## Abstract

Photonic crystal nanobeam cavities are versatile platforms of interest for optical communications, optomechanics, optofluidics, cavity QED, etc. In a previous work [Appl. Phys. Lett. **96**, 203102 (2010)], we proposed a deterministic method to achieve ultrahigh *Q* cavities. This follow-up work provides systematic analysis and verifications of the deterministic design recipe and further extends the discussion to air-mode cavities. We demonstrate designs of dielectric-mode and air-mode cavities with *Q* > 10^{9}, as well as dielectric-mode nanobeam cavities with both ultrahigh-*Q* (> 10^{7}) and ultrahigh on-resonance transmissions (*T* > 95%).

©2011 Optical Society of America

## 1. Introduction

High quality factor (*Q*), small mode volume (*V*) [1] optical cavities provide powerful means for modifying the interactions between light and matter [2], and have many exciting applications including quantum information processing [3], nonlinear optics [4], optomechanics [5, 6], optical trapping [7] and optofluidics [8]. Photonic crystal cavities (PhC) [9, 10] have demonstrated numerous advantages over other cavity geometries due to their wavelength-scale mode volumes and over-million *Q*-factors [11–31]. Although small mode volumes of PhC cavities can be easily achieved by design, ultrahigh *Q* factors are typically obtained using extensive parameter search and optimization. In a previous work [32], we proposed a deterministic method to design an ultrahigh *Q* PhC nanobeam cavity and verified our designs experimentally. The proposed method does not rely on any trial-and-error based parameter search and does not require any hole shifting, re-sizing and overall cavity re-scaling. The key design rules we proposed, that result in ultrahigh *Q* cavities, are (i) zero cavity length (*L* = 0), (ii) constant length of each mirror segment (‘period’=*a*) and (iii) a Gaussian-like field attenuation profile, provided by linear increase in the mirror strength.

In this follow-up work, we provide numerical proof of the proposed principles, and systematically optimize the design recipe to realize a radiation limited cavity and waveguide coupled cavity. Furthermore, we extend the recipe to the design of air-mode cavities, whose optical energies are concentrated in the low-index region of the structure.

Nanobeam cavities have recently emerged as a powerful alternative to the slab-based 2-D PhC cavities [17–22]. Nanobeams can achieve *Q*s on par with those found in slab-based geometries, but in much smaller footprints, and are the most natural geometries for integration with waveguides [23–31]. Our deterministically designed cavities have similar structures to the mode-gap cavities proposed by Notomi et al. [23], and later demonstrated experimentally by Kuramochi et al. [33], as well as our own work [34]. We note that the same design principle discussed here could be directly applied to realize ultra-high *Q* cavities based on dielectric stacks that are of interest for realization of vertical-cavity surface emitting lasers (VCSELs) and sharp filters. Finally, it is important to emphasize that while our method is based on the framework of Fourier space analysis [35–37], alternative approach, based on phase-matching between different mirror segments, could also be used to guide the design, as well as to explain the origin of deterministic ultra-high *Q*-factors in our devices [38, 39].

## 2. Numerical verification of the deterministic design approach

In this section, we use finite-difference time-domain (FDTD) simulations to systematically study the design principles proposed in [32]. Figure 1(a) shows the schematics of the proposed cavity structure [32]. It consists of an array of air-holes in decreasing radii, etched into a ridge waveguide. The hole-to-hole distances (“periodicity”) are constant. The structure is symmetric with respect to the dashed line in Fig. 1(a). In contrast to the majority of other cavity designs, current structure has no additional cavity length inserted between the two mirrors (*L*=0), that is the hole-to-hole spacing between the two central holes is the same as the rest of the structure (*a*). This minimizes the cavity loss and the mode volume simultaneously. The cavity loss is composed of the radiation loss into the free space (characterized by *Q*
_{rad}) and the coupling loss to the feeding waveguide (*Q*
_{wg}). *Q*
_{wg} can be increased simply by adding more gratings along the waveguide. *Q*
_{rad} can be increased by minimizing the spatial Fourier harmonics of the cavity mode inside the lightcone, achieved by creating a Gaussian-like attenuation profile [32]. The optical energy is concentrated in the dielectric region in the middle of the cavity (Fig. 1(b)). In order to achieve the Gaussian-like attenuation, we proposed to use a linearly increasing mirror strength along the waveguide [32], which was achieved by tapering the hole radii.

