Abstract

Diffraction gratings that redirect light propagating in a channel waveguide to an on-chip slab are emerging as important building blocks in integrated photonics. Such distributed Bragg deflectors enable precise shaping of slab confined beams for a variety of applications, including wavelength multiplexing, optical phased array feeding, and coupling interfaces for on-chip point-to-point communications. However, these deflectors suffer from significant losses caused by off-chip radiation. In this Letter, we show, for the first time, to the best of our knowledge, that off-chip radiation can be dramatically reduced by using the single-beam phase matching condition and subwavelength metamaterial refractive index engineering. We present a deflector design with losses below 0.3 dB, opening a path toward new applications of distributed Bragg deflectors in integrated photonics.

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

Silicon photonics (SiP) is a leading technology for photonic integrated circuits as its high degree of miniaturization and CMOS compatibility enable low cost mass production [1]. Subwavelength grating (SWG) metamaterial engineering [24] has significantly advanced SiP technology as it allows the local engineering of waveguide optical properties directly by lithography, providing new degrees of freedom in SiP design. Precise shaping of slab confined beams is required in many applications ranging from beam expansion [5,6] to on-chip particle acceleration [7]. Diffractive gratings are often used in SiP devices, e.g., for out-of-plane coupling between a waveguide mode and a free-space beam in fiber-chip couplers [8] and optical antennas [9]. In distributed Bragg deflectors (DBDs), originally proposed in [10], light propagating in a channel waveguide is coupled to a vertically confined beam in a slab adjacent to the channel waveguide, as illustrated in Fig. 1. The DBD comprises four regions: a channel waveguide with a diffractive sidewall grating, an SWG slab, a graded-index (GRIN) adaptation region and a slab waveguide. Light propagating in the channel waveguide is diffracted in-plane into the SWG metamaterial region and subsequently coupled to the slab waveguide through the index-matching GRIN structure. DBDs enable the generation of slab beams with precisely controlled amplitude and phase through grating apodization. Extreme beam expansion [11,12], optical spectrum analysis [13], and wavelength (de)multiplexing [1416] have been reported based on DBDs, with other interesting applications emerging, including thermal control of diffracted beams, optical phased array feeding, and on-chip photonic interconnects [17]. However, a significant drawback hampering a wider use of DBDs in integrated photonics has been their comparatively high on-chip losses reported experimentally [11,16]. Recently, a detailed simulation study [12] showed that the loss penalty in DBDs primarily originates in off-chip radiation, with the latter typically exceeding 2 dB.

 figure: Fig. 1.

Fig. 1. Schematics of a distributed Bragg deflector. (a) Complete device including input waveguide with an adiabatic taper and the apodized Bragg deflector. Red arrows indicate the local energy flow (Poynting vector) direction. The silicon dioxide cladding is not shown for the sake of clarity. (b) Top view of a periodic section of the Bragg deflector. Schematic representation of the deflector geometry under (c) the 3D homogeneous model and (d) the 2D homogeneous effective index model (the anisotropic SWG metamaterial slab region is shown in green).

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Here we report a new DBD design strategy that effectively mitigates radiation losses by using momentum matching to obtain a single-beam radiation condition. To demonstrate this new concept, we design a DBD beam expander that couples the Si-wire waveguide mode to a slab-propagating collimated Gaussian beam with 50 µm mode field diameter (MFD), i.e., achieving over 100 times beam width expansion, with less than 0.3 dB loss penalty.

Figure 1 shows a schematic view of the device, including the input waveguide, the input adiabatic taper, and the apodized Bragg deflector. Light propagating along the channel waveguide is diffracted by the sinusoidal sidewall grating into the SWG region and then coupled to the silicon slab via a GRIN matching region [15]. The main electromagnetic properties of the DBD can be described by the model shown in Fig. 1(c), in which the SWG region is represented as an equivalent anisotropic homogeneous metamaterial characterized by its diagonal permittivity tensor $\overline {\overline {{\varepsilon _{{\rm SWG}}}}} = \overline {\overline {n_{{\rm SWG}}^2}} = {\rm diag}[{n_\parallel ^2,n_\parallel ^2,n_ \bot ^2}]$ [18] (the GRIN region and Si slab are not included in this model).

First, we will deduce the single-beam condition using the effective index 2D model shown in Fig. 1(d). Then we will demonstrate by 3D finite-difference time-domain (FDTD) simulations that this condition applies also to a full 3D structure. To illustrate the operation of the device, we consider a TE-polarized Floquet–Bloch mode that propagates through the periodic waveguide shown in Fig. 1(d) with propagation constant ${{\beta}_{B}}$ and effective mode index ${{n}_{B}}= {{\beta}_{B}}/{{k}_{0}}$, where ${{k}_{0}}$ is the free-space propagation constant. Floquet–Bloch modes can be represented as a superposition of spatial harmonics with their momenta related to the grating period as [19]

$${k_{z,m}}/{k_0} = {n_B} + m\lambda /\Lambda ,$$
where $m$ is the order of diffraction and ${k_{z,m}}$ is the momentum of the $ m $th harmonic of the Floquet–Bloch mode. Furthermore, free propagating beams in the isotropic cladding and anisotropic SWG medium fulfill the following dispersion relations:
$${\left({{k_x}/\!\left({{k_0}{n_{{{{\rm SiO}}_{2}}}}} \right)} \right)^2} + {\left({{k_z}/\!\left({{k_0}{n_{{{{\rm SiO}}_{2}}}}} \right)} \right)^2} = 1,$$
$${\left({{k_x}/\!\left({{k_0}n_ \bot ^{{\rm eff}}} \right)} \right)^2} + {\left({{k_z}/\!\left({{k_0}n_\parallel ^{{\rm eff}}} \right)} \right)^2} = 1,$$
where $n_\parallel ^{{\rm eff}}$ and $n_ \bot ^{{\rm eff}}$ are the effective indices of the SWG slab TE-polarized mode propagating in the $z$ and $x$ directions, respectively.

