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1. Characterization of field-effect mobility at optical frequency by microring resonators

   https://doi.org/10.1364/PRJ.416656  

   Hsu, Wei-Che ; Li, Erwen ; Zhou, Bokun ; Wang, Alan X. ( March 2021 , Photonics Research)

   A novel characterization method is proposed to extract the optical frequency field-effect mobility (${\mu }_{\mathrm{op},\mathrm{FE}}$) of transparent conductive oxide (TCO) materials by a tunable silicon microring resonator with a heterogeneously integrated titanium-doped indium oxide$(\mathrm{ITiO})/{\mathrm{SiO}}_2/\mathrm{silicon}$metal–oxide–semiconductor (MOS) capacitor. By operating the microring in the accumulation mode, the quality factor and resonance wavelength shift are measured and subsequently used to derive the${\mu }_{\mathrm{op},\mathrm{FE}}$in the ultra-thin accumulation layer. Experimental results demonstrate that the${\mu }_{\mathrm{op},\mathrm{FE}}$of ITiO increases from 25.3 to$38.4{\mathrm{cm}}^2\text{⋅}{\mathrm{V}}^{-1}\text{⋅}{\mathrm{s}}^{-1}$with increasing gate voltages, which shows a similar trend as that at the electric frequency.

    

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2. Direct chip-scale optical frequency divider via regenerative harmonic injection locking

   https://doi.org/10.1364/OL.413335  

   Bustos-Ramirez, Ricardo ; Trask, Lawrence_R ; Bhardwaj, Ashish ; Hoefler, Gloria_E ; Kish, Fred_A ; Delfyett, Peter_J ( February 2021 , Optics Letters)

   A novel optical frequency division technique, called regenerative harmonic injection locking, is used to transfer the timing stability of an optical frequency comb with a repetition rate in the millimeter wave range ($\sim <\#comment/>300\mathrm{G}\mathrm{H}\mathrm{z}$) to a chip-scale mode-locked laser with a$\sim <\#comment/>10\mathrm{G}\mathrm{H}\mathrm{z}$repetition rate. By doing so, the 300 GHz optical frequency comb is optically divided by a factor of$30\times <\#comment/>$to 10 GHz. The stability of the mode-locked laser after regenerative harmonic injection locking is$\sim <\#comment/>{10}^{-<\#comment/>12}$at 1 s with a$1/\mathrm{\tau <\#comment/>}$trend. To facilitate optical frequency division, a coupled opto-electronic oscillator is implemented to assist the injection locking process. This technique is exceptionally power efficient, as it uses less than$100\text{µ<\#comment/>}\mathrm{W}$of optical power to achieve stable locking.

    

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3. Exceptional point of sixth-order degeneracy in a modified coupled-resonator optical waveguide

   https://doi.org/10.1364/JOSAB.385198  

   Nada, Mohamed_Y ; Capolino, Filippo ( July 2020 , Journal of the Optical Society of America B)

   We demonstrate for the first time, to our knowledge, the occurrence of asixth-order exceptional point of degeneracy (EPD) in a realistic multimode optical photonic structure by using a modified periodic coupled-resonator optical waveguide (CROW) at the optical wavelength${\mathrm{\lambda <\#comment/>}}_e=1550\text{nm}$. The sixth-order EPD is obtained in a CROW without the need of loss or gain, and such an EPD corresponds to a very special band edge of the periodic photonic structure where six eigenmodes coalesce, so we refer to it as the sixth-order degenerate band edge (6DBE). Moreover, we report a new scaling law of the quality factor$Q$of an optical cavity made of such a periodic 6DBE-CROW with cavity length as$Q\propto <\#comment/>N^7$, when operating near the 6DBE with$N$being the number of unit cells in the periodic finite-length CROW. Furthermore, we elaborate on the application of the 6DBE to ultralow-threshold lasers. We present a novel scaling law of the lasing threshold that scales as$N^{-<\#comment/>7}$when operating near the 6DBE. Also, we show the superiority of the threshold scaling of the 6DBE-CROW to the scaling of another CROW with the same size operating near a fourth-order EPD that is often referred to as the degenerate band edge (DBE). The lasing threshold scaling of the DBE-CROW laser is shown here for the first time to our knowledge. We also discuss the high sensitivity of the proposed 6DBE-CROW to perturbations, which may find applications in sensors, modulators, optical switches, nonlinear devices, and$Q$-switching cavities.