First, we analyze the ideal tapering profile using plane wave expansion method and verify the results with 3D FDTD simulations. The dielectric profile of the structure in the middle plane of the cavity can be expressed as

**r**

*=*

_{j}*j*·

*ax̂*,

*a*is the period, and

*j*= ±1, ±2... are integers.

*R*is the radius of the hole. Using plain wave expansion method [40] in the beam direction (

*x̂*), where

*G*= 2

*π*/

*a*. The zeroth (

*κ*

_{0}) and first (

*κ*

_{1}) order Fourier components can be expressed as [40]

*J*

_{1}is the first order Bessel function. Filling fraction

*f*=

*πR*

^{2}/

*ab*is the ratio of the area of the air-hole to the area of the unit cell. We note that the above expressions are calculated assuming nanobeam cavity has infinite thickness (i.e 2D equivalent case). Better estimation can be obtained by replacing

*ε*

_{air}and

*ε*

_{Si}with the effective permittivities.

The dispersion relation can be obtained by solving the master equation [41]:

*k*) for a given frequency (

*ω*) is a complex number, whose imaginary part denotes the mirror strength (

*γ*). For solutions near the band-edge, of interest for high-

*Q*cavity design [32], the frequency can be written as $\omega =(1-\delta )\sqrt{{\kappa}_{0}}\pi c/a$ (

*δ*is the detuning from the mid-gap frequency) and the wavevector as

*k*= (1 +

*iγ*)

*π*/

*a*. Substituting this into the master equation, we obtain ${\delta}^{2}+{\gamma}^{2}={\kappa}_{1}^{2}/4{\kappa}_{0}^{2}$. The cavity resonance asymptotes to the dielectric band-edge of the center mirror segment: ${w}_{\text{res}}\to (1-{\kappa}_{1}^{j=1}/2{\kappa}_{0}^{j=1})\sqrt{{\kappa}_{0}^{j=1}}\pi c/a$ (

*j*represents the

*j*

^{th}mirror segment counted from the center), at which point the mirror strength

*γ*

^{j}^{=1}= 0.

*γ*increases with

*j*. With

*ε*

_{air}= 1 and

*ε*

_{Si}= 3.46

^{2}, we calculate in Fig. 2(a) the

*γ*–

*j*relation for different tapering profiles. It can be seen that quadratic tapering profile results in linearly increasing mirror strengths, needed for Gaussian field attenuation [32]. To verify this, we perform FDTD simulation and obtain the band diagram (Fig. 2(b)) and

*γ*–

*f*relation (Fig. 2(c)). As shown in Fig. 2(d), linearly increasing mirror strength is indeed achieved after quadratic tapering.

Next, with the optimized tapering profile, the cavity is formed by putting two such mirrors back to back, leaving a cavity length *L* in between (Fig. 1(a)). Figure 2(e) shows the simulated *Q*-factors for various *L*s. Highest *Q*
_{rad} is achieved at zero cavity length (*L*=0), which supports the prediction in [32] based on 1D model.

Third, we verify that the cavity mode has a Gaussian-like attenuation profile. Figure 2(g) shows the *H _{z}*-field distribution in the plane right above the cavity, obtained from 3D FDTD simulation. As shown in Fig. 2(h), this field distribution can be ideally fitted with

*H*= sin(

_{z}*πx*/

*a*)exp(−

*σx*

^{2})exp(−

*ξy*

^{2}), with

*a*= 0.33,

*σ*= 0.14 and

*ξ*= 14. The fitted value

*a*agrees with the “period”, and

*σ*agrees with that extracted value from Fig. 2(d): $\sigma =\frac{\text{d}\gamma}{\text{d}x}\frac{\pi}{a}=0.13$. Figure 2(i) shows

*H*distribution along the dashed line in Figs. 2(g) and 2(h). Therefore, we conclude that zero cavity length, fixed periodicity and a quadratic tapering of the filling fraction results in a Gaussian field profile, which leads to a high-

_{z}*Q*cavity [32].