The phase matching condition at the boundaries of the periodic waveguide and surrounding media implies that diffracted orders will be allowed when Eqs. (1) and (2) are fulfilled simultaneously, as schematically represented in Fig. 2. The blue semicircle and the green ellipse represent the k-diagrams in the ${{\rm SiO} _2}$ cladding and SWG medium, respectively. The SWG slab effective index $n_\parallel ^{{\rm eff}}$ is larger than the cladding index ${n_{{\rm SiO}2}}$ while the index difference ${n}_\parallel ^{{\rm eff}}{- {n}_{{\rm SiO}2}}$ can be readily controlled by modifying the SWG duty cycle (${{\rm DC}_{{\rm SWG}}}$). The arrows in Fig. 2 indicate propagation directions of diffracted beams for two wavelength-to-pitch ratios $\lambda /{{\Lambda}_1}$ and $\lambda /{{\Lambda}_2}$. In general, several diffraction orders are allowed in the SWG and cladding media depending on the $\lambda /{\Lambda}$.

 figure: Fig. 2.

Fig. 2. K-space diagram for the grating shown in Fig. 1(d), including the single-beam diffraction window (shaded in gray). Red arrows show propagation vectors of diffractive beams for a conventional DBD design [12] when the period ${{\Lambda}_1}$ does not fulfill the single-beam condition. The blue arrow indicates the propagation direction vector of the single beam permitted for the period ${{\Lambda}_2}$, hence mitigating radiation into the cladding.

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Bragg deflector designs typically use the $-1$st diffraction order ($m = - 1$) in a near-normal direction [${\lambda}/{{\Lambda}_1}$], as indicated by the red arrows in Fig. 2. In such a case, propagating diffraction orders are allowed in both media, and a blazed grating [20] can be used to maximize the diffraction efficiency toward the SWG [13,15]. However, this approach has demonstrated limited success for DBD in silicon waveguides [12,16] mainly due to radiation losses into the cladding and the substrate. Here, we effectively mitigate these diffraction losses by enforcing the deflector operation within the single-beam diffraction regime. In our device, diffraction to the cladding material is frustrated (single-beam window in the $ K $-space diagram of Fig. 2). This results in a single $m = - 1$ diffraction order propagating in the SWG material providing $\lambda /{\Lambda}$ is properly chosen (Fig. 2, blue arrow). This results in an effective suppression of off-chip diffraction loss. As can be deduced from Fig. 2, the condition for single-beam diffraction into the SWG region is given by

$${-}n_\parallel ^{{\rm eff}} \lt {n_B} - \lambda /\Lambda \lt - {n_{{\rm SiO}2}}.$$

According to Eq. (4), within the single-beam operation window, diffraction is frustrated to media with refractive index $\le {{n}_{{\rm SiO}2}}$. This condition does not depend on a specific grating shape, and a single diffraction order can be achieved even without blazing. This condition also ensures that no diffraction orders are allowed in the ${\rm SiO}_2$ bottom and upper claddings.

 figure: Fig. 3.

Fig. 3. Floquet–Bloch analysis for a 3D homogeneous anisotropic model. Right axis, diffraction efficiency $\eta$ for the ${-}{1}$st diffraction order in the SWG slab. Left axis, effective index for the ${-}{1}$st diffraction order as a function of the wavelength-to-pitch ratio ($\lambda /{\Lambda}$). Red lines indicate the limiting values for the single-beam operation. ${W_g} = 600\;{\rm nm} $, ${\rm Dg} = 0.25$, $\lambda = 1.55\; {\unicode{x00B5}{\rm m}}$, ${{\Lambda}_{{\rm SWG}}} = 200\;{\rm nm} $, and ${{\rm DC}_{{\rm SWG}}} = 0.5$.

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We design our device for a 220-nm-thick silicon-on-insulator (SOI) platform with a 2 µm buried oxide (BOX) layer, for a central wavelength of $\lambda = 1550\;{\rm nm} $. The SWG grating pitch and duty cycle have been set as ${{\Lambda}_{{\rm SWG}}} = 200\;{\rm nm} $ and ${{\rm DC}_{{\rm SWG}}} = 0.5$. Using SWG homogenization through the anisotropic model [18] yields ${{n}_ \bot}= 1.94$ and ${{n}_\parallel}= 2.78$. We consider a typical waveguide sidewall grating geometry, ${{ W }_{g}}= 600\;{\rm nm}$ and ${{ D }_{g}}= 0.25$, as shown in Fig. 1. For this geometry, ${{n}_{B}} \approx 2.6$, ${n}_{\parallel }^{{\rm eff}}= 2.2$, and ${n}_{\bot }^{{\rm eff}}= 1.6$. We analyzed the structure by means of the anisotropic homogeneous model shown in Fig. 1(c) and Floquet–Bloch modal analysis based on 3D FDTD simulations with periodic boundary conditions [12]. Figure 3 shows the computed diffraction efficiency $ \eta $ of the $m = - 1$ diffraction order in the SWG slab (right axis) and the effective index corresponding to the $m = - 1$ diffraction order (left axis) as a function of $\lambda /\Lambda$. The diffraction efficiency to the ${-}{1}$st diffraction order in the SWG region approaches 100% for $\lambda /{\Lambda} \in [4.05$, 4.8], which is in excellent agreement with the single-beam condition [Eq. (4)], as shown in Fig. 3. The grating period is set to 360 nm to fulfill the single-beam condition near the central wavelength of 1550 nm. The width of the single-beam window is controlled with the duty cycle ${{\rm DC}_{{\rm SWG}}}$ and the corresponding SWG indices $n_{{\parallel}}^{{\rm eff}}$ and $n_{{\bot}}^{{\rm eff}}$. With increasing $n_\parallel ^{{\rm eff}}$, the single-beam window broadens, and a wide operation wavelength range can be achieved. On the other hand, ensuring the lateral confinement of the channel waveguide mode requires $n_\parallel ^{{\rm eff}} \lt {n_B}$.

