation for the PhC characterized by “B” is catastrophic; the resonance occurs exactly at the target wavelength, and the PhC slab becomes a perfect transmitter. While these resonances cannot be avoided entirely, their impact can be minimized by choosing judiciously from the locus of zero-transmission points shown in Fig. 2(b). In particular, point “D” has the spectrum shown in Fig. 2(e); an additional resonance is evident, but the tails of the resonances give rise to a transmission at *λ* = 1.56 *μ*m of only *T* = 2.1 × 10^{−4} and thus have a far smaller impact on the reflectivity at the target wavelength.

### B. Exploring parameter space

*ω*

_{0}and again use RCWA to make a color plot of transmission

*T*for a Gaussian beam with this waist size, analogous to that shown in Fig. 2(a). To this end, we use Fourier analysis to decompose the incident Gaussian beam on a plane wave basis,

*k*= |

**k**| = 2

*π*/

*λ*, and $k\u22c5\epsilon \u0302k=0$. The intensity distribution is found to be

*S*

^{4}for a plane wave incident on the slab with wavevector

**k**and polarization $\epsilon \u0302k$. For the moment, we continue to ignore absorption, so that

*T*= 1 −

*R*.

Color plots made in this way for hexagonal and square lattices are shown in Figs. 3(a) and 3(b), respectively, where we have taken *ω*_{0} = 25 *μ*m, and once again use *t* = 200 nm, *λ* = 1.56 *μ*m, and *n* = 2.14. Additional structure is evident in the color plot shown in Fig. 3(a) compared to that in Fig. 2(b); rather than having a one-dimensional continuum of points with zero transmission, we now identify three zones of low transmission, each containing nonzero local minima labeled by “A,” “B,” and “C.” The square lattice has a single low-transmission zone at smaller values of the period and hole radius; the local minimum is indicated with a cross labeled D shown in Fig. 3(b). To find the exact location of the minima, we employ the Downhill Simplex algorithm,^{25} a simple and robust approach for finding the extrema of functions of multiple variables. This algorithm has the advantage of requiring only evaluations of the function, and not derivatives.

The color plots of the transmission at other thicknesses are qualitatively similar to those shown in Figs. 3(a) and 3(b). We can thus apply the Downhill Simplex algorithm to membranes with thicknesses slightly above or below 200 nm, using the optimal values of (*a*, *r*) at *t* = 200 nm as initial guesses. Continuing in this way, we are able to efficiently track the transmission minima over a range 20 nm < *t* < 600 nm and obtain the periods and radii shown in Fig. 3(c), and the transmission curves shown in blue in Fig. 3(d). For clarity, only the evolution of the minimum labeled by “B” is shown for the hexagonal lattice, as it exhibits a consistently better performance (lower transmission) than the minima labeled by “A” and “C.” In addition, the transmission of an unpatterned membrane is shown in black.

It is straightforward to repeat this analysis for Gaussian beams with different waist sizes and dielectric membranes with a different index of refraction. Figure 3(d) shows the results for both hexagonal (solid lines) and square (dashed lines) lattices. The red curves show the transmission for the same index of refraction *n* = 2.14 discussed previously, but for an incident Gaussian beam with a waist of *ω*_{0} = 60 *μ*m. The green curves show the transmission for the case of stoichiometric Si_{3}N_{4} (*n* = 1.98), again with a waist *ω*_{0} = 60 *μ*m. The refractive index values used for stoichiometric Si_{3}N_{4} and Si-rich SiN were measured using ellipsometry for thin films deposited on silicon wafers.

A number of interesting and useful conclusions may be inferred from these curves. The minimum transmission is, for the most part, a monotonically decreasing function of the membrane thickness. For lower thicknesses, the hexagonal lattice has a lower minimum transmission than the square lattice, but the curves cross over for increasing thickness, resulting in lower transmission for the square lattice. This crossover thickness depends on the index of refraction and the beam waist. For *ω*_{0} = 60 *μ*m, it is ∼200 nm for *n* = 2.14 and ∼300 nm for *n* = 1.98. Curiously, the minimum transmission of the square lattice rises abruptly for thicknesses *t* > 400 nm (data for *t* > 442 nm suppressed for clarity), while that of the hexagonal lattice continues to drop monotonically as the thickness is raised.