Finally, as we have pointed out in [32], current method results in a cavity whose resonance is asymptotically approaching the dielectric band-edge frequency of the central mirror segment (circled in Fig. 2(b)). The deviation from the band-edge frequency can be calculated using perturbation theory [41, 42]:

**E**

_{||}is the component of

**E**that is parallel to the side wall surfaces of the holes and

**D**

_{⊥}is the component of

**D**that is perpendicular to the side wall surfaces of the holes. Under Gaussian distribution, the major field component

*D*= cos(

_{y}*π*/

*ax*)exp(−

*σx*

^{2})exp(−

*ξy*

^{2}),

*δε*perturbation occurs at

*r*= ±(

*j*− 1/2)

*a*+

*R*, where ${R}_{j}=\sqrt{{f}_{j}ab/\pi}$ denotes the radius of the

_{j}*j*

^{th}hole (counted the center), with

*j*=2,3...

*N*,

*N*is the total number of mirror segments at each side. Since the cavity mode has a Gaussian profile, $1/\sqrt{\sigma}$ characterizes the effective length of the cavity mode, and scales linearly with

*N*, with a nonzero intercept due to diffraction limit. For large

*N*, the intercept can be neglected, and thus ${\sigma}_{N}=\sqrt{20\times {0.14}^{2}/N}$. Plug the perturbation induced by the quadratic tapering from

*f*= 0.2 to

*f*= 0.1 into Eq. (6), the frequency offset

*δλ*/

*λ*v.s

*N*can be obtained. Figure 2(f) shows the frequency offset for different total number of mirror pairs (

*N*), calculated from the perturbation theory, as well as using FDTD simulations. It can be seen that the deviation decreases as the number of modulated mirror segments increases, and is below 1% for

*N*> 15.

Therefore, we verify that an ultrahigh-*Q*, dielectric-mode cavity resonant at a target frequency can be designed using the following algorithm:

- Determine a target frequency. For example in our case we want
*f*_{target}= 200THz. Since the cavity resonant frequency is typically 1% smaller than the dielectric band-edge of the central segment, estimated using the perturbation theory, we shift-up the target frequency by 1%, i.e.*f*_{adjusted}= 202THz. - Pick the thickness of the nanobeam - this is often pre-determined by the choice of the wafer. For example, in our case, the thickness of the nanobeam is 220nm, determined by the thickness of the device layer of our silicon-on-insulator (SOI) wafer.
- Choose periodicity according to
*a*=*λ*_{0}/2*n*_{eff}, where*n*_{eff}is effective mode index of the cavity and can be estimated by numerical modeling of a strip waveguide that nanobeam cavity is based on. However, we found that the absolute value of the periodicity is not crucial in our design, as long as there exists a bandgap. Therefore, we pick*n*_{eff}= 2.23, which is a median value of possible effective indices in the case of free standing silicon nanobeam (*n*_{eff}∈ (1, 3.46)). This results in*a*= 330nm. - Set nanobeam width. Large width increases the effective index of the cavity mode, pulls the mode away from the light line, thus reducing the in-plane radiation loss. On the other hand, a large beam width will allow for higher order modes with the same symmetry as the fundamental mode of interest. Using band diagram simulations, we found that the width of 700nm is good trade-off between these two conditions (Fig. 2(b)).
- Set the filling fraction of the first mirror section such that its dielectric band-edge is at the adjusted frequency: 202THz in our case. Band diagram calculations based on unit cells are sufficient for this analysis. We found that an optimal filling fraction in our case is
*f*_{start}= 0.2 (Fig. 2(b)). - Find the filling fraction that produces the maximum mirror strength for the target frequency. This involves calculating the mirror strength for several filling fractions (Fig. 2(c)), each of which takes one or two minutes on a laptop computer. In our case we found that
*f*_{end}= 0.1. - Pick the number of mirror segments (
*N*) to construct the Gaussian mirror: we found that*N*≥ 15 (on each side) are generally good to achieve high radiation-*Q*s. - Create the Gaussian mirror by tapering the filling fractions quadratically from
*f*_{start}(=0.2 in our case) to*f*_{end}(=0.1) over the period of*N*segments. From the above analysis, the mirror strengths can be linearized through quadric tapering (Fig. 2(d)). - Finally, the cavity is formed by putting two Gaussian mirrors back to back, with no additional cavity length in between (
*L*= 0). To achieve a radiation-limited cavity (*Q*_{wg}>>*Q*_{rad}), 10 additional mirrors with the maximum mirror strength are placed on both ends of the Gaussian mirror. We will show in the next section, no additional mirrors are needed to achieve a waveguide-coupled cavity (*Q*_{rad}>>*Q*_{wg}).