Finally, we design a full DBD device operating in the single-beam condition. Besides the suppression of radiation losses, a near-Gaussian diffracted field will be synthesized. This is achieved by judiciously controlling the grating strength (coupling coefficient) $\alpha$ along the propagation direction (apodization). The grating strength is defined as the fraction of the transmitted power radiated per unit length [19] and is readily obtained as the imaginary part of the Floquet–Bloch mode complex index $\alpha = - {{k}_0}{\rm Im}\{{n_B} \}$. Assuming the slowly varying amplitude approximation, each diffractive element can be individually designed by Floquet–Bloch analysis [13]. The grating strength $\alpha$ primarily depends on the waveguide width ${W_g}$ and the grating modulation depth $D_g$. Here we assume that the width of the SWG region and the GRIN adaptation region are sufficiently large to avoid evanescent leakage from the channel waveguide into the silicon slab and minimize reflections at the SWG/slab interface [12]. To ensure this, we set ${W_{{\rm SWG}}} = 2\; {\unicode{x00B5}{\rm m}}$ and ${W_{{\rm GRIN}}} = 2\; {\unicode{x00B5}{\rm m}}$. The waveguide width ${W_g} = 600\;{\rm nm} $ has been chosen sufficiently wide to accommodate the sidewall grating modulation depth while maintaining single mode operation. The varying modulation depth ${D_g}$ is used to control the grating strength $\alpha$ along the structure. Figure 4(a) shows the calculated grating coupling coefficient and Floquet–Bloch refractive index as a function of the grating modulation. It is observed that by varying the modulation depth the grating strength can be efficiently controlled while the Floquet–Bloch mode index ${n_B}$ changes are comparatively small. An excellent agreement is observed comparing the simulation results for the full SWG structure [Fig. 4(a), black curves] and the simplified homogeneous anisotropic model (ibid, red curves). This confirms that the anisotropic model, albeit much simpler than the full 3D structure, comprises all the relevant design information. While the simulation domain required for the full structure [Fig. 1(b)] Floquet–Bloch mode computation window is set to fit a whole number of sidewall grating and SWG periods, the homogenized structure [Fig. 1(c)] only includes a single sidewall grating period, with the latter being far more efficient to simulate. Finally, for the sake of completeness, we have also included the results obtained for the homogeneous isotropic model (${n_ \bot} = {n_\parallel} = 2.78$), which is often used in SWG research papers.

 figure: Fig. 4.

Fig. 4. (a) Grating strength (coupling coefficient) $\alpha$ and Floquet–Bloch index ${n_B}$, as a function of modulation depth ${D_g}$ for the structure with an SWG slab (black curves), the homogenized anisotropic slab (red curves), and the homogenized isotropic slab (blue curves), calculated via Floquet–Bloch modal analysis. ${W_g} = 600\;{\rm nm} $, $\lambda = 1.55\; {\unicode{x00B5}{\rm m}}$, ${{\Lambda}_{{\rm SWG}}} = 200\;{\rm nm} $, ${{\rm DC}_{{\rm SWG}}} = 0.5$, and ${\Lambda} = 360\;{\rm nm} $. (b) Target beam profile and the corresponding radiation strength $\alpha$. (c) Grating modulation depth along the deflector structure.

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As discussed above, grating apodization is required to generate the specific field distribution in the slab waveguide. The coupling coefficient variation along the grating can be readily calculated for a target intensity profile [20]. Figure 4(b) shows our target field, a 50 µm MFD Gaussian beam, $g(z) = \exp ({- {{({2z/{\rm MFD}})}^2} - j{k_{z, - 1}}z})$, with the corresponding grating strength distribution $\alpha (z)$, assuming 99.5% radiation factor (0.5% power remaining in the channel waveguide) [13]. The modulation apodization ${D_g}(z)$ shown in Fig. 4(c) was obtained by mapping the required grating strength profile $\alpha (z)$ using Fig. 4(a). The resulting ${D_g}(z)$ ranges from 0.1 to 0.33, corresponding to a peak-to-valley sinusoidal grating modulation from 120 nm to 400 nm, hence yielding feature sizes sufficiently large for compatibility with deep-UV photolithography. Since the apodization slightly changes the Floquet–Bloch mode effective index [Fig. 4(a), left axis], we compensate these index changes by chirping the pitch along the grating by ${\Delta \Lambda} = - {{\Lambda}^2}/{{\lambda}_0}{\Delta}{n_B}$. Finally, an input taper was designed to connect the silicon wire input waveguide with the deflector [Fig. 1(a)]. The taper is implemented by gradually introducing the SWG structure along a length of 20 µm to minimize excitation of higher order modes.