Importantly, the minimum transmission is considerably lower when using a larger beam waist. The transmission of the hexagonal lattice in a 100 nm thickness of Si-rich SiN, for example, goes from *T* = 2.58 × 10^{−2} to *T* = 1.0 × 10^{−3} when the beam waist is increased from *ω*_{0} = 25 *μ*m to *ω*_{0} = 60 *μ*m. Qualitatively, this of course reflects the fact that the more tightly focused beam contains plane waves with larger angles of incidence.

Next, a lower minimum transmission is achieved with a higher index of refraction, as seen by comparing the curves for Si-rich SiN (*n* = 2.14) and stoichiometric Si_{3}N_{4} (*n* = 1.98). However, this conclusion is premature, as material absorption has been neglected in these simulations, and Si-rich SiN has more absorption than stoichiometric Si_{3}N_{4}. The impact of loss from absorption on the achievable reflectivity will be taken up in Sec. II D.

Finally, the curves in Fig. 3(c) show that while the optimal period for the square lattice approaches the target wavelength *λ* = 1.56 *μ*m from below, the optimal period for the hexagonal lattice reaches a value of ∼1.8 *μ*m. Naively, this would seem to violate the subwavelength condition expected to be required to suppress the nonzero diffraction orders. For a 2D lattice, however, the condition for suppression of orders beyond the zeroth order is more complicated and must be derived on the basis of the reciprocal lattice.^{26,27} One finds that while for the square lattice, the condition is the same as that for a 1D grating, namely, *a* < *λ*, for a hexagonal lattice, the condition is $a<2\lambda /3\u22481.15\lambda $, which is equal to 1.8 *μ*m for *λ* = 1.56 *μ*m. Thus, in both cases, as the thickness of the membrane diminishes, the optimal period approaches the maximum possible period that is consistent with zeroth-order diffraction. In fact, the optimal periods for the hexagonal lattice are shown in Fig. 3(c) to be larger than those for the square lattice in all cases, which may make the hexagonal lattice less susceptible to fabrication imperfections.

### C. Wavelength sweeps

Figure 4 shows the transmission as a function of wavelength for hexagonal PhC structures using optimized values of the period and radius, for values of the membrane thickness from *t* = 20 nm to *t* = 400 nm. Here, the incident beam spot size is *ω*_{0} = 60 *μ*m, and the index of refraction is *n* = 1.98. For the thinner membranes, Fig. 4(b) also shows, at the edges, the transmission that is found for an unpatterned membrane of the same thickness. The curve corresponding to the largest thickness, *t* = 400 nm, has the broadest minimum and shows no sign (in the wavelength range shown) of the resonances shown in Fig. 2(e), arising from the off-normal plane wave components of the incident beam. As the thickness is reduced, the width of the high-reflectivity zone diminishes, and the resonances anticipated in Fig. 2(e) start to appear. As the thickness continues to drop, the minimum transmission rises, and the resonances move toward the wavelength of minimum transmission and become more numerous. As the thickness is reduced further, the minimum at the target wavelength becomes weaker and narrower, and by the time a thickness of *t* = 20 nm is reached, the structure has broken up into a number of small, isolated minima. Similarly, color plots (not shown) such as those shown in Fig. 3 tend to break up into “islands” containing local minima, and the Downhill Simplex algorithm may “jump” from one local minimum to another; this is clearly shown in Fig. 3(c) where the radius “jumps” at values of the thickness in the vicinity of *t* = 45 nm. Nevertheless, Fig. 4 shows clearly that for certain values of the wavelength, the reflectivity of the PhC structure is substantially enhanced relative to that of the unpatterned membrane for membranes as thin as *t* = 20 nm. While the simulations used to generate Fig. 4 were made for a hexagonal lattice, the behavior is qualitatively the same for square lattice geometry.