Besides the structure that were proposed in [32] (Fig. 1), the alternative
structure which has the air-hole in the symmetry plane, as shown in Fig. 3(a), also satisfies (i)–(ix). Both
structures result in dielectric-mode cavities, since the bandgap of each mirror
segments red-shifts away from the center of the cavity, and thus a potential well is
created for the dielectric band-edge mode of the central segment. The difference is
that the energy maximum in the air-hole centered cavity is no longer located in the
middle of the structure, but instead in the dielectric region next to the central
hole (Fig. 3(b)). Figure 3(c) shows the *H _{z}* field
profile in the plane right above the cavity, obtained from FDTD simulation. Figure 3(d) shows the fitted field profile
using the same parameters that are used in the original structure shown in Figs. 2(g)–2(i), but with sine function
replaced by cosine function. Figure 3(e)
shows the

*H*distribution along the dashed line in Figs. 3(c) and 3(d).

_{z}Armed with the analytical field profile of the cavities: ${H}_{z}^{odd}(x)=\text{sin}(\pi x/a)\text{exp}(-\sigma {x}^{2})$ (Fig. 2) and ${H}_{z}^{even}(x)=\text{cos}(\pi x/a)\text{exp}(-\sigma {x}^{2})$ (Fig. 3), we
can obtain the radiation losses and far fields of the cavities using the Fourier
space analysis [35]. The
Fourier transforms can be analytically obtained $\text{FT}\left({H}_{z}^{odd}\right)=(\text{exp}(-{(k+\pi /a)}^{2}/4\sigma )-\text{exp}(-{(k-\pi /a)}^{2}/4\sigma ))/i\sqrt{8\sigma}$ and $\text{FT}\left({H}_{z}^{even}\right)=(\text{exp}(-{(k+\pi /a)}^{2}/4\sigma )+\text{exp}(-{(k-\pi /a)}^{2}/4\sigma ))/\sqrt{8\sigma}$. Under *σa*
^{2}
<< 1, both distributions have their Fourier components strongly
localized at *k* =
±*π*/*a*, as is verified by FDTD
simulations in Fig. 4(a) and 4(b). Since ${H}_{z}^{odd}(x)$ is an odd function, it always has a zero Fourier
component at *k* = 0. Therefore, dielectric-centered cavities
(Fig. 1) should have higher
*Q*-factors. However, in high-Q cavity designs,
*σa*
^{2} << 1 is satisfied and
thus both dielectric-centered and air-centered cavities have comparable
*Q*-factors. FDTD simulation shows that the above ${H}_{z}^{odd}$ and ${H}_{z}^{even}$ cavities have *Q*
_{tot}
= 3.8 × 10^{8} and *Q*
_{tot}
= 3.5 × 10^{8} respectively. The mode volume of the ${H}_{z}^{odd}$ cavity is
0.67(*λ*
_{res}/*n*
_{Si})^{3},
smaller than the ${H}_{z}^{even}$ cavity (*V* =
0.76(*λ*
_{res}/*n*
_{Si})^{3}).

The far field radiation patterns (obtained using FDTD simulations) of the two cavities are shown in Figs. 4(c) and 4(d). The powers, in both cases, are radiated at shallow angles (> 70° zenith angle) to the direction of the waveguide. The *H ^{odd}* cavity has even less radiated power at small zenith angles, consistent with the above analysis. By integrating the zenith and azimuth angle dependent far field emission, we found that 32% and 63% of the power emitted to +

*ẑ*direction can be collected by a NA=0.95 lens, respectively for

*H*cavity and

^{odd}*H*cavity.

^{even}## 3. Ultra-high *Q*, dielectric-mode photonic crystal nanobeam cavities

#### 3.1. Radiation-Q limited and waveguide-coupled cavities

Since the dielectric-centered ${H}_{z}^{odd}$ cavity has smaller *V* than the ${H}_{z}^{even}$ one, we focus our discussion in the ${H}_{z}^{odd}$ case. Using the above design algorithm, we
design the Gaussian mirror and put 10 additional mirrors with the maximum mirror
strength on both ends of the Gaussian mirror to obtain the radiation-limited
cavity (*Q*
_{wg} >>
*Q*
_{tot}). We find in Fig. 5 that *Q*
_{tot} increases
exponentially and *V* increases linearly as the total number of
mirror pairs in the Gaussian mirror (*N*) increases. A record
ultra-high *Q* of 5.0 × 10^{9} is achieved while
maintaining the small mode volume of 0.9 ×
(*λ*
_{res}/*n*
_{Si})^{3}
at *N* = 30.