The complete device was simulated with 3D FDTD. Figure 5(a) shows the calculated magnetic field ${{ H }_{y}}$ (cross section taken at the center of the silicon layer). It is observed that the Si-wire waveguide mode is progressively radiated in the positive $ x $ direction, while radiation in the negative $ x $ direction is effectively suppressed. A small amount of the light ($\lt\!0.5\%$) remains in the waveguide at the end of the grating, as designed. Figure 5(b) shows a transverse cross section of the field near the center of the grating. It is observed that the field is directed into the silicon slab with high directionality while the off-chip radiation is suppressed. Furthermore, while the field confinement in the $ y $ direction slowly increases through the GRIN region, there is no significant radiation loss at this transition. The diffracted field profile along the blue dashed line in Fig. 5(a) is shown in Fig. 5(c). It is observed that the simulated diffracted beam profile [Fig. 5(c), light blue curve] matches the target field [Fig. 5(c), red curve]. We observe a small ripple on the simulated field that is caused by spurious reflections and radiation induced by the variation of the grating geometry along the propagation direction. However, the ripple influence on the coupling efficiency is negligible as the overlap integral of the simulated and target field exceeds 97%. The total insertion loss along the entire path from the input Si-wire waveguide to the collimated slab beam is below 0.3 dB. Using power monitors placed above and below the chip, we estimate that less than 0.15 dB of the losses are due to off-chip radiation, while the remaining 0.15 dB are attributed to all other sources, including field overlaps, mode conversion, and numerical losses. Finally, to investigate the tolerance to fabrication errors, we studied the effect of $\pm 20\;{\rm nm}$ over/under etching. This error resulted in less than 0.5 dB of additional loss.

 figure: Fig. 5.

Fig. 5. (a) 3D FDTD simulation of ${H_y}$ field evolution along the full DBD device. (b) Transversal cross section of ${H_y}$ field in an $XY$ plane near the center of the grating [${z} = 50\;{\unicode{x00B5}{\rm m}}$, dashed green line in (a)]. (c) ${H_y}$ field distribution in the slab waveguide (along the dashed blue line in a). The target profile is shown for comparison.

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In summary, we have presented a new strategy for implementing distributed Bragg grating deflectors in silicon photonics. We showed that off-chip radiation losses can be mitigated by leveraging a single-beam momentum matching condition and subwavelength metamaterial refractive index engineering. To demonstrate this new concept, we have designed a grating deflector beam expander using a computationally efficient Floquet–Bloch analysis of individual elements. The 3D FDTD simulation of the complete device yields excellent agreement with target design and confirmed a low total insertion loss of 0.3 dB. We believe that the ability to precisely shape slab confined beams with negligible loss as demonstrated in this work paves the way for practical implementations of SWG metamaterial engineered Bragg deflectors in integrated photonics.

Funding

Ministerio de Educación, Cultura y Deporte (FPU-16/03401); Ministerio de Economía y Competitividad (PID2019-106747RB-I00); Junta de Andalucía (UMA18-FEDERJA-219); Consejería de Economía, Conocimiento, Empresas y Universidad, Junta de Andalucía (P18-RT-1453, P18-RT-793); Universidad de Málaga.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

REFERENCES

1. D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016). [CrossRef]  

2. P. Cheben, D.-X. Xu, S. Janz, and A. Densmore, Opt. Express 14, 4695 (2006). [CrossRef]  

3. P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, Nature 560, 565 (2018). [CrossRef]  

4. R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, Proc. IEEE 106, 2144 (2018). [CrossRef]  

5. W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021). [CrossRef]  

6. S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, Light Sci. Appl. 7, 72 (2018). [CrossRef]  

7. N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020). [CrossRef]  

8. R. Marchetti, C. Lacava, L. Carroll, K. Gradkowski, and P. Minzioni, Photon. Res. 7, 201 (2019). [CrossRef]  

9. P. Ginel-Moreno, D. Pereira-Martín, A. Hadij-ElHouati, W. N. Ye, D. Melati, D.-X. D.-X. Xu, S. Janz, A. Ortega-Moñux, J. G. G. Wangüemert-Pérez, R. Halir, Í. Molina-Fernández, J. H. J. H. Schmid, and P. Cheben, Opt. Lett. 45, 5668 (2020). [CrossRef]  

10. H. M. Stoll, Appl. Opt. 17, 2562 (1978). [CrossRef]  

11. R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004). [CrossRef]  

12. A. Hadij-ElHouati, P. Cheben, A. Ortega-Moñux, J. G. Wangüemert-Pérez, R. Halir, J. H. Schmid, and Í. Molina-Fernández, Opt. Express 27, 33180 (2019). [CrossRef]  

13. C. K. Madsen, J. Wagener, T. A. Strasser, D. Muehlner, M. A. Milbrodt, E. J. Laskowski, and J. DeMarco, IEEE J. Sel. Top. Quantum Electron. 4, 925 (1998). [CrossRef]  

14. Y. Hao, Y. Wu, J. Yang, X. Jiang, and M. Wang, Opt. Express 14, 8630 (2006). [CrossRef]  

15. P. J. Bock, P. Cheben, A. Delâge, J. H. Schmid, D.-X. Xu, S. Janz, and T. J. Hall, Opt. Express 16, 17616 (2008). [CrossRef]  

16. P. J. Bock, P. Cheben, J. H. Schmid, A. V. Velasco, A. Delâge, S. Janz, D.-X. Xu, J. Lapointe, T. J. Hall, and M. L. Calvo, Opt. Express 20, 19882 (2012). [CrossRef]  

17. A. Hadij-ElHouati, I. Molina-Fernández, R. Halir, J. G. Wangüemert-Pérez, and A. Ortega-Moñux, “Deflector lateral de haz único, multiplexor/demultiplexor y dispositivo alimentador de antena óptica que incorporan el deflector, y métodos que los utilizan,” Spain Patent Application  P202031204 (2  December  2020).