### D. Losses

Until now, the discussion has concerned idealized photonic crystals, in which loss mechanisms, such as material absorption and scattering, have been ignored. In practice, such loss mechanisms must limit the performance of the PhC. It is straightforward to include material absorption in the form of a nonzero extinction coefficient (imaginary part *n*_{I} of the index of refraction). Denoting PhC absorption by *A*, we then have *R* + *T* + *A* = 1. Measurements using membranes^{24,28,29} of the extinction coefficient of Si-rich SIN have given values in the range 1.48 × 10^{−5} < *n*_{I} < 1.6 × 10^{−4} at wavelengths of *λ* = 1064 nm and *λ* = 1550 nm. Stoichiometric Si_{3}N_{4} has been found to have a lower loss, where a recent review of the literature shows that 3.7 × 10^{−9} < *n*_{I} < 2 × 10^{−5} at *λ* = 1550 nm.^{30,31} Scattering from random geometrical defects is a more difficult loss mechanism to calculate, and we do not attempt to do so; the absorption-limited performance discussed here must be understood as the limiting performance achievable if scattering losses can be made negligible. A final loss mechanism limiting device performance, particularly within an optical cavity, arises from the limited spatial size of the PhC itself. We defer this matter to Sec. IV B.

To illustrate the effect of material absorption on the reflectivity of a PhC membrane, we consider a hexagonal photonic crystal made of Si-rich SiN with the thickness of *t* = 220 nm and nominal index of refraction *n* = 2.14. The lowest curve in Fig. 5(a) shows 1 − *R* for such a photonic crystal in the absence of material absorption (*n*_{I} = 0) for all spot sizes in the range 15 *µ*m < *ω*_{0} < 75 *µ*m. The other curves in Fig. 5(a) show 1 − *R* for various values of the extinction coefficient *n*_{I}, taken over a range 10^{−5} < *n*_{I} < 3 × 10^{−4} slightly larger than the range found in experimental studies of Si-rich SiN.^{24,28,29} Not surprisingly, the reflectivity of the PhC is degraded as the material absorption increases. This effect is less noticeable for small values of the spot size *ω*_{0}, where the reflectivity is already more strongly limited by the spread in wavevectors present in the incident beam. For nonzero *n*_{I}, as the spot size *ω*_{0} increases, the reflectivity asymptotically approaches a limiting value beyond which increasing the spot size is of no benefit.

In addition to understanding the role that material absorption plays in limiting the reflectivity of a PhC, it can be useful to consider how the PhC absorption *A* compares to the PhC transmission *T*. When used as the end mirror of an optical cavity, both the transmission and absorption act as loss mechanisms limiting the amount of light circulating. In fact, only the reflectivity *R* is relevant to the cavity finesse; *A* and *T* can be changed at will, provided that *A* + *T* stays constant, and the finesse will remain unchanged. However, if the PhC is to be used as the input coupler to the cavity, its transmission *T* will govern how much of the incident power can be coupled into the cavity, and if it is used as the output coupler, its transmission will determine how much cavity transmission is available. In both cases, it is generally desirable to have *T* ≫ *A*. If used as the central reflective element in a “membrane in the middle” setup, it is the PhC transmission *T* that governs the extent to which the two sub-cavities are coupled. Figure 5(b) shows how the absorption and transmission are related, as a function of spot size, for the hexagonal lattice whose reflectivity is shown in Fig. 5(a). One sees that as the extinction coefficient *n*_{I} is raised, not only does the reflectivity drop, as shown in Fig. 5(a), but also the ratio of the absorption to the transmission becomes increasingly large. Once again, this effect is exacerbated with larger spot sizes *ω*_{0}.

At this point, it is of interest to recall, as shown in Fig. 3(d), that when absorption is neglected, better reflection is obtained for Si-rich SiN (*n* = 2.14) than for stoichiometric Si_{3}N_{4} (*n* = 1.98) and ask how the situation changes if absorption is included. The answer is shown in Fig. 6, taking for concreteness the hexagonal lattices of Fig. 3(d), optimized over period and radius for maximum reflectivity with a spot size *ω*_{0} = 60 *μ*m. Once again, the extinction coefficients are taken to be in the range 10^{−5} < *n*_{I} < 3 × 10^{−4} for Si-rich SiN. For stoichiometric Si_{3}N_{4}, we take *n*_{I} = 10^{−6} as a realistic value based on the most recent literature.^{30} The effect of absorption for the case of Si-rich SiN is dramatic for the larger thicknesses. Indeed, even with an extinction coefficient of *n*_{I} = 10^{−5}, the advantage conferred by the higher index of refraction for Si-rich SiN is lost for thicknesses above *t* = 166 nm, where the reflectivity obtained with both materials is *R* ≈ 0.9998.

