Exceptional coupling in photonic anisotropic metamaterials for extremely low waveguide crosstalk

Md Borhan Mia; Syed Z. Ahmed; Ishtiaque Ahmed; Yun Jo Lee; Minghao Qi; Sangsik Kim; Sangsik Kim

doi:10.1364/OPTICA.394987

Exceptional coupling in photonic anisotropic metamaterials for extremely low waveguide crosstalk

Md Borhan Mia, Syed Z. Ahmed, Ishtiaque Ahmed, Yun Jo Lee, Minghao Qi, and Sangsik Kim

Optica
Vol. 7,
Issue 8,
pp. 881-887
(2020)
•https://doi.org/10.1364/OPTICA.394987

Get Citation
Copy Citation Text
Md Borhan Mia, Syed Z. Ahmed, Ishtiaque Ahmed, Yun Jo Lee, Minghao Qi, and Sangsik Kim, "Exceptional coupling in photonic anisotropic metamaterials for extremely low waveguide crosstalk," Optica 7, 881-887 (2020)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (3149 KB)
Set citation alerts
Save article

Accessible

Open Access

Abstract

Electromagnetic coupling is ubiquitous in photonic systems and transfers optical signals from one device to the other, creating crosstalk between devices. While this allows the functionality of some photonic components such as couplers, it limits the integration density of photonic chips, and many approaches have been proposed to reduce the crosstalk. However, due to the wave nature of light, complete elimination of crosstalk between closely spaced, identical waveguides is believed to be impossible and has not been observed experimentally. Here we show an exceptional coupling that can completely suppresses the crosstalk utilizing highly anisotropic photonic metamaterials. The anisotropic dielectric perturbations in the metamaterial mutually cancel the couplings from different field components, resulting in an infinitely long coupling length. We demonstrate the extreme suppression of crosstalk via exceptional coupling on a silicon-on-insulator platform, which is compatible with a complementary metal-oxide-semiconductor process. The idea of exceptional coupling with anisotropic metamaterials can be applied to many other electromagnetic devices, and it could drastically increase the integration density of photonic chips.

Corrections

MD Borhan Mia, Syed Z. Ahmed, Ishtiaque Ahmed, Yun Jo Lee, Minghao Qi, and Sangsik Kim, "Exceptional coupling in photonic anisotropic metamaterials for extremely low waveguide crosstalk: publisher’s note," Optica 7, 1408-1408 (2020)
http://proxy.osapublishing.org/optica/abstract.cfm?uri=optica-7-10-1408

29 July 2020: A typographical correction was made to paragraph 2 of page 885.

1. INTRODUCTION

Photonic chips can confine light down to hundreds of nanometer-scale waveguide cross sections, miniaturizing bulky tabletop optical systems into a single tiny chip. A compact, chip-scale device size allows the use of photonic systems outside of the laboratory, and the integration of photonic chips with electronic circuitry leads to a broad range of applications in high-speed optical communication [1,2], biochemical sensing [3,4], high-precision spectroscopy [5,6], and light detection and ranging [7–9]. The large index contrast between silicon (Si) and silicon dioxide (${{\rm SiO}_2}$) allows the integration of highly complicated photonic integrated circuits (PICs) on a silicon-on-insulator (SOI) wafer, and its compatibility with a complementary metal-oxide-semiconductor (CMOS) manufacturing process offers an opportunity for low-cost solutions with large-scale device fabrication. Furthermore, recent advances with high-$Q$ microresonators significantly enhance the light–matter interaction and have revolutionized research in nonlinear and quantum photonics [10–13], atomic physics [14,15], and time/frequency metrology [16,17]. In many applications, realizing a high-density photonic chip integration is highly desired, as more building blocks provide more functionalities on a single chip. It also could reduce the unit cost of mass production and lower the power consumption within the chip. However, further miniaturization of photonic chips is hampered by the wave nature of light, i.e., the evanescent wave in the cladding causes the waveguide crosstalk.

The waveguide crosstalk is the power transfer of a light signal between the adjacent waveguides due to the evanescent waves. In a typical photonic chip, to avoid the crosstalk, waveguides are required to be separated large enough, and this limits the integration density of photonic chips. To overcome this limit, there have been many research efforts directed toward reducing the crosstalk [18–27]. Plasmonics have been explored with their ability to confine light in the subwavelength scale [18–21]; however, metallic losses remain to be considered. Topological approaches that use a waveguide super-lattice [22,23], inverse design [24,25], or transformations optics [26] have been proposed, yet these approaches add more complexity or phase variations in the design and often lead to higher scattering losses. Recently, an extreme skin-depth (e-skid) waveguide that utilizes all-dielectric metamaterial claddings has been proposed to reduce the crosstalk [27]. The subwavelength multilayers effectively work as an anisotropic metamaterial [27–29] and could suppress the evanescent waves in the cladding. This reduces the crosstalk, and an $\approx 30$ times longer coupling length has been demonstrated compared to typical strip waveguides. However, even with the reduced skin-depth, there still is some degree of crosstalk, and the further question remains as to if it is possible to suppress the crosstalk completely.

Fig. 1. On-chip coupled waveguide configurations and exceptional coupling in coupled extreme skin-depth (e-skid) waveguides. (a)–(c) Schematic cross sections, geometric parameters, and mode profiles of the coupled silicon waveguides: (a) strip, (b) practical e-skid with subwavelength multilayers, and (c) ideal e-skid with effective medium theory (EMT). (d)–(f) Numerically simulated effective indices of the symmetric ${n_{\rm s}}$ (yellow solid) and anti-symmetric ${n_{\rm a}}$ (blue dashed) modes, and (g)–(i) their corresponding normalized coupling lengths ${L_{\rm c}}/{\lambda _0} = 1/(2|{n_{\rm s}} - {n_{\rm a}}|)$ (blue dots): (d) and (g) strip, (e) and (h) e-skid with multilayers, and (f) and (i) e-skid with EMT. All the simulations are performed as a function of the core width $w$, while fixing the other parameters as $h = 220\;{\rm nm} $, $\Lambda = 100\;{\rm nm} $, $\rho = 0.5$, and $N = 5$. The free space wavelength is ${\lambda _0} = 1550\;{\rm nm} $. The inset boxes of (d)–(f) show the zoomed-in view of each mode. The red-shaded areas in e-skid couplings show the non-trivial coupling regimes, where ${n_{\rm s}} \lt {n_{\rm a}}$, which cannot be observed in a typical strip coupling. The red arrows in e-skid couplings indicate the exceptional coupling points, where ${n_{\rm s}} \approx {n_{\rm a}}$, thus causing the ${L_{\rm c}} \to \infty$.

Download Full Size | PPT Slide | PDF

In this paper, we explore the limit of the waveguide crosstalk and show that there exists an exceptional coupling with anisotropic metamaterial cladding where the crosstalk can be suppressed completely. Our coupled mode analysis reveals that the anisotropic dielectric perturbation of metamaterial cladding causes a non-trivial coupled regime, where the modal index of the anti-symmetric mode is higher than that of the symmetric mode, and an exceptional coupling at the transition to the non-trivial coupling regime. We experimentally demonstrate such exceptional couplings on an SOI wafer, achieving extremely long coupling lengths. The engineering capability of the exceptional coupling is also presented both theoretically and experimentally.

2. EXCEPTIONAL COUPLING IN COUPLED EXTREME SKIN-DEPTH WAVEGUIDES

To push the limit of chip integration density, we explore the exceptional coupling phenomena in the e-skid waveguides by evaluating coupling lengths of the three different coupled waveguide schemes. Figures 1(a)–1(c) show the cross sections and geometric parameters of the three coupled waveguide configurations: (a) typical strip waveguides, (b) practical e-skid waveguides with subwavelength-scale multilayer claddings, and (c) ideal e-skid waveguides with anisotropic metamaterial claddings using the effective medium theory (EMT). All configurations are implemented on an SOI platform, i.e., Si and ${{\rm SiO}_2}$ as a core and a substrate, respectively. Throughout the paper, we explore the coupled modes between the two identical fundamental quasi-transverse-electric (quasi-${{\rm TE}_0}$) modes, and the electric field profiles ${\rm Re}({E_x})$ and ${\rm Im}({E_z})$ of the coupled symmetric (sym) and anti-symmetric (anti) modes are also plotted in Figs. 1(a)–1(c). The multilayer e-skid in Fig. 1(b) is a practical structure that can be fabricated with the current electron-beam lithography [27] and CMOS technology [30,31]. The EMT e-skid in Fig. 1(c) is an equivalent model with an anisotropic metamaterial, and its permittivities (${\varepsilon _x} = {\varepsilon _ \bot}$ and ${\varepsilon _y} = {\varepsilon _z} = {\varepsilon _\parallel}$) follow [32]

(1a)$${\varepsilon _\parallel} = \rho {\varepsilon _{{\rm Si}}} + (1 - \rho){\varepsilon _{{\rm air}}},$$(1b)$${\varepsilon _ \bot} = \frac{{{\varepsilon _{{\rm Si}}}{\varepsilon _{{\rm air}}}}}{{\rho {\varepsilon _{{\rm air}}} + (1 - \rho){\varepsilon _{{\rm Si}}}}},$$

where ${\varepsilon _{{\rm Si}}}$ and ${\varepsilon _{{\rm air}}}$ are the permittivities of Si and air, respectively. $\rho$ is the filling fraction of Si. Note that, due to the large index contrast between Si and air, a huge anisotropy can appear, and its anisotropy can be engineered by controlling the $\rho$. With the increased anisotropy, the skin-depth in the cladding can be reduced, lowering the crosstalk [27,33]. We set the periodicity $\Lambda$ and $\rho$ of the multilayers to be 100 nm and 0.5, respectively, considering the minimum feature size of 50 nm. Other parameters are set to $h = 220\;{\rm nm} $, $g = 550\;{\rm nm} $, and $N = 5$, unless otherwise specified.