    

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4. Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

   https://doi.org/10.1007/s00440-021-01070-4  

   Miller, Jason ; Sheffield, Scott ; Werner, Wendelin ( June 2021 , Probability Theory and Related Fields)

   We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$for$$\kappa '$$${\kappa }^{\prime }$in (4, 8) that is drawn on an independent$$\gamma $$$\gamma$-LQG surface for$$\gamma ^2=16/\kappa '$$${\gamma }^2=16/{\kappa }^{\prime }$. The results are similar in flavor to the ones from our companion paper dealing with$$\hbox {CLE}_{\kappa }$$${\text{CLE}}_{\kappa }$for$$\kappa $$$\kappa$in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled “CLE Percolations” described the law of interfaces obtained when coloring the loops of a$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$independently into two colors with respective probabilitiespand$$1-p$$$1-p$. This description was complete up to one missing parameter$$\rho $$$\rho$. The results of the present paper about CLE on LQG allow us to determine its value in terms ofpand$$\kappa '$$${\kappa }^{\prime }$. It shows in particular that$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$and$$\hbox {CLE}_{16/\kappa '}$$${\text{CLE}}_{16/{\kappa }^{\prime }}$are related via a continuum analog of the Edwards-Sokal coupling between$$\hbox {FK}_q$$${\text{FK}}_q$percolation and theq-state Potts model (which makes sense even for non-integerqbetween 1 and 4) if and only if$$q=4\cos ^2(4\pi / \kappa ')$$$q=4{cos}^2(4\pi /{\kappa }^{\prime })$. This provides further evidence for the long-standing belief that$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$and$$\hbox {CLE}_{16/\kappa '}$$${\text{CLE}}_{16/{\kappa }^{\prime }}$represent the scaling limits of$$\hbox {FK}_q$$${\text{FK}}_q$percolation and theq-Potts model whenqand$$\kappa '$$${\kappa }^{\prime }$are related in this way. Another consequence of the formula for$$\rho (p,\kappa ')$$$\rho (p,{\kappa }^{\prime })$is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

    

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5. Multipliers on bi-parameter Haar system Hardy spaces

   https://doi.org/10.1007/s00208-024-02887-9  

   Lechner, R. ; Motakis, P. ; Müller, P_F_X ; Schlumprecht, Th ( May 2024 , Mathematische Annalen)

   Let$$(h_I)$$$\left(h_I\right)$denote the standard Haar system on [0, 1], indexed by$$I\in \mathcal {D}$$$I\in D$, the set of dyadic intervals and$$h_I\otimes h_J$$$h_I\otimes h_J$denote the tensor product$$(s,t)\mapsto h_I(s) h_J(t)$$$(s,t)\mapsto h_I\left(s\right)h_J\left(t\right)$,$$I,J\in \mathcal {D}$$$I,J\in D$. We consider a class of two-parameter function spaces which are completions of the linear span$$\mathcal {V}(\delta ^2)$$$V\left({\delta }^2\right)$of$$h_I\otimes h_J$$$h_I\otimes h_J$,$$I,J\in \mathcal {D}$$$I,J\in D$. This class contains all the spaces of the formX(Y), whereXandYare either the Lebesgue spaces$$L^p[0,1]$$$L^p[0,1]$or the Hardy spaces$$H^p[0,1]$$$H^p[0,1]$,$$1\le p < \infty $$$1\le p<\infty$. We say that$$D:X(Y)\rightarrow X(Y)$$$D:X\left(Y\right)\to X\left(Y\right)$is a Haar multiplier if$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$$D(h_I\otimes h_J)=d_{I,J}{h}_I\otimes h_J$, where$$d_{I,J}\in \mathbb {R}$$$d_{I,J}\in R$, and ask which more elementary operators factor throughD. A decisive role is played by theCapon projection$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$$C:V\left({\delta }^2\right)\to V\left({\delta }^2\right)$given by$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$$C{h}_I\otimes h_J=h_I\otimes h_J$if$$|I|\le |J|$$$\left|I\right|\le \left|J\right|$, and$$\mathcal {C} h_I\otimes h_J = 0$$$C{h}_I\otimes h_J=0$if$$|I| > |J|$$$\left|I\right|>\left|J\right|$, as our main result highlights: Given any bounded Haar multiplier$$D:X(Y)\rightarrow X(Y)$$$D:X\left(Y\right)\to X\left(Y\right)$, there exist$$\lambda ,\mu \in \mathbb {R}$$$\lambda ,\mu \in R$such that$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$$\begin{array}{c}\lambda C+\mu (\text{Id}-C)\text{approximately 1-projectionally factors through}D,\end{array}$i.e., for all$$\eta > 0$$$\eta >0$, there exist bounded operatorsA, Bso thatABis the identity operator$${{\,\textrm{Id}\,}}$$$\text{Id}$,$$\Vert A\Vert \cdot \Vert B\Vert = 1$$$\Vert A\Vert ·\Vert B\Vert =1$and$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta $$$\Vert \lambda C+\mu (\text{Id}-C)-ADB\Vert <\eta$. Additionally, if$$\mathcal {C}$$$C$is unbounded onX(Y), then$$\lambda = \mu $$$\lambda =\mu$and then$${{\,\textrm{Id}\,}}$$$\text{Id}$either factors throughDor$${{\,\textrm{Id}\,}}-D$$$\text{Id}-D$.

    

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