As will be discussed in Sec. IV B, our experimental results with Si-rich SiN are consistent with the reflectivity primarily being limited by material absorption, with *n*_{I} ≈ 1.5 × 10^{−5}. Figure 6 shows that absorption should be less of an issue if stoichiometric Si_{3}N_{4} is used. In practice, factors other than optical loss are likely to influence the type of silicon nitride that is selected for a given application, including the desired mechanical quality factor and film thickness, as well as the required fabrication yield and ease of resonator release from the substrate.

### E. Sensitivity to fabrication imperfections; second derivatives

Fabricated devices will never have the exact geometry and index of refraction for which they were designed, so it is of interest to explore the sensitivity of the design to small deviations in the design parameters. At the transmission minima (*a*_{0}, *r*_{0}) located by the Downhill Simplex routine, the derivatives *∂T*/*∂a* and *∂T*/*∂r* must vanish, and in the vicinity of the minima, the dependence of the transmission on the period *a* and radius *r* will be governed by the second derivatives *∂*^{2}*T*/*∂a*^{2}, *∂*^{2}*T*/*∂r*^{2}, and *∂*^{2}*T*/*∂a∂r*. The thickness *t*, index of refraction *n*, and wavelength *λ* were kept fixed for the optimization procedure, but for the geometry specified by (*a*_{0}, *r*_{0}), the transmission will vary as the parameters *t*, *n*, and *λ* are varied. While the first derivatives, *∂T*/*∂t*, *∂T*/*∂n*, and *∂T*/*∂λ*, need not vanish completely at (*a*_{0}, *r*_{0}), they must be very small, as the transmission is bounded from below by zero. Thus, in the vicinity of (*a*_{0}, *r*_{0}), the dependence of the transmission on *a*, *r*, *t*, *n*, and *λ* is governed by the fifteen second derivatives, such as *∂*^{2}*T*/*∂a*^{2}, *∂*^{2}*T*/*∂r*^{2}, *∂*^{2}*T*/*∂λ*^{2}, *∂*^{2}*T*/*∂a∂r*, and *∂*^{2}*T*/*∂a∂n*.

We have found numerically that the second derivatives at the minima of the hexagonal lattice are all smaller than the corresponding ones for the square lattice, so we expect that the reflectivity of the hexagonal lattice should be less sensitive to deviations of the actual fabricated values of the period and radius from the target values than the square lattice. It is straightforward to compare the wavelength sensitivity of the two structures embodied in *∂*^{2}*T*/*∂λ*^{2} in experiment, a point to which we will return in Sec. IV A.

## III. EXPERIMENTAL METHODS

To demonstrate the effectiveness of the PhC optimization procedure described in Sec. II, both square and hexagonal PhCs were designed for an Si-rich SiN membrane with a nominal thickness of 220 nm and a square planar geometry that is 800 *μ*m on a side [see Fig. 1(a)]. The thickness was chosen to be close to the crossover point shown in Fig. 3(d) where the square and hexagonal lattices are predicted to give similar reflectivity. The refractive index of a silicon nitride film is dependent on the growth conditions; that of the Si-rich SiN film employed here was found to be *n* = 2.093 when measured using ellipsometry before fabrication of the membrane, slightly different from the value *n* = 2.14 found in our earlier work^{24} and used in the simulations. The optimized design values were calculated to be *a* = 1.274 *μ*m and *r* = 0.469 *μ*m for the square PhC and *a* = 1.510 *μ*m and *r* = 0.515 *μ*m for the hexagonal PhC, where *a* is the lattice period and *r* is the hole radius. In addition to these nominal design values, an array of PhCs was fabricated with the same value of the period but with hole radii that are reduced by values ranging from 5 to 40 nm. Nanofabrication of the etched features often introduces an increase in critical dimensions due to both the lithographic and etching steps. Fabricating PhCs with varying radii provides a method for correcting for these dimensional inaccuracies.