The crosstalk between the two adjacent waveguides is assessed by the coupling length ${L_{\rm c}}$, which quantifies the length that transfers the optical power completely from one waveguide to the other waveguide, i.e., the crosstalk is lower for a longer ${L_{\rm c}}$, and it is the opposite for a shorter ${L_{\rm c}}$. To compare the coupling lengths of each configuration, effective refractive indices of the coupled symmetric (${n_{\rm s}}$, yellow solid) and anti-symmetric (${n_{\rm a}}$, blue dashed) modes are simulated in Figs. 1(d)–1(f), and their corresponding normalized coupling lengths (blue dots) are plotted in Figs. 1(g)–1(i): (d,g) strip, (e,h) e-skid with multilayer, and (f,i) e-skid with EMT. Each coupling length is normalized by the free-space wavelength at ${\lambda _0} = 1550\;{\rm nm} $, and the ${L_{\rm c}}$ of the two identical waveguides can be calculated by [34,35]

(2)$$\frac{{{L_{\rm c}}}}{{{\lambda _0}}} = \frac{1}{{2\Delta n}} = \frac{1}{{2|{n_{\rm s}} - {n_{\rm a}}|}},$$

where $\Delta n = |{n_{\rm s}} - {n_{\rm a}}|$ is the magnitude of the index difference between ${n_{\rm s}}$ and ${n_{\rm a}}$. The inset boxes in Figs. 1(d)–1(f) show the zoomed-in view of ${n_{\rm s}}$ and ${n_{\rm a}}$ at different regimes. Notice that, in the coupled e-skids of Figs. 1(e) and 1(f), there are non-trivial coupling regimes where ${n_{\rm s}} \lt {n_{\rm a}}$ (red-shaded region). This non-trivial coupling is not observable in typical strip waveguides [Fig. 1(d)], and this is not observable with other types of waveguides either. More importantly, at the transition from a typical coupling regime (${n_{\rm s}} \gt {n_{\rm a}}$) to the non-trivial coupling regime (${n_{\rm s}} \lt {n_{\rm a}}$), there is an exceptional coupling where $\Delta n$ approaches zero (i.e., ${n_{\rm s}} \approx {n_{\rm a}}$). As shown in Figs. 1(h) and 1(i), at these exceptional coupling points, the coupling length approaches to infinity, i.e., the crosstalk is suppressed completely. It is worth noting that our system does not involve any gain or loss (thus, a Hermitian system), but exchanges energy between the two waveguides and shows a singularity that completely cancels out the coupling [36]; thus, an exceptional coupling. For multilayer and EMT cases, the exceptional couplings appear at different $w$. This is due to the deviations of effective ${\varepsilon _ \bot}$ and ${\varepsilon _\parallel}$ of the multilayer compared to those of ideal EMT. As the $\Lambda$ is reduced, the effective indices of the multilayer approach those of ideal EMT, and the results in Figs. 1(e) and 1(h) will approach those in Figs. 1(f) and 1(i) (see Supplement 1).

3. ANISOTROPIC COUPLED MODE ANALYSIS ON THE EXCEPTIONAL COUPLING

To understand the underlying mechanism of the exceptional coupling, we analyzed each configuration using the anisotropic coupled mode analysis. In a quasi-${{\rm TE}_0}$ mode, an ${E_x}$ component is dominant, but there is an ${E_z}$ component as well. Thus, to address the coupled modes correctly, the anisotropic coupling coefficients from all the field components (i.e., ${\kappa _x},{\kappa _y}$, and ${\kappa _z}$) should be considered [34,35]:

(3)$${{\kappa }_{i}}=\frac{\omega {{\varepsilon }_{0}}}{4}\iint\Delta {{\varepsilon }_{i}}(x,y){{E}_{1i}}(x,y)E_{2i}^{*}(x,y){\rm d}x{\rm d}y,$$

where the subscript $i = x,y$, and $z$. ${E_{1i}}$ and ${E_{2i}}$ are the normalized electric fields of isolated (without coupling) quasi-${{\rm TE}_0}$ modes at each side, and $\Delta {\varepsilon _i}$ is the dielectric perturbation between them. Note that, for isotropic media, as in typical strip waveguides, all the dielectric perturbation components are the same (i.e., $\Delta {\varepsilon _x} = \Delta {\varepsilon _y} = \Delta {\varepsilon _z}$); however, for anisotropic cases, as in e-skid waveguides, they are different (i.e., $\Delta {\varepsilon _x} \ne \Delta {\varepsilon _y} = \Delta {\varepsilon _z}$), causing the non-trivial coupling regime. The overall coupling coefficient $|\kappa |$ can be obtained by adding each component together (i.e., $|\kappa | = |{\kappa _x} + {\kappa _y} + {\kappa _z}|$), and the coupling length of the two same waveguides is the following [34,35]:

(4)$${L_{\rm c}} = \frac{\pi}{{2|\kappa |}}.$$

Figures 2(a)–2(c) show the normalized coupling coefficients of each component ${\kappa _x}$ (blue dashed), ${\kappa _y}$ (orange dashed), and ${\kappa _z}$ (yellow dashed), and their corresponding overall coupling coefficient $|\kappa |$ (orange solid) and the normalized coupling length ${L_{\rm c}}/{\lambda _0}$ (blue dots) are plotted in Figs. 2(d)–2(f) and Figs. 2(g)–2(i), respectively: (a,d,g) coupled strip, and coupled e-skid with (b,e,h) the multilayer and (c,f,i) EMT. In every case, as the $w$ increases, the coupling coefficients are reduced, and the coupling lengths are increased, due to the higher confinement in the core and less overlap between the modes. In Fig. 2(a), the ${\kappa _x}$ is clearly dominant than the other components, even with a non-negligible ${\kappa _z}$. The sign of ${\kappa _z}$ is negative due to the imaginary ${E_z}$, and it counteracts with the ${\kappa _x}$ in determining the $|\kappa |$. In coupled strip waveguides, the magnitude of ${\kappa _x}$ is always greater than that of ${\kappa _z}$ (i.e., $\kappa \gt 0$) as the ${E_x}$ is dominant in the quasi-${{\rm TE}_0}$ mode. Figure 2(d) shows the overall $|\kappa |$ with the actual unit, and its corresponding normalized coupling length in Fig. 2(g) exactly matches the result from the full numerical simulation in Fig. 1(g). In case of coupled strip waveguides, the dielectric perturbation is isotropic (i.e., $\Delta {\varepsilon _x} = \Delta {\varepsilon _y} = \Delta {\varepsilon _z}$) and the magnitude of ${\kappa _z}$ would be always lower than that of ${\kappa _x}$. In cases of e-skids, as shown in the red-shaded regions of Figs. 2(b) and 2(c), there are non-trivial coupling regimes, where the magnitude of ${\kappa _z}$ is greater than that of ${\kappa _x}$ (i.e., $\kappa \lt 0$). These non-trivial coupling regimes in the coupled e-skids are due to the anisotropic dielectric perturbations $\Delta {\varepsilon _i}$ of the scheme (i.e., $\Delta {\varepsilon _x} \ne \Delta {\varepsilon _y} = \Delta {\varepsilon _z}$), allowing the ${\kappa _z}$ to compensate for the ${\kappa _x}$. The overall $|\kappa |$ approaches to zero at the transition points, resulting in infinitely long coupling lengths (i.e., ${L_{\rm c}} \to \infty$). The peaks in Figs. 2(h) and 2(i) correspond to these exceptional couplings, and these match well with the full simulation results in Figs. 1(h) and 1(i). Again, the anisotropic dielectric perturbation in the metamaterial cladding (i.e., $\Delta {\varepsilon _z} \gt \Delta {\varepsilon _x}$) is the fundamental origin of such non-trivial coupled regime ($\kappa \lt 0$) and the exceptional coupling ($|\kappa | \approx 0$), by reducing the coupling via ${E_x}$ component (${\kappa _x}$), while increasing the counteractive coupling via ${E_z}$ component (${\kappa _z}$). In the previous demonstration of e-skid in [27], this exceptional coupling was not observed due to a lower anisotropic perturbation with an upper cladding and different geometric parameters.