Our design strategy has an additional important advantage over other types of photonic crystal
cavities [17–31], that is: the cavity naturally
couples to the feeding waveguide, as the hole radii decrease away from the
center of the cavity. High-*Q* and high transmissions
(*T*) cavities are possible with the above design steps
(i)–(ix), with no additional “coupling sections” needed.
We study *T* and *Q*
_{total} dependence on
the total number of mirror pair segments in the Gaussian mirror
(*N*) in Fig. 5(b).
Partial *Q*-factors (*Q*
_{rad},
*Q*
_{wg}) are obtained from FDTD simulations, and
*T* is obtained using $T={Q}_{\text{total}}^{2}/{Q}_{\text{wg}}^{2}$ [41]. As shown in Fig.
5(b), we achieve a nanobeam cavity with *Q* =
1.3 × 10^{7}, *T* = 97% at
*N* = 25.

#### 3.2. Higher order modes of the dielectric-mode cavity

The ultra-high *Q* mode that we deterministically designed is the fundamental mode of the cavity. Meanwhile, higher order cavity modes also exist. The number of higher order modes depends on the width of the photonic band gap and total number of mirror segments in the Gaussian mirror. To reduce the simulation time, we study the higher order modes of a waveguide-coupled cavity, that has a total number of 12 mirror pair segments, possessing a moderate *Q*-factor. Figure 6(a) shows the transmission spectrum obtained from FDTD simulation, by exciting the input waveguide with a waveguide mode, and monitoring the transmission through the cavity at the output waveguide. The band-edge modes are observed at wavelengths longer than 1.6*μ*m and shorter than 1.3*μ*m. Figures 6(b)–6(d) shows the major field-component (*E _{y}*) distribution of the three cavity modes. As expected, the eigenmodes alternate between symmetric and anti-symmetric modes. Symmetry plane is defined perpendicular to the beam direction, in the middle of the cavity (dashed line in Fig. 6). The total

*Q*-factors of modes I–III are 10,210, 1,077 and 286 respectively. Effective mode volumes of them are 0.55, 0.85 and 1.06 respectively. We note that transversely odd modes are well separated from the transversely symmetric cavity modes, hence were not considered in Fig. 6.

## 4. Ultra-high *Q*, air-mode photonic crystal nanobeam cavities

#### 4.1. Radiation-Q limited cavity

An air-mode cavity concentrates the optical energy in the low index region of the cavity.
Therefore, these cavities are of interest for applications where strong
interactions between light and material placed in the low index region of the
cavity is required, including nonlinear optics [4], optical trapping [7], biochemical sensing [8] and light-atom interaction
[43]. The ultra-high
*Q* air-mode nanobeam cavity is realized by pulling the
air-band mode of photonic crystal into its bandgap, which can also be designed
using the same design principles that we developed for dielectric-mode cavities.
In contrast to the dielectric-mode case, the resonant frequency of the air-mode
cavity is determined by the air band-edge frequency of the central mirror
segment. Then, to create the Gaussian confinement, the bandgaps of the mirror
segments should shift to higher frequencies as their distances from the center
of the cavity increase. This can be achieved by progressively increasing the
filling fractions of the mirror segments away from the center of the structure
(instead of decreasing in the dielectric-mode cavity case). One way to
accomplish this is to increase the size of the holes away from the center of the
cavity. While this may be suitable for non-waveguide coupled
(radiation-*Q* limited) cavities, it is not ideal for a
waveguide-coupled cavity. For this reason, we employ the design that relies on
tapering of the waveguide width instead of the hole size. Similar geometry was
recently proposed by Ahn et. al. [44] for the design of a dielectric-mode photonic crystal
laser.