The PhC membranes were fabricated using a two-sided process with a double-side polished 〈100〉 silicon substrate that is 525 *μ*m thick. Si-rich SiN was deposited on the substrate using low-pressure chemical vapor deposition (LPCVD), resulting in a thickness of 219.0(6) nm and a nominal residual stress in the range of 400 MPa. PhCs were patterned on the front side using electron-beam lithography and transferred into the silicon nitride through reactive ion etching (RIE). A low-temperature silicon oxide (LTO) film with a thickness of 1.5 *μ*m was then deposited on the wafer to protect the PhCs during subsequent steps. The backside of the wafer was patterned with square openings centered on each PhC using a combination of maskless optical lithography, RIE, and deep reactive ion etching (DRIE), with a goal of etching the silicon substrate to within 40 *μ*m of the top surface. The substrate was then diced, the resulting chips were cleaned, and the remaining silicon under the PhCs and the silicon oxide on top of the PhCs were etched away simultaneously using a timed chip-by-chip etch in KOH. The combination of DRIE and KOH etching for the release of the membrane significantly reduces the etch time in KOH, which improves the process yield and reduces the potential for expanding the hole radii and thinning the membrane. A representative fabricated hexagonal PhC membrane is shown in Figs. 1(b)–1(d).

To study the optical characteristics of the fabricated PhCs, the setup shown in Fig. 7 was used. Linearly polarized light from a widely tunable external cavity diode laser centered at 1550 nm was passed through an electro-optic intensity modulator and then collimated. A half-wave plate is available to rotate the polarization of the collimated light, and a lens focused the light onto the PhC. For initial experiments, light transmitted by the PhC was focused onto a photodetector, enabling measurements of the transmission spectra, such as those shown in Fig. 4.

*L*<

*r*

_{c}, where

*L*is the cavity length and

*r*

_{c}is the radius of curvature of the dielectric mirror. The spot size

*ω*

_{0}of the confined Gaussian mode on the PhC is given by

^{32}

*L*is easily made by sweeping the laser wavelength, observing the transmission resonances, and measuring the mode spacing (free spectral range) Δ

*ν*=

*c*/(2

*L*), where

*c*is the speed of light. In these experiments, the radius of curvature of the dielectric mirror is 25 mm, and we used cavity lengths in the range 19.8 mm $<L<$ 24.9 mm, corresponding to spot sizes 28

*µ*m <

*ω*

_{0}< 71

*µ*m.

*R*

_{1}and

*R*

_{2}are the reflectivities of the input and output couplers, respectively, so that the PhC reflectivity is

*R*

_{2}= 0.999 985(5), based on the data provided by the manufacturer and our own experience constructing cavities with these mirrors.

In practice, we measure the finesse by using the ringdown method,^{12} using the electro-optic intensity modulator to provide fast switching of the input beam. By measuring the free spectral range and ringdown times associated with the transmission resonances, we are thus able to infer the spot size *ω*_{0} of the Gaussian mode on the PhC and the PhC reflectivity. Finally, we use the half-wave plate shown in Fig. 7 to study polarization mode splitting in the cavity induced by the PhC.

## IV. EXPERIMENTAL RESULTS

In the following, we describe four experiments that investigate different aspects of the PhC devices constructed according to the prescription given in Sec. II. First, measurement results for the achievable cavity finesse and corresponding PhC reflectivity are presented. Next, the dependence of the finesse on the spot size *ω*_{0} at the waist of the cavity mode is explored, augmenting the idealized RCWA analysis presented earlier with the unavoidable loss due to the finite size of the PhC itself. We then study the sensitivity of the PhC center wavelength to variations in the hole radius introduced during nanofabrication. Finally, we present the unanticipated polarization mode splitting that we observe in the transmission spectra of all of our PhC cavities. This is a phenomenon of practical importance that arises from a broken symmetry, possibly nonuniform stress in the SiN membrane, which was not included in the RCWA analysis.

### A. Cavity finesse and PhC reflectivity

The transmission spectra of hexagonal and square PhCs, both with a 300 *μ*m diameter for the patterned area and the optimized design values described in the previous section, were measured using the optical setup shown in Fig. 7, where the concave dielectric mirror M was initially not employed. The free-space transmission spectra offer a direct comparison with the numerical calculations. This broadband sweep provides a qualitative assessment of differences between hexagonal and square lattices. As shown in Fig. 8(a), both PhCs exhibit near-zero transmission minima in the vicinity of 1550 nm. For wavelengths near those giving the transmission minima, the wavelength dependence of the transmission is governed by a quadratic term proportional to *d*^{2}*T*/*dλ*^{2}. The magnitude of this dependence is noticeably smaller for the hexagonal PhC, providing a broader zone of high reflectivity. The simulated transmission spectra from *S*^{4}, shown in the inset of Fig. 8(a), predict this distinction between the optical responses of these two crystal lattices. As discussed in Sec. II E, the transmission of the hexagonal lattice about its minimum is found in simulations to exhibit less sensitivity than the square lattice to all the independent variables, but only the dependence on the wavelength, embodied in *∂*^{2}*T*/*∂λ*^{2}, can be readily demonstrated experimentally.