Fig. 2. Anisotropic coupled mode analysis on the exceptional coupling in coupled e-skid waveguides. (a)–(c) Normalized anisotropic coupling coefficients ${\kappa _x}$ (blue dashed), ${\kappa _y}$ (orange dashed), and ${\kappa _z}$ (yellow dashed) of the coupled (a) strip, (b) e-skid with multilayer, and (c) e-skid with EMT waveguides. Geometric parameters and the wavelength are the same as in Figs. 1. (d)–(f) Magnitude of the total coupling coefficient $|\kappa | = |{\kappa _x} + {\kappa _y} + {\kappa _z}|$ (orange solid), and (g)–(i) their corresponding normalized coupling lengths ${L_{\rm c}}/{\lambda _0} = \pi /(2|\kappa |{\lambda _0})$ (blue dots) for each configuration: (d) and (g) strip, (e) and (h) e-skid with multilayer, and (f) and (i) e-skid with EMT. The normalized coupling lengths in (g)–(i), which are obtained with anisotropic coupled mode analysis, match with those results in Figs. 1(g)–1(i) from the full numerical simulations. The red-shaded areas in e-skid couplings show the non-trivial coupling regimes where $\kappa \lt 0$, which cannot be observed in typical strip waveguide coupling. The red arrows in e-skid couplings indicate the exceptional coupling points where $|\kappa | \approx 0$, thus causing the ${L_{\rm c}} \to \infty$. As shown in (b) and (c), the anisotropic nature of e-skid waveguides can cause a larger ${\kappa _z}$, which results in the non-trivial coupling regime ($\kappa \lt 0$) and the exceptional coupling ($\kappa \approx 0$) at the transition.

Download Full Size | PPT Slide | PDF

4. EXPERIMENTAL DEMONSTRATION OF THE EXCEPTIONAL COUPLING AND EXTREME SUPPRESSION OF THE CROSSTALK

To confirm our theoretical finding, we fabricated the coupled e-skid (multilayer) and strip waveguides, and the crosstalks of each configuration were measured and compared. Figures 3(a) and 3(b) show the schematic views of the coupled e-skid and strip waveguides, respectively, and Fig. 3(c) shows the scanning electron microscope (SEM) images of the fabricated devices: (top) wide-view of the scheme, and zoomed-in views of the (left) coupled e-skids and (right) adiabatic transition from strip to e-skid (see Supplement 1). The ${I_0}$ indicates the optical power at the input port, and the ${I_1}$ and ${I_2}$ are the output powers at the through and coupled ports, respectively. The crosstalk is defined as the power ratio ${I_2}/{I_1}$, and it is related to the coupling length ${L_{\rm c}}$, following [34]

(5)$$\frac{{{I_2}}}{{{I_1}}} = \mathop {\tan}\nolimits^2 \left({\frac{{\pi\! L}}{{2{L_{\rm c}}}}} \right),$$

where $L$ is the physical length of the coupled waveguides. To measure the crosstalk, we sent a light signal to the input port ${I_0}$ through a fiber-coupled grating coupler. Then, output signals ${I_1}$ and ${I_2}$ were measured simultaneously (see Supplement 1). Figure 3(d) shows the experimentally measured crosstalk (in dB), and Fig. 3(e) is the corresponding normalized coupling length. Solid and dashed lines are the cases of the coupled e-skid and strip waveguides, respectively, and each color represents different core widths when $w = 420\;{\rm nm} $ (blue), 430 nm (orange), 440 nm (yellow), and 450 nm (purple). Figures 3(f) and 3(g) are the simulation results that correspond to Figs. 3(d) and 3(e), respectively. Note that the dips in the crosstalk and the peaks in the ${L_{\rm c}}/\lambda$ indicate the exceptional couplings. In Fig. 3(d), notice that the crosstalks of the coupled e-skids are suppressed down to about ${-}60\;{\rm dB} $, which is approximately 50 dB lower than that of the standard strip waveguides. As to the coupling length, the peak ${L_{\rm c}}/\lambda$ of the coupled e-skids are in the order of ${10^5}$, which is approximately 500 times longer than the case of strip waveguides. As shown in Figs. 3(f) and 3(g), in an ideal case of exceptional coupling, the crosstalk can be suppressed completely with an infinitely long coupling length. However, in real experiments, the minimum crosstalk is limited by the scattering from the waveguide sidewall roughness and the cross-coupling at the transition between strip to e-skid waveguides (see Supplement 1). Still, the crosstalk that we achieved here is extremely low, and to the best of our knowledge, these results demonstrate the longest coupling length, spanning about ${10^5}$ of free-space wavelengths. Note that the exceptional coupling phenomenon occurs on top of the reduced skin-depth effect of e-skid. Typical crosstalk suppression of e-skid without exceptional coupling in our scheme is around 20 dB (e.g., with different numbers of multilayer as in Supplement 1), and the extreme suppression of crosstalk can be achieved only with the exceptional coupling. While the reduced skin-depth effect of e-skid is almost independent of wavelength, there is a trade-off between the bandwidth $\Delta \lambda$ and the degree of crosstalk suppression $\Delta {\rm Crosstalk}$ in exceptional coupling. Still, the bandwidth of exceptional coupling is within the reasonable range of operation, i.e., $\Delta \lambda \approx 62.9 \pm 3.7\;{\rm nm} $ for $\Delta {\rm Crosstalk}= 30\;{\rm dB} $ and $\Delta \lambda \approx 16.0 \pm 2.1\;{\rm nm} $ for $\Delta {\rm Crosstalk} = 40\;{\rm dB} $ (see Supplement 1 for the full bandwidth analysis).

Fig. 3. Experimental demonstration of the exceptional coupling in coupled e-skid waveguides. Schematic view of the coupled (a) e-skid (multilayer) and (b) strip waveguides. ${I_0},{I_1}$, and ${I_2}$ indicate the optical powers at input, through, and coupled ports, respectively. (c) SEM images of the fabricated devices. Zoomed-in images show (left) the coupled e-skid waveguides and (right) the adiabatic transition from strip to e-skid waveguides. (d) Experimentally measured waveguide crosstalk and (e) the corresponding normalized coupling length of the coupled e-skid (solid) and strip (dashed) waveguides: $w = 420\;{\rm nm} $ (blue), 430 nm (orange), 440 nm (yellow), and 450 nm (purple). Numerically simulated (f) crosstalk and (g) normalized coupling length that correspond to the experimental results in (d) and (e), respectively. Geometric parameters are $h = 220\;{\rm nm} $, $\rho = 0.5$, $\Lambda = 100\;{\rm nm} $, $N = 5$, and $L = 100\,\,\unicode{x00B5}$m. Map plots of the measured crosstalk as functions of $\lambda$ and $w$ for the coupled (h) e-skid and (j) strip waveguides; (i) and (k) are their corresponding simulation results, respectively. Dark regions in (h) and (i) indicate the exceptional couplings in coupled e-skid waveguides.

Download Full Size | PPT Slide | PDF

The full map plots of the measured crosstalk, as functions of $\lambda$ and $w$, for the coupled e-skid and strip waveguides are plotted in Figs. 3(h) and 3(j), respectively, clearly showing much lower crosstalk with the e-skid waveguides. Figures 3(i) and 3(k) are their simulation results, respectively. The dark regions in Figs. 3(h) and 3(i) indicate the exceptional coupling, which can be observed only with the coupled e-skid waveguides. Since the exceptional coupling occurs at the point where the coupling coefficient ${\kappa _z}$ compensates the ${\kappa _x}$, the exceptional coupling can be engineered by controlling the modal overlaps between the two waveguides. For example, as shown in Figs. 3(h) and 3(i), increasing the $w$ shifts the exceptional coupling to a longer wavelength; this is because a wider $w$ increases the light confinement and reduces the modal overlap between the two coupled e-skids, while a longer wavelength works the opposite way. Similarly, changing the other geometric parameters $h,g,$ and $\rho$ shifts the exceptional coupling point, and we also observed exceptional couplings with different numbers of e-skid layers $N$ and filling fraction $\rho$, both numerically and experimentally (see Supplement 1).

5. CONCLUSION

In summary, we have presented exceptional couplings in the coupled e-skid waveguides that can achieve extremely low waveguide crosstalk. Our coupled mode analysis reveals that the unique anisotropic dielectric perturbation of the e-skid is the fundamental origin of the non-trivial coupling regime that can cause the exceptional coupling at the transition. We experimentally demonstrated the exceptional couplings on an SOI platform, which is low loss, low cost, and compatible with the CMOS foundry. At the exceptional coupling wavelength, the crosstalk suppression of $\approx 50\;{\rm dB} $ was achieved, which corresponds to $\approx 500$ times longer coupling length than the case of strip waveguides. There is a tradeoff between the bandwidth and the degree of crosstalk suppression, but within the range of practical operation. The exceptional coupling can be engineered with geometric parameters and the filling fraction of the e-skids. Our approach of using the exceptional coupling in the coupled e-skid waveguides drastically increases the photonic chip integration density and can be applied to design other ultracompact photonic devices realizing highly dense PICs.