The same design steps can be followed as in the dielectric-mode cavity case, with the following changes: First, the adjusted frequency (198THz) is 1% lower than the target frequency (200THz). (The thickness of the nanobeam is 220nm and period is 330nm, same as previous case.) Second, the nanobeam width at the center of the cavity is *w*
_{start} = 1*μ*m (Fig. 7(a)), with the hole radii kept constant at 100nm. Third, to create the Gaussian mirror, the beam widths are quadratically tapered from *w*
_{start} = 1*μ*m to *w*
_{end} = 0.7*μ*m, which produces the maximum mirror strength (band diagrams shown in Fig. 7(a)). This procedure involves calculating the mirror strength for several beam widths (Fig. 7(b)), each takes one or two minutes on a laptop computer. As shown in Fig. 7(c), the mirror strengths are linearized after the quadratic tapering. In order to achieve a radiation-*Q* limited cavity, 10 additional mirror segments are placed at both ends of the Gaussian mirror that has beam width *w*
_{end} = 0.7*μ*m.

Similar in the dielectric-mode cavity cases, ${H}_{z}^{odd}$ and ${H}_{z}^{even}$ air-mode cavities can be formed by placing the
air and dielectric in the central symmetric plane of the cavity, respectively.
Again, we will focus on ${H}_{z}^{odd}$, air-mode cavities and the conclusions will be
valid to the ${H}_{z}^{even}$ cavities as well. Figure 8(a) shows the total *Q* of
nanobeam cavities that have different total number of mirror pair segments in
the Gaussian mirrors. We have achieved a record ultra-high *Q* of
1.4 × 10^{9}, air-mode nanobeam cavity. As shown in Fig. 8(a), the effective mode volumes of
the air-mode cavities are much larger than the dielectric-mode cavities.

#### 4.2. Cavity strongly coupled to the feeding waveguide

As we have pointed out, the tapering-width approach (as compared to taping hole radii) offers a
natural way of coupling the nanobeam air-mode cavity to the feeding waveguide.
Since the width of the beam is decreasing, the cavity naturally couples to the
feeding waveguide. We study *T* and
*Q*
_{total} dependence on the total number of mirror
pair segments in the Gaussian mirror (*N*) using FDTD
simulations. As shown in Fig. 8(b), we
achieve nanobeam cavity with *Q* = 3.0 ×
10^{6}, *T* = 96% at
*N* = 25.

#### 4.3. Higher order modes of the air-mode cavity

The ultra-high *Q* cavity that we were able to design is the fundamental mode of the cavity. Higher order modes coexist with the fundamental modes inside the band gap. Fig. 9(a) shows the transmission spectrum of a waveguide-coupled air-mode nanobeam cavity, that has 15 mirror pair segments in the Gaussian mirror. The band-edge modes are observed at wavelengths longer than 1.6*μ*m. The modes in the range of 1.2*μ*m to 1.35*μ*m are formed by the higher order band modes in Fig. 7(a). Figures 9(b)–9(c) show the major field-component distribution (*E _{y}*) of the two cavity modes inside the bandgap. The total

*Q*-factors of these two modes are 23,935 and 5,525 respectively. The effective mode volumes are 2.32 and 3.01 respectively.

## 5. Conclusion

We have presented a detailed analysis and a deterministic design of the ultra-high
*Q* photonic crystal nanobeam cavities. With this method,
*Q* > 10^{9} radiation-limited cavity, and
*Q* > 10^{7}, *T* >
95% waveguide-coupled cavity are deterministically designed. These
*Q*-factors are comparable with those found in whispering gallery
mode (WGM) cavities [45–47]. Meanwhile, the mode volumes are typically
two or three orders of magnitude smaller than WGM ones. Furthermore, energy maximum
can be localized in either the dielectric region or air region with this method.
Although we demonstrate designs for TE-like, transversely symmetric cavity modes,
the design method is universal, and can be applied to realize nanobeam cavities that
support TM-polarized modes, as well as line-defect 2D photonic crystal cavities. We
believe that the proposed method will greatly ease the processes of high
*Q* nanobeam cavity design, and thus enable both fundamental
studies in strong light and matter interactions, and practical applications in novel
light sources, functional optical components (filters, delay lines, sensors) and
densely integrated photonic circuits.

## Acknowledgments

We acknowledge numerous fruitful discussions with M.W. MuCutcheon and P. B. Deotare. This work is supported by NSF Grant No. ECCS-0701417 and NSF CAREER grant.

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