To more precisely characterize the reflectivity of the PhCs in the vicinity of the wavelength of minimum transmission identified above, we insert the curved dielectric mirror M shown in Fig. 7 in order to realize Fabry–Perot cavities with both hexagonal and square PhC input couplers. As discussed in Sec. III, stable cavities are possible with cavity lengths *L* in the range 0 < *L* < 25 mm, with the choice of length determining the spot size *ω*_{0}, given by Eq. (4), of the confined mode on the PhC. The choice of *ω*_{0} has important consequences on the achievable finesse, as will be discussed in the following section. After optimizing the cavity finesse by tuning the cavity length to force a spot size *ω*_{0} ≈ 60 *μ*m, ringdown measurements^{12} were performed for a number of cavity modes near the PhC center wavelength. A representative cavity ringdown measurement is shown in Fig. 8(b), where a fit to the data provides the photon lifetime. From these measurements, the cavity finesse near the center wavelength was calculated, as shown in Fig. 8(c), where the maximum measured finesse exceeds 30 000 for both hexagonal and square PhCs. A Lorentzian function was fit to both sets of measured finesse data as a function of wavelength, yielding an estimated maximum finesse of 33 000(3000) and 33 000(2000) for the hexagonal and square PhCs, respectively. The uncertainty of each data point is given by the standard deviation of five consecutive measurements. The uncertainty of the estimated maximum finesse is given by the parameter standard error from the fitting algorithm.

From these values, the reflectivities of the PhC membranes were calculated from Eq. (6) to be 0.999 82(2). These results further confirm the best reflectivity achieved with a PhC membrane to date.^{12} The FWHM linewidths of the Lorentzian fits for the hexagonal and square PhCs are 1.9(1) and 1.5(1) nm, respectively, again demonstrating that the bandwidth around the center wavelength is greater for the hexagonal PhC.

### B. Finite size photonic crystal loss

As shown in Fig. 5(a), RCWA simulations show that the reflectivity of an infinitely large PhC mirror increases monotonically with increasing waist spot size *ω*_{0} of the incident beam. In practice, the spatial extent of a fabricated PhC is not infinite, and optical power incident outside the PhC will experience a far smaller reflectivity. Such “finite size” effects cannot be captured in RCWA, although they have been addressed theoretically for 1D structures.^{21}

*D*, as shown in Fig. 9(a). Assuming for simplicity that no light is reflected outside the PhC area, the fractional loss of light due to clipping,

*L*

_{c}, can be calculated to be

*L*

_{c}as a function of

*ω*

_{0}for

*D*= 150

*μ*m and

*D*= 300

*μ*m.

*R*

_{1}in Eq. (5) as

*R*

_{2}= 1, is then

*T*

_{1}+

*A*

_{1}are shown in Fig. 5(a) for various values of the extinction coefficient

*n*

_{I}as a function of spot size

*ω*

_{0}. A similar curve for

*n*

_{I}= 1.5 × 10

^{−5}is shown (black dashed line) along with the clipping losses

*L*

_{c}shown in Fig. 9(b), making it clear that the two cavity loss mechanisms have opposite dependencies on the spot size.

The calculated upper bounds to the finesse *F*_{UB} given in Eq. (9), using the cavity losses shown in Fig. 9(b) as functions of spot size, are shown in Fig. 9(c). We find optimal values of the spot size to be *ω*_{0} = 33.0 *μ*m and *ω*_{0} = 59.4 *μ*m for the PhCs with diameters *D* = 150 *μ*m and *D* = 300 *μ*m, respectively. The black dashed line in Fig. 9(c) shows that, for the level of material absorption *n*_{I} = 1.5 × 10^{−5} considered here, even taking a PhC of infinite spatial extent would only give a modest increase in the achievable finesse over the value $F\u224835000$ found for *D* = 300 *μ*m.