Funding

National Science Foundation (ECCS-1930784); U.S. Department of Energy (DE-NA-0003525).

Acknowledgment

This work was performed, in part, at the Center for Integrated Nanotechnologies (CINT), an Office of Science User Facility operated for the U.S. Department of Energy Office of Science by Los Alamos National Laboratory and Sandia National Laboratories. The authors thank John Nogan and Anthony R. James of the CINT Integration Lab for their technical support in device fabrication.

Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

REFERENCES

1. P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, M. H. Anderson, R. Rosenberger, K. Vijayan, W. Freude, T. J. Kippenberg, and C. Koos, “Microresonator-based solitons for massively parallel coherent optical communications,” Nature 546, 274–279 (2017). [CrossRef]

2. A. H. Atabaki, S. Moazeni, F. Pavanello, H. Gevorgyan, J. Notaros, L. Alloatti, M. T. Wade, C. Sun, S. A. Kruger, H. Meng, K. A. Qubaisi, I. Wang, B. Zhang, A. Khilo, C. V. Baiocco, M. A. Popović, V. M. Stojanović, and R. J. Ram, “Integrating photonics with silicon nanoelectronics for the next generation of systems on a chip,” Nature 556, 349–354 (2018). [CrossRef]

3. X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics 5, 591–597 (2011). [CrossRef]

4. S. Lin and K. B. Crozier, “Trapping-assisted sensing of particles and proteins using on-chip optical microcavities,” ACS Nano 7, 1725–1730 (2013). [CrossRef]

5. M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016). [CrossRef]

6. A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018). [CrossRef]

7. S. A. Miller, Y.-C. Chang, C. T. Phare, M. C. Shin, M. Zadka, S. P. Roberts, B. Stern, X. Ji, A. Mohanty, O. A. J. Gordillo, U. D. Dave, and M. Lipson, “Large-scale optical phased array using a low-power multi-pass silicon photonic platform,” Optica 7, 3–6 (2020). [CrossRef]

8. J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013). [CrossRef]

9. S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, “Photonic waveguide to free-space Gaussian beam extreme mode converter,” Light Sci. Appl. 7, 72 (2018). [CrossRef]

10. Q. Li, M. Davanço, and K. Srinivasan, “Efficient and low-noise single-photon-level frequency conversion interfaces using silicon nanophotonics,” Nat. Photonics 10, 406–414 (2016). [CrossRef]

11. Z. Wang, T. Li, A. Soman, D. Mao, T. Kananen, and T. Gu, “On-chip wavefront shaping with dielectric metasurface,” Nat. Commun. 10, 3547 (2019). [CrossRef]

12. B. Hausmann, I. Bulu, V. Venkataraman, P. Deotare, and M. Lončar, “Diamond nonlinear photonics,” Nat. Photonics 8, 369–374 (2014). [CrossRef]

13. S. Kim, K. Han, C. Wang, J. A. Jaramillo-Villegas, X. Xue, C. Bao, Y. Xuan, D. E. Leaird, A. M. Weiner, and M. Qi, “Dispersion engineering and frequency comb generation in thin silicon nitride concentric microresonators,” Nat. Commun. 8, 372 (2017). [CrossRef]

14. M. T. Hummon, S. Kang, D. G. Bopp, Q. Li, D. A. Westly, S. Kim, C. Fredrick, S. A. Diddams, K. Srinivasan, V. Aksyuk, and J. E. Kitching, “Photonic chip for laser stabilization to an atomic vapor with 10-11 instability,” Optica 5, 443–449 (2018). [CrossRef]

15. K. K. Mehta, C. D. Bruzewicz, R. McConnell, R. J. Ram, J. M. Sage, and J. Chiaverini, “Integrated optical addressing of an ion qubit,” Nat. Nanotech. 11, 1066–1070 (2016). [CrossRef]

16. T. J. Kippenberg, R. Holzwarth, and S. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011). [CrossRef]

17. D. Spencer, T. Drake, T. Briles, J. Stone, L. Sinclair, C. Fredrick, Q. Li, D. Westly, B. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018). [CrossRef]

18. R. F. Oulton, V. J. Sorger, D. Genov, D. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008). [CrossRef]

19. S. Kim and M. Qi, “Mode-evolution-based polarization rotation and coupling between silicon and hybrid plasmonic waveguides,” Sci. Rep. 5, 18378 (2015). [CrossRef]

20. C. Haffner, W. Heni, Y. Fedoryshyn, J. Niegemann, A. Melikyan, D. L. Elder, B. Baeuerle, Y. Salamin, A. Josten, U. Koch, C. Hoessbacher, F. Ducry, J. Juchli, A. Emboras, D. Hillerkuss, M. Kohl, L. R. Dalton, C. Hafner, and J. Leuthold, “All-plasmonic Mach–Zehnder modulator enabling optical high-speed communication at the microscale,” Nat. Photonics 9, 525–528 (2015). [CrossRef]

21. S. Kim and M. Qi, “Polarization rotation and coupling between silicon waveguide and hybrid plasmonic waveguide,” Opt. Express 23, 9968–9978 (2015). [CrossRef]

22. W. Song, R. Gatdula, S. Abbaslou, M. Lu, A. Stein, W. Y. Lai, J. Provine, R. F. W. Pease, D. N. Christodoulides, and W. Jiang, “High-density waveguide superlattices with low crosstalk,” Nat. Commun. 6, 7027 (2015). [CrossRef]

23. R. Gatdula, S. Abbaslou, M. Lu, A. Stein, and W. Jiang, “Guiding light in bent waveguide superlattices with low crosstalk,” Optica 6, 585–591 (2019). [CrossRef]

24. B. Shen, R. Polson, and R. Menon, “Metamaterial-waveguide bends with effective bend radius λ₀/2,” Opt. Lett. 40, 5750–5753 (2015). [CrossRef]

25. B. Shen, R. Polson, and R. Menon, “Increasing the density of passive photonic-integrated circuits via nanophotonic cloaking,” Nat. Commun. 7, 13126 (2016). [CrossRef]

26. L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012). [CrossRef]

27. S. Jahani, S. Kim, J. Atkinson, J. C. Wirth, F. Kalhor, A. A. Noman, W. D. Newman, P. Shekhar, K. Han, V. Van, R. G. DeCorby, L. Chrostowski, M. Qi, and Z. Jacob, “Controlling evanescent waves using silicon photonic all-dielectric metamaterials for dense integration,” Nat. Commun. 9, 1893 (2018). [CrossRef]

28. P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2018). [CrossRef]

29. R. Halir, P. Cheben, J. M. Luque-González, J. D. Sarmiento-Merenguel, J. H. Schmid, G. Wangüemert-Pérez, D.-X. Xu, S. Wang, A. Ortega-Moñux, and Í. Molina-Fernández, “Ultra-broadband nanophotonic beamsplitter using an anisotropic sub-wavelength metamaterial,” Laser Photon. Rev. 10, 1039–1046 (2016). [CrossRef]

30. J. S. Orcutt, A. Khilo, C. W. Holzwarth, M. A. Popović, H. Li, J. Sun, T. Bonifield, R. Hollingsworth, F. X. Kärtner, H. I. Smith, V. Stojanović, and R. J. Ram, “Nanophotonic integration in state-of-the-art CMOS foundries,” Opt. Express 19, 2335–2346 (2011). [CrossRef]

31. V. Stojanović, R. J. Ram, M. Popović, S. Lin, S. Moazeni, M. Wade, C. Sun, L. Alloatti, A. Atabaki, F. Pavanello, N. Mehta, and P. Bhargava, “Monolithic silicon-photonic platforms in state-of-the-art CMOS SOI processes,” Opt. Express 26, 13106–13121 (2018). [CrossRef]

32. G. W. Milton, The Theory of Composites (Cambridge University, 2002).

33. S. Jahani and Z. Jacob, “Transparent subdiffraction optics: nanoscale light confinement without metal,” Optica 1, 96–100 (2014). [CrossRef]

34. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2006).