18. J. M. Luque-González, A. Herrero-Bermello, A. Ortega-Moñux, Í. Molina-Fernández, A. V. Velasco, P. Cheben, J. H. Schmid, S. Wang, and R. Halir, Opt. Lett. 43, 4691 (2018). [CrossRef]  

19. T. Tamir and S. T. Peng, Appl. Phys. 14, 235 (1977). [CrossRef]  

20. R. Waldhäusl, B. Schnabel, P. Dannberg, E.-B. Kley, A. Bräuer, and W. Karthe, Appl. Opt. 36, 9383 (1997). [CrossRef]  

References

  • View by:

  1. D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
    [Crossref]
  2. P. Cheben, D.-X. Xu, S. Janz, and A. Densmore, Opt. Express 14, 4695 (2006).
    [Crossref]
  3. P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, Nature 560, 565 (2018).
    [Crossref]
  4. R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, Proc. IEEE 106, 2144 (2018).
    [Crossref]
  5. W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
    [Crossref]
  6. S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, Light Sci. Appl. 7, 72 (2018).
    [Crossref]
  7. N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
    [Crossref]
  8. R. Marchetti, C. Lacava, L. Carroll, K. Gradkowski, and P. Minzioni, Photon. Res. 7, 201 (2019).
    [Crossref]
  9. P. Ginel-Moreno, D. Pereira-Martín, A. Hadij-ElHouati, W. N. Ye, D. Melati, D.-X. D.-X. Xu, S. Janz, A. Ortega-Moñux, J. G. G. Wangüemert-Pérez, R. Halir, Í. Molina-Fernández, J. H. J. H. Schmid, and P. Cheben, Opt. Lett. 45, 5668 (2020).
    [Crossref]
  10. H. M. Stoll, Appl. Opt. 17, 2562 (1978).
    [Crossref]
  11. R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004).
    [Crossref]
  12. A. Hadij-ElHouati, P. Cheben, A. Ortega-Moñux, J. G. Wangüemert-Pérez, R. Halir, J. H. Schmid, and Í. Molina-Fernández, Opt. Express 27, 33180 (2019).
    [Crossref]
  13. C. K. Madsen, J. Wagener, T. A. Strasser, D. Muehlner, M. A. Milbrodt, E. J. Laskowski, and J. DeMarco, IEEE J. Sel. Top. Quantum Electron. 4, 925 (1998).
    [Crossref]
  14. Y. Hao, Y. Wu, J. Yang, X. Jiang, and M. Wang, Opt. Express 14, 8630 (2006).
    [Crossref]
  15. P. J. Bock, P. Cheben, A. Delâge, J. H. Schmid, D.-X. Xu, S. Janz, and T. J. Hall, Opt. Express 16, 17616 (2008).
    [Crossref]
  16. P. J. Bock, P. Cheben, J. H. Schmid, A. V. Velasco, A. Delâge, S. Janz, D.-X. Xu, J. Lapointe, T. J. Hall, and M. L. Calvo, Opt. Express 20, 19882 (2012).
    [Crossref]
  17. A. Hadij-ElHouati, I. Molina-Fernández, R. Halir, J. G. Wangüemert-Pérez, and A. Ortega-Moñux, “Deflector lateral de haz único, multiplexor/demultiplexor y dispositivo alimentador de antena óptica que incorporan el deflector, y métodos que los utilizan,” Spain Patent Application P202031204 (2 December 2020).
  18. J. M. Luque-González, A. Herrero-Bermello, A. Ortega-Moñux, Í. Molina-Fernández, A. V. Velasco, P. Cheben, J. H. Schmid, S. Wang, and R. Halir, Opt. Lett. 43, 4691 (2018).
    [Crossref]
  19. T. Tamir and S. T. Peng, Appl. Phys. 14, 235 (1977).
    [Crossref]
  20. R. Waldhäusl, B. Schnabel, P. Dannberg, E.-B. Kley, A. Bräuer, and W. Karthe, Appl. Opt. 36, 9383 (1997).
    [Crossref]

2021 (1)

W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
[Crossref]

2020 (2)

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

P. Ginel-Moreno, D. Pereira-Martín, A. Hadij-ElHouati, W. N. Ye, D. Melati, D.-X. D.-X. Xu, S. Janz, A. Ortega-Moñux, J. G. G. Wangüemert-Pérez, R. Halir, Í. Molina-Fernández, J. H. J. H. Schmid, and P. Cheben, Opt. Lett. 45, 5668 (2020).
[Crossref]

2019 (2)

2018 (4)

S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, Light Sci. Appl. 7, 72 (2018).
[Crossref]

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, Nature 560, 565 (2018).
[Crossref]

R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, Proc. IEEE 106, 2144 (2018).
[Crossref]

J. M. Luque-González, A. Herrero-Bermello, A. Ortega-Moñux, Í. Molina-Fernández, A. V. Velasco, P. Cheben, J. H. Schmid, S. Wang, and R. Halir, Opt. Lett. 43, 4691 (2018).
[Crossref]

2016 (1)

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

2012 (1)

2008 (1)

2006 (2)

2004 (1)

R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004).
[Crossref]

1998 (1)

C. K. Madsen, J. Wagener, T. A. Strasser, D. Muehlner, M. A. Milbrodt, E. J. Laskowski, and J. DeMarco, IEEE J. Sel. Top. Quantum Electron. 4, 925 (1998).
[Crossref]

1997 (1)

1978 (1)

1977 (1)

T. Tamir and S. T. Peng, Appl. Phys. 14, 235 (1977).
[Crossref]

Agrawal, A.

W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
[Crossref]

Aksyuk, V.