Using hexagonal PhCs of diameters *D* = 150 and 300 *μ*m, we used the ringdown method^{12} to measure the finesse $F$ for various values of *ω*_{0} around the optimal values given above. After an initial determination of the cavity length by means of the free spectral range, as discussed in Sec. III, the length was varied with a micrometer-controlled stage, and the corresponding spot size *ω*_{0} was determined from Eq. (4). The resulting finesse as a function of *ω*_{0} is shown in Fig. 9(d). The values of *ω*_{0} optimizing the finesse agree well with expectations for both PhCs. It is clear that a greater PhC area and larger *ω*_{0} can dramatically improve the achievable finesse. It is also notable that the highest values of the finesse that we find in Fig. 9(d) are consistent with the values of *n*_{I} only slightly larger than the value *n*_{I} = 1.5 × 10^{−5} used in the curves shown in Fig. 9(c), which, in turn, is at the level of the lowest values of the extinction coefficient for Si-rich SiN reported elsewhere.^{24,28,29} We thus conclude that material absorption is the dominant loss mechanism limiting the reflectivity that we achieve. For stoichiometric Si_{3}N_{4} characterized by an extinction coefficient of *n*_{I} = 10^{−6}, the upper bound to the achievable finesse using such a PhC with a thickness of *t* = 220 nm and diameter *D* = 300 *μ*m is $F\u224893000$. In practice, scattering losses would likely reduce the achievable finesse below this value.

### C. Sensitivity of PhC center wavelength

It is frequently important to match the center wavelength of a PhC device to the tuning range of a laser or the optical frequency of an atomic transition. One approach to mitigate the effect of fabrication imperfections is to take a “shotgun” approach and prepare a large number of samples with a range of values of some fabrication parameter, with the expectation that an acceptable center wavelength will be found in at least one of them. To this end, we fabricated a number of samples with the same period but with varying hole radii and measured their transmission spectra. Both the simulated and measured center wavelengths as a function of the target hole radii are shown in Fig. 10, for both hexagonal and square PhCs. Linear fits to the experimental results yield a slope of −0.35 nm/nm for the hexagonal PhCs and −0.73 nm/nm for the square PhCs, in reasonable agreement with the results from simulations of −0.40 and −0.83 nm/nm, respectively. These results indicate that the hexagonal lattice is less sensitive to variations in the hole radius introduced by nanofabrication processes, thereby making it easier to achieve a desired center wavelength.

As shown in Fig. 10, the measured center wavelength is shifted down from the simulated results for all values of hole radius, where the shift ranges from 17 to 26 nm. This shift is due to a combination of an increase in hole radius and decrease in membrane thickness during the nanofabrication process,^{8} as well as uncertainty in the refractive index measurement. Nevertheless, using a range of radii with target values below the optimal simulated radius can be an effective approach to achieving the central wavelength desired in a fabricated device.

### D. Polarization eigenmode splitting

It is well known that astigmatism and birefringence in mirror materials can lead to polarization eigenmode splitting.^{3,33,34} We have, in fact, observed polarization mode splitting in all the cavities, using both hexagonal and square PhCs that we have constructed. We characterize the mode splitting by observing the cavity transmission spectrum for various values of the polarization of the input light, using the half-wave plate shown in Fig. 7. Figure 11 shows the mode splitting of a hexagonal PhC when the polarization of the input light is set to equally excite both polarization eigenmodes while scanning the laser over a cavity resonance near 1551.3 nm. The modes are well resolved, and the frequency splitting of the modes is measured to be 2.03(5) MHz by means of Lorentzian fits to the resonances. The insets in Fig. 11 show the measured transverse mode profiles of the transmitted beam when the laser is tuned to the corresponding modes, confirming that they are both TEM_{00} modes. Two additional hexagonal PhCs and four square PhCs were also measured, yielding a mean frequency splitting of 3 MHz with a standard deviation of 1.36 MHz for the hexagonal PhCs and a mean splitting of 4.83 MHz with standard deviation of 3.73 MHz for the square PhCs. The fact that the splitting varies when different PhCs are used implies that it is the PhCs, rather than the dielectric mirror, which are primarily responsible for the observed mode splitting. Since the samples were all on the same chip, our data suggest that the square lattice exhibits more polarization mode splitting than the hexagonal one for similar levels of sample nonuniformity, but the sample size is too small to make a definite statement to this effect. Since either mode can be uniquely accessed by rotating the input polarization, the mode splitting does not limit the use of the PhC membranes in cavity optomechanics and may be useful in some pump-probe type experiments.^{35} We, in fact, rotated the input polarization to excite only one polarization eigenmode for all of the data shown in this paper, except for that in the present section.