35. W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994). [CrossRef]

36. M. A. Miri and A. Alu, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019). [CrossRef]

References

View by:
Article Order
|
Year
|
Author
|
Publication

P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, M. H. Anderson, R. Rosenberger, K. Vijayan, W. Freude, T. J. Kippenberg, and C. Koos, “Microresonator-based solitons for massively parallel coherent optical communications,” Nature 546, 274–279 (2017).
[Crossref]
A. H. Atabaki, S. Moazeni, F. Pavanello, H. Gevorgyan, J. Notaros, L. Alloatti, M. T. Wade, C. Sun, S. A. Kruger, H. Meng, K. A. Qubaisi, I. Wang, B. Zhang, A. Khilo, C. V. Baiocco, M. A. Popović, V. M. Stojanović, and R. J. Ram, “Integrating photonics with silicon nanoelectronics for the next generation of systems on a chip,” Nature 556, 349–354 (2018).
[Crossref]
X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics 5, 591–597 (2011).
[Crossref]
S. Lin and K. B. Crozier, “Trapping-assisted sensing of particles and proteins using on-chip optical microcavities,” ACS Nano 7, 1725–1730 (2013).
[Crossref]
M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref]
A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018).
[Crossref]
S. A. Miller, Y.-C. Chang, C. T. Phare, M. C. Shin, M. Zadka, S. P. Roberts, B. Stern, X. Ji, A. Mohanty, O. A. J. Gordillo, U. D. Dave, and M. Lipson, “Large-scale optical phased array using a low-power multi-pass silicon photonic platform,” Optica 7, 3–6 (2020).
[Crossref]
J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]
S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, “Photonic waveguide to free-space Gaussian beam extreme mode converter,” Light Sci. Appl. 7, 72 (2018).
[Crossref]
Q. Li, M. Davanço, and K. Srinivasan, “Efficient and low-noise single-photon-level frequency conversion interfaces using silicon nanophotonics,” Nat. Photonics 10, 406–414 (2016).
[Crossref]
Z. Wang, T. Li, A. Soman, D. Mao, T. Kananen, and T. Gu, “On-chip wavefront shaping with dielectric metasurface,” Nat. Commun. 10, 3547 (2019).
[Crossref]
B. Hausmann, I. Bulu, V. Venkataraman, P. Deotare, and M. Lončar, “Diamond nonlinear photonics,” Nat. Photonics 8, 369–374 (2014).
[Crossref]
S. Kim, K. Han, C. Wang, J. A. Jaramillo-Villegas, X. Xue, C. Bao, Y. Xuan, D. E. Leaird, A. M. Weiner, and M. Qi, “Dispersion engineering and frequency comb generation in thin silicon nitride concentric microresonators,” Nat. Commun. 8, 372 (2017).
[Crossref]
M. T. Hummon, S. Kang, D. G. Bopp, Q. Li, D. A. Westly, S. Kim, C. Fredrick, S. A. Diddams, K. Srinivasan, V. Aksyuk, and J. E. Kitching, “Photonic chip for laser stabilization to an atomic vapor with 10-11 instability,” Optica 5, 443–449 (2018).
[Crossref]
K. K. Mehta, C. D. Bruzewicz, R. McConnell, R. J. Ram, J. M. Sage, and J. Chiaverini, “Integrated optical addressing of an ion qubit,” Nat. Nanotech. 11, 1066–1070 (2016).
[Crossref]
T. J. Kippenberg, R. Holzwarth, and S. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]
D. Spencer, T. Drake, T. Briles, J. Stone, L. Sinclair, C. Fredrick, Q. Li, D. Westly, B. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]
R. F. Oulton, V. J. Sorger, D. Genov, D. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[Crossref]
S. Kim and M. Qi, “Mode-evolution-based polarization rotation and coupling between silicon and hybrid plasmonic waveguides,” Sci. Rep. 5, 18378 (2015).
[Crossref]
C. Haffner, W. Heni, Y. Fedoryshyn, J. Niegemann, A. Melikyan, D. L. Elder, B. Baeuerle, Y. Salamin, A. Josten, U. Koch, C. Hoessbacher, F. Ducry, J. Juchli, A. Emboras, D. Hillerkuss, M. Kohl, L. R. Dalton, C. Hafner, and J. Leuthold, “All-plasmonic Mach–Zehnder modulator enabling optical high-speed communication at the microscale,” Nat. Photonics 9, 525–528 (2015).
[Crossref]
S. Kim and M. Qi, “Polarization rotation and coupling between silicon waveguide and hybrid plasmonic waveguide,” Opt. Express 23, 9968–9978 (2015).
[Crossref]
W. Song, R. Gatdula, S. Abbaslou, M. Lu, A. Stein, W. Y. Lai, J. Provine, R. F. W. Pease, D. N. Christodoulides, and W. Jiang, “High-density waveguide superlattices with low crosstalk,” Nat. Commun. 6, 7027 (2015).
[Crossref]
R. Gatdula, S. Abbaslou, M. Lu, A. Stein, and W. Jiang, “Guiding light in bent waveguide superlattices with low crosstalk,” Optica 6, 585–591 (2019).
[Crossref]
B. Shen, R. Polson, and R. Menon, “Metamaterial-waveguide bends with effective bend radius λ0/2,” Opt. Lett. 40, 5750–5753 (2015).
[Crossref]
B. Shen, R. Polson, and R. Menon, “Increasing the density of passive photonic-integrated circuits via nanophotonic cloaking,” Nat. Commun. 7, 13126 (2016).
[Crossref]
L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
[Crossref]
S. Jahani, S. Kim, J. Atkinson, J. C. Wirth, F. Kalhor, A. A. Noman, W. D. Newman, P. Shekhar, K. Han, V. Van, R. G. DeCorby, L. Chrostowski, M. Qi, and Z. Jacob, “Controlling evanescent waves using silicon photonic all-dielectric metamaterials for dense integration,” Nat. Commun. 9, 1893 (2018).
[Crossref]
P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2018).
[Crossref]
R. Halir, P. Cheben, J. M. Luque-González, J. D. Sarmiento-Merenguel, J. H. Schmid, G. Wangüemert-Pérez, D.-X. Xu, S. Wang, A. Ortega-Moñux, and Í. Molina-Fernández, “Ultra-broadband nanophotonic beamsplitter using an anisotropic sub-wavelength metamaterial,” Laser Photon. Rev. 10, 1039–1046 (2016).
[Crossref]
J. S. Orcutt, A. Khilo, C. W. Holzwarth, M. A. Popović, H. Li, J. Sun, T. Bonifield, R. Hollingsworth, F. X. Kärtner, H. I. Smith, V. Stojanović, and R. J. Ram, “Nanophotonic integration in state-of-the-art CMOS foundries,” Opt. Express 19, 2335–2346 (2011).
[Crossref]
V. Stojanović, R. J. Ram, M. Popović, S. Lin, S. Moazeni, M. Wade, C. Sun, L. Alloatti, A. Atabaki, F. Pavanello, N. Mehta, and P. Bhargava, “Monolithic silicon-photonic platforms in state-of-the-art CMOS SOI processes,” Opt. Express 26, 13106–13121 (2018).
[Crossref]
G. W. Milton, The Theory of Composites (Cambridge University, 2002).
S. Jahani and Z. Jacob, “Transparent subdiffraction optics: nanoscale light confinement without metal,” Optica 1, 96–100 (2014).
[Crossref]
A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2006).
W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[Crossref]
M. A. Miri and A. Alu, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

2020 (1)

S. A. Miller, Y.-C. Chang, C. T. Phare, M. C. Shin, M. Zadka, S. P. Roberts, B. Stern, X. Ji, A. Mohanty, O. A. J. Gordillo, U. D. Dave, and M. Lipson, “Large-scale optical phased array using a low-power multi-pass silicon photonic platform,” Optica 7, 3–6 (2020).
[Crossref]

2019 (3)

Z. Wang, T. Li, A. Soman, D. Mao, T. Kananen, and T. Gu, “On-chip wavefront shaping with dielectric metasurface,” Nat. Commun. 10, 3547 (2019).
[Crossref]

R. Gatdula, S. Abbaslou, M. Lu, A. Stein, and W. Jiang, “Guiding light in bent waveguide superlattices with low crosstalk,” Optica 6, 585–591 (2019).
[Crossref]

M. A. Miri and A. Alu, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

2018 (8)

V. Stojanović, R. J. Ram, M. Popović, S. Lin, S. Moazeni, M. Wade, C. Sun, L. Alloatti, A. Atabaki, F. Pavanello, N. Mehta, and P. Bhargava, “Monolithic silicon-photonic platforms in state-of-the-art CMOS SOI processes,” Opt. Express 26, 13106–13121 (2018).
[Crossref]

S. Jahani, S. Kim, J. Atkinson, J. C. Wirth, F. Kalhor, A. A. Noman, W. D. Newman, P. Shekhar, K. Han, V. Van, R. G. DeCorby, L. Chrostowski, M. Qi, and Z. Jacob, “Controlling evanescent waves using silicon photonic all-dielectric metamaterials for dense integration,” Nat. Commun. 9, 1893 (2018).
[Crossref]

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2018).
[Crossref]

S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, “Photonic waveguide to free-space Gaussian beam extreme mode converter,” Light Sci. Appl. 7, 72 (2018).
[Crossref]