W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
[Crossref]

Aksyuk, V. A.

S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, Light Sci. Appl. 7, 72 (2018).
[Crossref]

Atwater, H. A.

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, Nature 560, 565 (2018).
[Crossref]

Barker, D. S.

W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
[Crossref]

Benedikovic, D.

R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, Proc. IEEE 106, 2144 (2018).
[Crossref]

Black, D. S.

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Bock, P. J.

Boeuf, F.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Bowers, J. E.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Bräuer, A.

Byer, R. L.

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Calvo, M. L.

Carroll, L.

Cassan, E.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Cheben, P.

Dannberg, P.

Davis, R. L.

R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004).
[Crossref]

Delâge, A.

DeMarco, J.

C. K. Madsen, J. Wagener, T. A. Strasser, D. Muehlner, M. A. Milbrodt, E. J. Laskowski, and J. DeMarco, IEEE J. Sel. Top. Quantum Electron. 4, 925 (1998).
[Crossref]

Densmore, A.

Eckel, S.

W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
[Crossref]

England, R. J.

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Fallahi, M.

R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004).
[Crossref]

Fédéli, J.-M.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Ginel-Moreno, P.

Gradkowski, K.

Hadij-ElHouati, A.

P. Ginel-Moreno, D. Pereira-Martín, A. Hadij-ElHouati, W. N. Ye, D. Melati, D.-X. D.-X. Xu, S. Janz, A. Ortega-Moñux, J. G. G. Wangüemert-Pérez, R. Halir, Í. Molina-Fernández, J. H. J. H. Schmid, and P. Cheben, Opt. Lett. 45, 5668 (2020).
[Crossref]

A. Hadij-ElHouati, P. Cheben, A. Ortega-Moñux, J. G. Wangüemert-Pérez, R. Halir, J. H. Schmid, and Í. Molina-Fernández, Opt. Express 27, 33180 (2019).
[Crossref]

A. Hadij-ElHouati, I. Molina-Fernández, R. Halir, J. G. Wangüemert-Pérez, and A. Ortega-Moñux, “Deflector lateral de haz único, multiplexor/demultiplexor y dispositivo alimentador de antena óptica que incorporan el deflector, y métodos que los utilizan,” Spain Patent Application P202031204 (2 December 2020).

Halir, R.

P. Ginel-Moreno, D. Pereira-Martín, A. Hadij-ElHouati, W. N. Ye, D. Melati, D.-X. D.-X. Xu, S. Janz, A. Ortega-Moñux, J. G. G. Wangüemert-Pérez, R. Halir, Í. Molina-Fernández, J. H. J. H. Schmid, and P. Cheben, Opt. Lett. 45, 5668 (2020).
[Crossref]

A. Hadij-ElHouati, P. Cheben, A. Ortega-Moñux, J. G. Wangüemert-Pérez, R. Halir, J. H. Schmid, and Í. Molina-Fernández, Opt. Express 27, 33180 (2019).
[Crossref]

J. M. Luque-González, A. Herrero-Bermello, A. Ortega-Moñux, Í. Molina-Fernández, A. V. Velasco, P. Cheben, J. H. Schmid, S. Wang, and R. Halir, Opt. Lett. 43, 4691 (2018).
[Crossref]

R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, Proc. IEEE 106, 2144 (2018).
[Crossref]

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, Nature 560, 565 (2018).
[Crossref]

A. Hadij-ElHouati, I. Molina-Fernández, R. Halir, J. G. Wangüemert-Pérez, and A. Ortega-Moñux, “Deflector lateral de haz único, multiplexor/demultiplexor y dispositivo alimentador de antena óptica que incorporan el deflector, y métodos que los utilizan,” Spain Patent Application P202031204 (2 December 2020).

Hall, T. J.

Hao, Y.

Hartmann, J.-M.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Herrero-Bermello, A.

Ho, J. G.

R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004).
[Crossref]

Janz, S.

Jiang, X.

Karthe, W.

Kim, S.

S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, Light Sci. Appl. 7, 72 (2018).
[Crossref]

Kley, E.-B.

Klimov, N.

W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
[Crossref]

Komljenovic, T.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Lacava, C.

Lam, T.

R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004).
[Crossref]

Lapointe, J.

Laskowski, E. J.

C. K. Madsen, J. Wagener, T. A. Strasser, D. Muehlner, M. A. Milbrodt, E. J. Laskowski, and J. DeMarco, IEEE J. Sel. Top. Quantum Electron. 4, 925 (1998).
[Crossref]

Leedle, K. J.

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Li, Q.

S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, Light Sci. Appl. 7, 72 (2018).
[Crossref]

Long, W.

R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004).
[Crossref]

Luque-González, J. M.

Madsen, C. K.

C. K. Madsen, J. Wagener, T. A. Strasser, D. Muehlner, M. A. Milbrodt, E. J. Laskowski, and J. DeMarco, IEEE J. Sel. Top. Quantum Electron. 4, 925 (1998).
[Crossref]

Marchetti, R.

Marris-Morini, D.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Mashanovich, G. Z.

R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, Proc. IEEE 106, 2144 (2018).
[Crossref]

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

McClelland, J. J.

W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
[Crossref]

McGehee, W. R.

W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
[Crossref]

Melati, D.

Miao, Y.

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Milbrodt, M. A.

C. K. Madsen, J. Wagener, T. A. Strasser, D. Muehlner, M. A. Milbrodt, E. J. Laskowski, and J. DeMarco, IEEE J. Sel. Top. Quantum Electron. 4, 925 (1998).
[Crossref]

Minzioni, P.

Mishechkin, O. V.

R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004).
[Crossref]

Molina-Fernandez, I.