## V. CONCLUSION

We have presented a practical guide for designing and fabricating photonic crystal membranes with ultrahigh reflectivity in Si-rich SiN and stoichiometric Si_{3}N_{4}. We show that important roles are played by the membrane thickness, spot size of the incident beam, loss beyond the edges of a finite-sized photonic crystal, and material absorption. By using the Downhill Simplex algorithm to optimize over the period and radius of the photonic crystal structure, we are able to determine the optimal geometry providing the highest possible reflectivity for a given membrane thickness. We have shown the importance of accounting for the angular spread of wavevectors in a localized Gaussian laser beam, leading to a pronounced dependence in the reflectivity on the waist spot size *ω*_{0}. For thin membranes (i.e., $<160$ nm), the higher refractive index of Si-rich SiN can provide a higher reflectivity than stoichiometric Si_{3}N_{4}, whereas for greater thicknesses, the expected lower absorption of stoichiometric Si_{3}N_{4} could produce greater reflectivity than Si-rich SiN. We find that relative to a square lattice, the hexagonal lattice has a larger period for a desired center wavelength, is less sensitive to fabrication imperfections, has a wider spectral zone of high reflectivity, and has higher reflectivity for membrane thicknesses less than 200 nm. Remarkably, the benefit of patterning a membrane with a PhC structure extends even to thicknesses as small as *t* = 20 nm, although as the thickness is reduced, the reflection spectrum breaks up into a set of narrow resonances.

PhC membranes with a thickness of 220 nm fabricated from Si-rich SiN according to the prescription described here were employed as input couplers in an optical cavity. Membranes with PhC patterns of 300 *μ*m diameter demonstrated exceptionally high finesse $F\u226533000$ and corresponding reflectivity *R* > 0.999 82. A larger reflection bandwidth is found for cavities made with hexagonal rather than square PhC structures, confirming expectations from simulations. The reflectivity achieved is compatible with material absorption close to the lowest value reported for Si-rich SiN, suggesting that, in fact, absorption is the dominant mechanism limiting the reflectivity. For stoichiometric Si_{3}N_{4} characterized by an extinction coefficient of *n*_{I} = 10^{−6}, the upper bound to the achievable finesse using such a PhC is $F\u224893000$, although scattering losses would likely reduce the achievable finesse below this value.

The nature of the RCWA simulations employed for the design is such that it is not able to account for the finite size of the PhC, and the losses associated with optical power being clipped at the edge of the PhC structure are critical to determining the reflectivity ultimately achieved. We have found, by using PhC devices of both 150 and 300 *μ*m diameters, that a simple model based on the power lost in the tails of a Gaussian beam predicts the optimal compromise between PhC size and mode waist spot size *ω*_{0} remarkably well.

Finally, we find that the cavities constructed with these membranes exhibit polarization mode splitting, which may be a useful resource in “pump–probe” experiments. The ultrahigh reflectivity of these membranes makes them well suited both for fundamental cavity optomechanical measurements, including ground state cooling, as well as for sensing elements in optomechanical transducers. Future work will experimentally explore the limits of PhC reflectivity for even thinner membranes, use stoichiometric Si_{3}N_{4} to minimize optical and mechanical losses, and incorporate these PhC mirrors into phononic shields.

## ACKNOWLEDGMENTS

F.Z. acknowledges support from the Joint Quantum Institute, NIST/University of Maryland, College Park, MD. Y.B. acknowledges support from the National Institute of Standards and Technology, Department of Commerce, USA (Grant No. 70NANB17H247). This research was performed in part in the NIST Center for Nanoscale Science and Technology Nanofab.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**F. Zhou**: Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (supporting). **Y. Bao**: Data curation (equal); Formal analysis (equal); Investigation (equal); Resources (equal); Writing – original draft (supporting); Writing – review & editing (supporting). **J. J. Gorman**: Investigation (supporting); Resources (equal); Supervision (equal); Writing – original draft (supporting); Writing – review & editing (equal). **J. R. Lawall**: Conceptualization (lead); Data curation (equal); Formal analysis (lead); Investigation (equal); Software (lead); Supervision (equal); Writing – original draft (lead); Writing – review & editing (lead).

## DATA AVAILABILITY

The data presented here are available from the corresponding author upon reasonable request.

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