M. T. Hummon, S. Kang, D. G. Bopp, Q. Li, D. A. Westly, S. Kim, C. Fredrick, S. A. Diddams, K. Srinivasan, V. Aksyuk, and J. E. Kitching, “Photonic chip for laser stabilization to an atomic vapor with 10-11 instability,” Optica 5, 443–449 (2018).
[Crossref]

D. Spencer, T. Drake, T. Briles, J. Stone, L. Sinclair, C. Fredrick, Q. Li, D. Westly, B. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018).
[Crossref]

A. H. Atabaki, S. Moazeni, F. Pavanello, H. Gevorgyan, J. Notaros, L. Alloatti, M. T. Wade, C. Sun, S. A. Kruger, H. Meng, K. A. Qubaisi, I. Wang, B. Zhang, A. Khilo, C. V. Baiocco, M. A. Popović, V. M. Stojanović, and R. J. Ram, “Integrating photonics with silicon nanoelectronics for the next generation of systems on a chip,” Nature 556, 349–354 (2018).
[Crossref]

2017 (2)

P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, M. H. Anderson, R. Rosenberger, K. Vijayan, W. Freude, T. J. Kippenberg, and C. Koos, “Microresonator-based solitons for massively parallel coherent optical communications,” Nature 546, 274–279 (2017).
[Crossref]

S. Kim, K. Han, C. Wang, J. A. Jaramillo-Villegas, X. Xue, C. Bao, Y. Xuan, D. E. Leaird, A. M. Weiner, and M. Qi, “Dispersion engineering and frequency comb generation in thin silicon nitride concentric microresonators,” Nat. Commun. 8, 372 (2017).
[Crossref]

2016 (5)

R. Halir, P. Cheben, J. M. Luque-González, J. D. Sarmiento-Merenguel, J. H. Schmid, G. Wangüemert-Pérez, D.-X. Xu, S. Wang, A. Ortega-Moñux, and Í. Molina-Fernández, “Ultra-broadband nanophotonic beamsplitter using an anisotropic sub-wavelength metamaterial,” Laser Photon. Rev. 10, 1039–1046 (2016).
[Crossref]

B. Shen, R. Polson, and R. Menon, “Increasing the density of passive photonic-integrated circuits via nanophotonic cloaking,” Nat. Commun. 7, 13126 (2016).
[Crossref]

M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref]

K. K. Mehta, C. D. Bruzewicz, R. McConnell, R. J. Ram, J. M. Sage, and J. Chiaverini, “Integrated optical addressing of an ion qubit,” Nat. Nanotech. 11, 1066–1070 (2016).
[Crossref]

Q. Li, M. Davanço, and K. Srinivasan, “Efficient and low-noise single-photon-level frequency conversion interfaces using silicon nanophotonics,” Nat. Photonics 10, 406–414 (2016).
[Crossref]

2015 (5)

B. Shen, R. Polson, and R. Menon, “Metamaterial-waveguide bends with effective bend radius λ0/2,” Opt. Lett. 40, 5750–5753 (2015).
[Crossref]

S. Kim and M. Qi, “Mode-evolution-based polarization rotation and coupling between silicon and hybrid plasmonic waveguides,” Sci. Rep. 5, 18378 (2015).
[Crossref]

C. Haffner, W. Heni, Y. Fedoryshyn, J. Niegemann, A. Melikyan, D. L. Elder, B. Baeuerle, Y. Salamin, A. Josten, U. Koch, C. Hoessbacher, F. Ducry, J. Juchli, A. Emboras, D. Hillerkuss, M. Kohl, L. R. Dalton, C. Hafner, and J. Leuthold, “All-plasmonic Mach–Zehnder modulator enabling optical high-speed communication at the microscale,” Nat. Photonics 9, 525–528 (2015).
[Crossref]

S. Kim and M. Qi, “Polarization rotation and coupling between silicon waveguide and hybrid plasmonic waveguide,” Opt. Express 23, 9968–9978 (2015).
[Crossref]

W. Song, R. Gatdula, S. Abbaslou, M. Lu, A. Stein, W. Y. Lai, J. Provine, R. F. W. Pease, D. N. Christodoulides, and W. Jiang, “High-density waveguide superlattices with low crosstalk,” Nat. Commun. 6, 7027 (2015).
[Crossref]

2014 (2)

S. Jahani and Z. Jacob, “Transparent subdiffraction optics: nanoscale light confinement without metal,” Optica 1, 96–100 (2014).
[Crossref]

B. Hausmann, I. Bulu, V. Venkataraman, P. Deotare, and M. Lončar, “Diamond nonlinear photonics,” Nat. Photonics 8, 369–374 (2014).
[Crossref]

2013 (2)

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]

S. Lin and K. B. Crozier, “Trapping-assisted sensing of particles and proteins using on-chip optical microcavities,” ACS Nano 7, 1725–1730 (2013).
[Crossref]

2012 (1)

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
[Crossref]

2011 (3)

J. S. Orcutt, A. Khilo, C. W. Holzwarth, M. A. Popović, H. Li, J. Sun, T. Bonifield, R. Hollingsworth, F. X. Kärtner, H. I. Smith, V. Stojanović, and R. J. Ram, “Nanophotonic integration in state-of-the-art CMOS foundries,” Opt. Express 19, 2335–2346 (2011).
[Crossref]

X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics 5, 591–597 (2011).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

2008 (1)

R. F. Oulton, V. J. Sorger, D. Genov, D. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[Crossref]

1994 (1)

W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[Crossref]

Abbaslou, S.

R. Gatdula, S. Abbaslou, M. Lu, A. Stein, and W. Jiang, “Guiding light in bent waveguide superlattices with low crosstalk,” Optica 6, 585–591 (2019).
[Crossref]

Aksyuk, V.

Aksyuk, V. A.

Alloatti, L.

Alu, A.

M. A. Miri and A. Alu, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

Anderson, M. H.

Atabaki, A.

Atabaki, A. H.

Atkinson, J.

Atwater, H. A.

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2018).
[Crossref]

Baeuerle, B.

Baiocco, C. V.

Bao, C.

Bhargava, P.

Bluestone, A.

Bonifield, T.

Bopp, D. G.

Bowers, J. E.

Brasch, V.

Briles, T.

Bruzewicz, C. D.

K. K. Mehta, C. D. Bruzewicz, R. McConnell, R. J. Ram, J. M. Sage, and J. Chiaverini, “Integrated optical addressing of an ion qubit,” Nat. Nanotech. 11, 1066–1070 (2016).
[Crossref]

Bulu, I.

B. Hausmann, I. Bulu, V. Venkataraman, P. Deotare, and M. Lončar, “Diamond nonlinear photonics,” Nat. Photonics 8, 369–374 (2014).
[Crossref]

Cardenas, J.

A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018).
[Crossref]

Chang, L.

Chang, Y.-C.

Cheben, P.

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2018).
[Crossref]

Chiaverini, J.

K. K. Mehta, C. D. Bruzewicz, R. McConnell, R. J. Ram, J. M. Sage, and J. Chiaverini, “Integrated optical addressing of an ion qubit,” Nat. Nanotech. 11, 1066–1070 (2016).
[Crossref]

Christodoulides, D. N.

Chrostowski, L.

Crozier, K. B.

S. Lin and K. B. Crozier, “Trapping-assisted sensing of particles and proteins using on-chip optical microcavities,” ACS Nano 7, 1725–1730 (2013).
[Crossref]

Dalton, L. R.

Davanço, M.

Q. Li, M. Davanço, and K. Srinivasan, “Efficient and low-noise single-photon-level frequency conversion interfaces using silicon nanophotonics,” Nat. Photonics 10, 406–414 (2016).
[Crossref]

Dave, U. D.

DeCorby, R. G.

Deotare, P.

B. Hausmann, I. Bulu, V. Venkataraman, P. Deotare, and M. Lončar, “Diamond nonlinear photonics,” Nat. Photonics 8, 369–374 (2014).
[Crossref]

Diddams, S.

T. J. Kippenberg, R. Holzwarth, and S. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

Diddams, S. A.

Drake, T.

Ducry, F.

Dutt, A.

A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018).
[Crossref]

Elder, D. L.

Emboras, A.

Fan, X.

X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics 5, 591–597 (2011).
[Crossref]

Fedoryshyn, Y.

Fredrick, C.

Freude, W.

Gabrielli, L. H.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
[Crossref]

Gaeta, A. L.

A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018).
[Crossref]

Gatdula, R.

R. Gatdula, S. Abbaslou, M. Lu, A. Stein, and W. Jiang, “Guiding light in bent waveguide superlattices with low crosstalk,” Optica 6, 585–591 (2019).
[Crossref]

Genov, D.

Gevorgyan, H.

Gordillo, O. A. J.

Gu, T.