R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, Proc. IEEE 106, 2144 (2018).
[Crossref]

Molina-Fernández, I.

A. Hadij-ElHouati, I. Molina-Fernández, R. Halir, J. G. Wangüemert-Pérez, and A. Ortega-Moñux, “Deflector lateral de haz único, multiplexor/demultiplexor y dispositivo alimentador de antena óptica que incorporan el deflector, y métodos que los utilizan,” Spain Patent Application P202031204 (2 December 2020).

Molina-Fernández, Í.

Muehlner, D.

C. K. Madsen, J. Wagener, T. A. Strasser, D. Muehlner, M. A. Milbrodt, E. J. Laskowski, and J. DeMarco, IEEE J. Sel. Top. Quantum Electron. 4, 925 (1998).
[Crossref]

Nachman, P. M.

R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004).
[Crossref]

Nedeljkovic, M.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

O’Brien, P.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Ortega-Monux, A.

R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, Proc. IEEE 106, 2144 (2018).
[Crossref]

Ortega-Moñux, A.

Peng, S. T.

T. Tamir and S. T. Peng, Appl. Phys. 14, 235 (1977).
[Crossref]

Pereira-Martín, D.

Poylio, J.

R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004).
[Crossref]

Reed, G. T.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Roxworthy, B. J.

S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, Light Sci. Appl. 7, 72 (2018).
[Crossref]

Sapra, N. V.

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Schmid, J. H.

A. Hadij-ElHouati, P. Cheben, A. Ortega-Moñux, J. G. Wangüemert-Pérez, R. Halir, J. H. Schmid, and Í. Molina-Fernández, Opt. Express 27, 33180 (2019).
[Crossref]

J. M. Luque-González, A. Herrero-Bermello, A. Ortega-Moñux, Í. Molina-Fernández, A. V. Velasco, P. Cheben, J. H. Schmid, S. Wang, and R. Halir, Opt. Lett. 43, 4691 (2018).
[Crossref]

R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, Proc. IEEE 106, 2144 (2018).
[Crossref]

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, Nature 560, 565 (2018).
[Crossref]

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

P. J. Bock, P. Cheben, J. H. Schmid, A. V. Velasco, A. Delâge, S. Janz, D.-X. Xu, J. Lapointe, T. J. Hall, and M. L. Calvo, Opt. Express 20, 19882 (2012).
[Crossref]

P. J. Bock, P. Cheben, A. Delâge, J. H. Schmid, D.-X. Xu, S. Janz, and T. J. Hall, Opt. Express 16, 17616 (2008).
[Crossref]

Schmid, J. H. J. H.

Schnabel, B.

Smith, D. R.

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, Nature 560, 565 (2018).
[Crossref]

Solgaard, O.

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Srinivasan, K.

S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, Light Sci. Appl. 7, 72 (2018).
[Crossref]

Stoll, H. M.

Strasser, T. A.

C. K. Madsen, J. Wagener, T. A. Strasser, D. Muehlner, M. A. Milbrodt, E. J. Laskowski, and J. DeMarco, IEEE J. Sel. Top. Quantum Electron. 4, 925 (1998).
[Crossref]

Su, L.

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Tamir, T.

T. Tamir and S. T. Peng, Appl. Phys. 14, 235 (1977).
[Crossref]

Thomson, D.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Trivedi, R.

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Velasco, A. V.

Vercruysse, D.

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Virot, L.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Vivien, L.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Vuckovic, J.

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Wagener, J.

C. K. Madsen, J. Wagener, T. A. Strasser, D. Muehlner, M. A. Milbrodt, E. J. Laskowski, and J. DeMarco, IEEE J. Sel. Top. Quantum Electron. 4, 925 (1998).
[Crossref]

Waldhäusl, R.

Wang, C.-J.

R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004).
[Crossref]

Wang, M.

Wang, S.

Wanguemert-Perez, J. G.

R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, Proc. IEEE 106, 2144 (2018).
[Crossref]

Wangüemert-Pérez, J. G.

A. Hadij-ElHouati, P. Cheben, A. Ortega-Moñux, J. G. Wangüemert-Pérez, R. Halir, J. H. Schmid, and Í. Molina-Fernández, Opt. Express 27, 33180 (2019).
[Crossref]

A. Hadij-ElHouati, I. Molina-Fernández, R. Halir, J. G. Wangüemert-Pérez, and A. Ortega-Moñux, “Deflector lateral de haz único, multiplexor/demultiplexor y dispositivo alimentador de antena óptica que incorporan el deflector, y métodos que los utilizan,” Spain Patent Application P202031204 (2 December 2020).

Wangüemert-Pérez, J. G. G.

Westly, D.

W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
[Crossref]

Westly, D. A.

S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, Light Sci. Appl. 7, 72 (2018).
[Crossref]

Wu, Y.

Xu, D.-X.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

P. J. Bock, P. Cheben, J. H. Schmid, A. V. Velasco, A. Delâge, S. Janz, D.-X. Xu, J. Lapointe, T. J. Hall, and M. L. Calvo, Opt. Express 20, 19882 (2012).
[Crossref]

P. J. Bock, P. Cheben, A. Delâge, J. H. Schmid, D.-X. Xu, S. Janz, and T. J. Hall, Opt. Express 16, 17616 (2008).
[Crossref]

P. Cheben, D.-X. Xu, S. Janz, and A. Densmore, Opt. Express 14, 4695 (2006).
[Crossref]

Xu, D.-X. D.-X.

Yang, J.

Yang, K. Y.

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Ye, W. N.

Yulaev, A.