Z. Wang, T. Li, A. Soman, D. Mao, T. Kananen, and T. Gu, “On-chip wavefront shaping with dielectric metasurface,” Nat. Commun. 10, 3547 (2019).
[Crossref]

Haffner, C.

Hafner, C.

Halir, R.

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2018).
[Crossref]

Han, K.

Hausmann, B.

B. Hausmann, I. Bulu, V. Venkataraman, P. Deotare, and M. Lončar, “Diamond nonlinear photonics,” Nat. Photonics 8, 369–374 (2014).
[Crossref]

Heni, W.

Hillerkuss, D.

Hoessbacher, C.

Hollingsworth, R.

Holzwarth, C. W.

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, and S. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

Hosseini, E. S.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]

Huang, W. P.

W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[Crossref]

Hummon, M. T.

Ilic, B.

Jacob, Z.

S. Jahani and Z. Jacob, “Transparent subdiffraction optics: nanoscale light confinement without metal,” Optica 1, 96–100 (2014).
[Crossref]

Jahani, S.

S. Jahani and Z. Jacob, “Transparent subdiffraction optics: nanoscale light confinement without metal,” Optica 1, 96–100 (2014).
[Crossref]

Jaramillo-Villegas, J. A.

Ji, X.

A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018).
[Crossref]

Jiang, W.

R. Gatdula, S. Abbaslou, M. Lu, A. Stein, and W. Jiang, “Guiding light in bent waveguide superlattices with low crosstalk,” Optica 6, 585–591 (2019).
[Crossref]

Johnson, S. G.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
[Crossref]

Joshi, C.

A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018).
[Crossref]

Josten, A.

Juchli, J.

Kalhor, F.

Kananen, T.

Z. Wang, T. Li, A. Soman, D. Mao, T. Kananen, and T. Gu, “On-chip wavefront shaping with dielectric metasurface,” Nat. Commun. 10, 3547 (2019).
[Crossref]

Kang, S.

Karpov, M.

Kärtner, F. X.

Kemal, J. N.

Khilo, A.

Kim, S.

S. Kim and M. Qi, “Polarization rotation and coupling between silicon waveguide and hybrid plasmonic waveguide,” Opt. Express 23, 9968–9978 (2015).
[Crossref]

S. Kim and M. Qi, “Mode-evolution-based polarization rotation and coupling between silicon and hybrid plasmonic waveguides,” Sci. Rep. 5, 18378 (2015).
[Crossref]

Kippenberg, T. J.

T. J. Kippenberg, R. Holzwarth, and S. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

Kitching, J. E.

Koch, U.

Kohl, M.

Komljenovic, T.

Koos, C.

Kordts, A.

Kruger, S. A.

Lai, W. Y.

Leaird, D. E.

Lee, S. H.

Leuthold, J.

Li, H.

Li, Q.

Q. Li, M. Davanço, and K. Srinivasan, “Efficient and low-noise single-photon-level frequency conversion interfaces using silicon nanophotonics,” Nat. Photonics 10, 406–414 (2016).
[Crossref]

Li, T.

Z. Wang, T. Li, A. Soman, D. Mao, T. Kananen, and T. Gu, “On-chip wavefront shaping with dielectric metasurface,” Nat. Commun. 10, 3547 (2019).
[Crossref]

Lin, S.

S. Lin and K. B. Crozier, “Trapping-assisted sensing of particles and proteins using on-chip optical microcavities,” ACS Nano 7, 1725–1730 (2013).
[Crossref]

Lipson, M.

A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018).
[Crossref]

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
[Crossref]

Liu, D.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
[Crossref]

Loncar, M.

B. Hausmann, I. Bulu, V. Venkataraman, P. Deotare, and M. Lončar, “Diamond nonlinear photonics,” Nat. Photonics 8, 369–374 (2014).
[Crossref]

Lu, M.

R. Gatdula, S. Abbaslou, M. Lu, A. Stein, and W. Jiang, “Guiding light in bent waveguide superlattices with low crosstalk,” Optica 6, 585–591 (2019).
[Crossref]

Luke, K.

A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018).
[Crossref]

Luque-González, J. M.

Mao, D.

Z. Wang, T. Li, A. Soman, D. Mao, T. Kananen, and T. Gu, “On-chip wavefront shaping with dielectric metasurface,” Nat. Commun. 10, 3547 (2019).
[Crossref]

Marin-Palomo, P.

McConnell, R.

K. K. Mehta, C. D. Bruzewicz, R. McConnell, R. J. Ram, J. M. Sage, and J. Chiaverini, “Integrated optical addressing of an ion qubit,” Nat. Nanotech. 11, 1066–1070 (2016).
[Crossref]

Mehta, K. K.

K. K. Mehta, C. D. Bruzewicz, R. McConnell, R. J. Ram, J. M. Sage, and J. Chiaverini, “Integrated optical addressing of an ion qubit,” Nat. Nanotech. 11, 1066–1070 (2016).
[Crossref]

Mehta, N.

Melikyan, A.

Meng, H.

Menon, R.

B. Shen, R. Polson, and R. Menon, “Increasing the density of passive photonic-integrated circuits via nanophotonic cloaking,” Nat. Commun. 7, 13126 (2016).
[Crossref]

B. Shen, R. Polson, and R. Menon, “Metamaterial-waveguide bends with effective bend radius λ0/2,” Opt. Lett. 40, 5750–5753 (2015).
[Crossref]

Miller, S. A.

Milton, G. W.

G. W. Milton, The Theory of Composites (Cambridge University, 2002).

Miri, M. A.

M. A. Miri and A. Alu, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

Moazeni, S.

Mohanty, A.

Molina-Fernández, Í.

Newbury, N. R.

Newman, W. D.

Niegemann, J.

Noman, A. A.

Norberg, E.

Notaros, J.

Oh, D. Y.

Okawachi, Y.

A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018).
[Crossref]

Orcutt, J. S.

Ortega-Moñux, A.

Oulton, R. F.

Papp, S. B.

Pavanello, F.

Pease, R. F. W.

Pfeiffer, M. H.

Pfeiffer, M. H. P.

Pfeifle, J.

Phare, C. T.

Pile, D.

Polson, R.

B. Shen, R. Polson, and R. Menon, “Increasing the density of passive photonic-integrated circuits via nanophotonic cloaking,” Nat. Commun. 7, 13126 (2016).
[Crossref]

B. Shen, R. Polson, and R. Menon, “Metamaterial-waveguide bends with effective bend radius λ0/2,” Opt. Lett. 40, 5750–5753 (2015).
[Crossref]

Popovic, M.

Popovic, M. A.

Provine, J.

Qi, M.

S. Kim and M. Qi, “Polarization rotation and coupling between silicon waveguide and hybrid plasmonic waveguide,” Opt. Express 23, 9968–9978 (2015).
[Crossref]

S. Kim and M. Qi, “Mode-evolution-based polarization rotation and coupling between silicon and hybrid plasmonic waveguides,” Sci. Rep. 5, 18378 (2015).
[Crossref]

Qubaisi, K. A.

Ram, R. J.

K. K. Mehta, C. D. Bruzewicz, R. McConnell, R. J. Ram, J. M. Sage, and J. Chiaverini, “Integrated optical addressing of an ion qubit,” Nat. Nanotech. 11, 1066–1070 (2016).
[Crossref]

Roberts, S. P.

Rosenberger, R.

Roxworthy, B. J.

Sage, J. M.

K. K. Mehta, C. D. Bruzewicz, R. McConnell, R. J. Ram, J. M. Sage, and J. Chiaverini, “Integrated optical addressing of an ion qubit,” Nat. Nanotech. 11, 1066–1070 (2016).
[Crossref]

Salamin, Y.

Sarmiento-Merenguel, J. D.

Schmid, J. H.

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2018).
[Crossref]

Shekhar, P.

Shen, B.

B. Shen, R. Polson, and R. Menon, “Increasing the density of passive photonic-integrated circuits via nanophotonic cloaking,” Nat. Commun. 7, 13126 (2016).
[Crossref]

B. Shen, R. Polson, and R. Menon, “Metamaterial-waveguide bends with effective bend radius λ0/2,” Opt. Lett. 40, 5750–5753 (2015).
[Crossref]

Shin, M. C.

Sinclair, L.

Smith, D. R.

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2018).
[Crossref]

Smith, H. I.

Soman, A.

Z. Wang, T. Li, A. Soman, D. Mao, T. Kananen, and T. Gu, “On-chip wavefront shaping with dielectric metasurface,” Nat. Commun. 10, 3547 (2019).
[Crossref]

Song, W.

Sorger, V. J.

Spencer, D.

Srinivasan, K.

Q. Li, M. Davanço, and K. Srinivasan, “Efficient and low-noise single-photon-level frequency conversion interfaces using silicon nanophotonics,” Nat. Photonics 10, 406–414 (2016).
[Crossref]

Stein, A.

R. Gatdula, S. Abbaslou, M. Lu, A. Stein, and W. Jiang, “Guiding light in bent waveguide superlattices with low crosstalk,” Optica 6, 585–591 (2019).
[Crossref]

Stern, B.

Stojanovic, V.

Stojanovic, V. M.

Stone, J.

Suh, M.-G.

M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref]

Sun, C.

Sun, J.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]

Theogarajan, L.

Timurdogan, E.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]

Trocha, P.

Vahala, K.

Vahala, K. J.

M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref]

Van, V.

Venkataraman, V.

B. Hausmann, I. Bulu, V. Venkataraman, P. Deotare, and M. Lončar, “Diamond nonlinear photonics,” Nat. Photonics 8, 369–374 (2014).
[Crossref]

Vijayan, K.

Volet, N.

Wade, M.

Wade, M. T.

Wang, C.

Wang, I.

Wang, S.

Wang, Z.

Z. Wang, T. Li, A. Soman, D. Mao, T. Kananen, and T. Gu, “On-chip wavefront shaping with dielectric metasurface,” Nat. Commun. 10, 3547 (2019).
[Crossref]

Wangüemert-Pérez, G.

Watts, M. R.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]

Weiner, A. M.

Westly, D.

Westly, D. A.

White, I. M.

X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics 5, 591–597 (2011).
[Crossref]

Wirth, J. C.

Wolf, S.

Xu, D.-X.

Xuan, Y.

Xue, X.

Yaacobi, A.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]

Yang, K. Y.

M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref]

Yang, Q.-F.

M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref]

Yariv, A.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2006).

Yeh, P.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2006).

Yi, X.

M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref]

Yulaev, A.

Zadka, M.

Zhang, B.

Zhang, X.

ACS Nano (1)

S. Lin and K. B. Crozier, “Trapping-assisted sensing of particles and proteins using on-chip optical microcavities,” ACS Nano 7, 1725–1730 (2013).
[Crossref]

J. Opt. Soc. Am. A (1)

W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[Crossref]

Laser Photon. Rev. (1)

Light Sci. Appl. (1)

Nat. Commun. (6)

Z. Wang, T. Li, A. Soman, D. Mao, T. Kananen, and T. Gu, “On-chip wavefront shaping with dielectric metasurface,” Nat. Commun. 10, 3547 (2019).
[Crossref]

B. Shen, R. Polson, and R. Menon, “Increasing the density of passive photonic-integrated circuits via nanophotonic cloaking,” Nat. Commun. 7, 13126 (2016).
[Crossref]

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
[Crossref]

Nat. Nanotech. (1)

K. K. Mehta, C. D. Bruzewicz, R. McConnell, R. J. Ram, J. M. Sage, and J. Chiaverini, “Integrated optical addressing of an ion qubit,” Nat. Nanotech. 11, 1066–1070 (2016).
[Crossref]

Nat. Photonics (5)

B. Hausmann, I. Bulu, V. Venkataraman, P. Deotare, and M. Lončar, “Diamond nonlinear photonics,” Nat. Photonics 8, 369–374 (2014).
[Crossref]

Q. Li, M. Davanço, and K. Srinivasan, “Efficient and low-noise single-photon-level frequency conversion interfaces using silicon nanophotonics,” Nat. Photonics 10, 406–414 (2016).
[Crossref]

X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics 5, 591–597 (2011).
[Crossref]

Nature (5)

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2018).
[Crossref]

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]

Sci. Adv. (1)

A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018).
[Crossref]

Sci. Rep. (1)

S. Kim and M. Qi, “Mode-evolution-based polarization rotation and coupling between silicon and hybrid plasmonic waveguides,” Sci. Rep. 5, 18378 (2015).
[Crossref]

Science (3)

M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

M. A. Miri and A. Alu, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

Other (2)

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2006).

G. W. Milton, The Theory of Composites (Cambridge University, 2002).

Supplementary Material (1)

Name	Description
» Supplement 1	Supplemental document

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

${n_{\rm s}}$

(yellow solid) and anti-symmetric

${n_{\rm a}}$

(blue dashed) modes, and (g)–(i) their corresponding normalized coupling lengths

${L_{\rm c}}/{\lambda _0} = 1/(2|{n_{\rm s}} - {n_{\rm a}}|)$

(blue dots): (d) and (g) strip, (e) and (h) e-skid with multilayers, and (f) and (i) e-skid with EMT. All the simulations are performed as a function of the core width

$w$

, while fixing the other parameters as

$h = 220\;{\rm nm} $

$\Lambda = 100\;{\rm nm} $

$\rho = 0.5$

, and

$N = 5$

. The free space wavelength is

${\lambda _0} = 1550\;{\rm nm} $

. The inset boxes of (d)–(f) show the zoomed-in view of each mode. The red-shaded areas in e-skid couplings show the non-trivial coupling regimes, where

${n_{\rm s}} \lt {n_{\rm a}}$

, which cannot be observed in a typical strip coupling. The red arrows in e-skid couplings indicate the exceptional coupling points, where

${n_{\rm s}} \approx {n_{\rm a}}$

, thus causing the

${L_{\rm c}} \to \infty$

View in Article | Download Full Size | PPT Slide | PDF

Fig. 2. Anisotropic coupled mode analysis on the exceptional coupling in coupled e-skid waveguides. (a)–(c) Normalized anisotropic coupling coefficients

${\kappa _x}$

(blue dashed),

${\kappa _y}$

(orange dashed), and

${\kappa _z}$

(yellow dashed) of the coupled (a) strip, (b) e-skid with multilayer, and (c) e-skid with EMT waveguides. Geometric parameters and the wavelength are the same as in Figs. 1. (d)–(f) Magnitude of the total coupling coefficient

$|\kappa | = |{\kappa _x} + {\kappa _y} + {\kappa _z}|$

(orange solid), and (g)–(i) their corresponding normalized coupling lengths

${L_{\rm c}}/{\lambda _0} = \pi /(2|\kappa |{\lambda _0})$

(blue dots) for each configuration: (d) and (g) strip, (e) and (h) e-skid with multilayer, and (f) and (i) e-skid with EMT. The normalized coupling lengths in (g)–(i), which are obtained with anisotropic coupled mode analysis, match with those results in Figs. 1(g)–1(i) from the full numerical simulations. The red-shaded areas in e-skid couplings show the non-trivial coupling regimes where

$\kappa \lt 0$

, which cannot be observed in typical strip waveguide coupling. The red arrows in e-skid couplings indicate the exceptional coupling points where

$|\kappa | \approx 0$

, thus causing the

${L_{\rm c}} \to \infty$

. As shown in (b) and (c), the anisotropic nature of e-skid waveguides can cause a larger

${\kappa _z}$

, which results in the non-trivial coupling regime (

$\kappa \lt 0$

) and the exceptional coupling (

$\kappa \approx 0$

) at the transition.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 3. Experimental demonstration of the exceptional coupling in coupled e-skid waveguides. Schematic view of the coupled (a) e-skid (multilayer) and (b) strip waveguides.

${I_0},{I_1}$

, and

${I_2}$

indicate the optical powers at input, through, and coupled ports, respectively. (c) SEM images of the fabricated devices. Zoomed-in images show (left) the coupled e-skid waveguides and (right) the adiabatic transition from strip to e-skid waveguides. (d) Experimentally measured waveguide crosstalk and (e) the corresponding normalized coupling length of the coupled e-skid (solid) and strip (dashed) waveguides:

$w = 420\;{\rm nm} $

(blue), 430 nm (orange), 440 nm (yellow), and 450 nm (purple). Numerically simulated (f) crosstalk and (g) normalized coupling length that correspond to the experimental results in (d) and (e), respectively. Geometric parameters are

$h = 220\;{\rm nm} $

$\rho = 0.5$

$\Lambda = 100\;{\rm nm} $

$N = 5$

, and

$L = 100\,\,\unicode{x00B5}$

m. Map plots of the measured crosstalk as functions of

$\lambda$

and

$w$

for the coupled (h) e-skid and (j) strip waveguides; (i) and (k) are their corresponding simulation results, respectively. Dark regions in (h) and (i) indicate the exceptional couplings in coupled e-skid waveguides.

View in Article | Download Full Size | PPT Slide | PDF

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

ε_{∥} = ρ ε_{S i} + (1 - ρ) ε_{a i r},

ε_{⊥} = \frac{ε_{S i} ε_{a i r}}{ρ ε_{a i r} + (1 - ρ) ε_{S i}},

\frac{L_{c}}{λ_{0}} = \frac{1}{2 Δ n} = \frac{1}{2 | n_{s} - n_{a} |},

κ_{i} = \frac{ω ε_{0}}{4} \iint Δ ε_{i} (x, y) E_{1 i} (x, y) E_{2 i}^{*} (x, y) d x d y,

L_{c} = \frac{π}{2 | κ |} .

\frac{I_{2}}{I_{1}} = \tan^{2} (\frac{π L}{2 L_{c}}),