W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
[Crossref]

S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, Light Sci. Appl. 7, 72 (2018).
[Crossref]

Zhu, W.

W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
[Crossref]

Zilkie, A.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Appl. Opt. (2)

Appl. Phys. (1)

T. Tamir and S. T. Peng, Appl. Phys. 14, 235 (1977).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

C. K. Madsen, J. Wagener, T. A. Strasser, D. Muehlner, M. A. Milbrodt, E. J. Laskowski, and J. DeMarco, IEEE J. Sel. Top. Quantum Electron. 4, 925 (1998).
[Crossref]

IEEE Photon. Technol. Lett. (1)

R. L. Davis, W. Long, C.-J. Wang, T. Lam, J. G. Ho, P. M. Nachman, J. Poylio, O. V. Mishechkin, and M. Fallahi, IEEE Photon. Technol. Lett. 16, 464 (2004).
[Crossref]

J. Opt. (1)

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, J. Opt. 18, 073003 (2016).
[Crossref]

Light Sci. Appl. (1)

S. Kim, D. A. Westly, B. J. Roxworthy, Q. Li, A. Yulaev, K. Srinivasan, and V. A. Aksyuk, Light Sci. Appl. 7, 72 (2018).
[Crossref]

Nature (1)

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, Nature 560, 565 (2018).
[Crossref]

New J. Phys. (1)

W. R. McGehee, W. Zhu, D. S. Barker, D. Westly, A. Yulaev, N. Klimov, A. Agrawal, S. Eckel, V. Aksyuk, and J. J. McClelland, New J. Phys. 23, 013021 (2021).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Photon. Res. (1)

Proc. IEEE (1)

R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, Proc. IEEE 106, 2144 (2018).
[Crossref]

Science (1)

N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle, D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao, O. Solgaard, R. L. Byer, and J. Vučković, Science 367, 79 (2020).
[Crossref]

Other (1)

A. Hadij-ElHouati, I. Molina-Fernández, R. Halir, J. G. Wangüemert-Pérez, and A. Ortega-Moñux, “Deflector lateral de haz único, multiplexor/demultiplexor y dispositivo alimentador de antena óptica que incorporan el deflector, y métodos que los utilizan,” Spain Patent Application P202031204 (2 December 2020).

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)

Fig. 1.
Fig. 1. Schematics of a distributed Bragg deflector. (a) Complete device including input waveguide with an adiabatic taper and the apodized Bragg deflector. Red arrows indicate the local energy flow (Poynting vector) direction. The silicon dioxide cladding is not shown for the sake of clarity. (b) Top view of a periodic section of the Bragg deflector. Schematic representation of the deflector geometry under (c) the 3D homogeneous model and (d) the 2D homogeneous effective index model (the anisotropic SWG metamaterial slab region is shown in green).
Fig. 2.
Fig. 2. K-space diagram for the grating shown in Fig. 1(d), including the single-beam diffraction window (shaded in gray). Red arrows show propagation vectors of diffractive beams for a conventional DBD design [12] when the period ${{\Lambda}_1}$ does not fulfill the single-beam condition. The blue arrow indicates the propagation direction vector of the single beam permitted for the period ${{\Lambda}_2}$ , hence mitigating radiation into the cladding.
Fig. 3.
Fig. 3. Floquet–Bloch analysis for a 3D homogeneous anisotropic model. Right axis, diffraction efficiency $\eta$ for the ${-}{1}$ st diffraction order in the SWG slab. Left axis, effective index for the ${-}{1}$ st diffraction order as a function of the wavelength-to-pitch ratio ( $\lambda /{\Lambda}$ ). Red lines indicate the limiting values for the single-beam operation. ${W_g} = 600\;{\rm nm} $ , ${\rm Dg} = 0.25$ , $\lambda = 1.55\; {\unicode{x00B5}{\rm m}}$ , ${{\Lambda}_{{\rm SWG}}} = 200\;{\rm nm} $ , and ${{\rm DC}_{{\rm SWG}}} = 0.5$ .
Fig. 4.
Fig. 4. (a) Grating strength (coupling coefficient) $\alpha$ and Floquet–Bloch index ${n_B}$ , as a function of modulation depth ${D_g}$ for the structure with an SWG slab (black curves), the homogenized anisotropic slab (red curves), and the homogenized isotropic slab (blue curves), calculated via Floquet–Bloch modal analysis. ${W_g} = 600\;{\rm nm} $ , $\lambda = 1.55\; {\unicode{x00B5}{\rm m}}$ , ${{\Lambda}_{{\rm SWG}}} = 200\;{\rm nm} $ , ${{\rm DC}_{{\rm SWG}}} = 0.5$ , and ${\Lambda} = 360\;{\rm nm} $ . (b) Target beam profile and the corresponding radiation strength $\alpha$ . (c) Grating modulation depth along the deflector structure.
Fig. 5.
Fig. 5. (a) 3D FDTD simulation of ${H_y}$ field evolution along the full DBD device. (b) Transversal cross section of ${H_y}$ field in an $XY$ plane near the center of the grating [ ${z} = 50\;{\unicode{x00B5}{\rm m}}$ , dashed green line in (a)]. (c)  ${H_y}$ field distribution in the slab waveguide (along the dashed blue line in a). The target profile is shown for comparison.

Equations (4)

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k z , m / k 0 = n B + m λ / Λ ,
( k x / ( k 0 n S i O 2 ) ) 2 + ( k z / ( k 0 n S i O 2 ) ) 2 = 1 ,
( k x / ( k 0 n e f f ) ) 2 + ( k z / ( k 0 n e f f ) ) 2 = 1 ,
n e f f < n B λ / Λ < n S i O 2 .

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