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1. Quasi-monolithic heterodyne laser interferometer for inertial sensing

   https://doi.org/10.1364/OL.473476  

   Zhang, Yanqi; Guzman, Felipe (January 2022, Optics Letters)

   We present a compact heterodyne laser interferometer developed for high-sensitivity displacement sensing applications. This interferometer consists of customized prisms and wave plates assembled as a quasi-monolithic unit to realize a miniaturized system. The interferometer design adopts a common-mode rejection scheme to provide a high rejection ratio to common environmental noise. Experimental tests in vacuum show a displacement sensitivity level of $11\mathrm{p}\mathrm{m}/\sqrt{\mathrm{H}\mathrm{z}}$at $100\mathrm{m}\mathrm{H}\mathrm{z}$and as low as $0.6\mathrm{p}\mathrm{m}/\sqrt{\mathrm{H}\mathrm{z}}$above $1\mathrm{p}\mathrm{m}$. The prototype unit is $20\mathrm{m}\mathrm{m}\times <\#comment/>20\mathrm{m}\mathrm{m}\times <\#comment/>10\mathrm{m}\mathrm{m}$in size and weighs $4.5\mathrm{g}$, allowing subsequent integration in compact systems. 

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2. Strong optomechanical interactions with long-lived fundamental acoustic waves

   https://doi.org/10.1364/OPTICA.476764  

   Xu, Wendao; Iyer, Arjun; Jin, Lei; Set, Sze Y.; Renninger, William H. (January 2023, Optica)

   Traveling-wave optomechanical interactions, known as Brillouin interactions, have now been established as a powerful and versatile resource for photonic sources, sensors, and radio-frequency processors. However, established Brillouin-based interactions with sufficient interaction strengths involve short phonon lifetimes, which critically limit their performance for applications, including radio-frequency filtering and optomechanical storage devices. Here, we investigate a new paradigm of optomechanical interactions with tightly confined fundamental acoustic modes, which enables the unique and desirable combination of high optomechanical coupling, long phonon lifetimes, tunable phonon frequencies, and single-sideband amplification. Using sensitive four-wave mixing spectroscopy controlling for noise and spatial mode coupling, optomechanical interactions with long $><\#comment/>2\text{µ<\#comment/>}\mathrm{s}$phonon lifetimes and strong $><\#comment/>400{\mathrm{W}}^{-<\#comment/>1}{\mathrm{m}}^{-<\#comment/>1}$coupling are observed in a tapered fiber. In addition, we demonstrate novel phonon self-interference effects resulting from the unique combination of an axially varying device geometry with long phonon lifetimes. A generalized theoretical model, in excellent agreement with experiments, is developed with broad applicability to inhomogeneous optomechanical systems. 

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3. Sums of linear transformations

   https://doi.org/10.1090/tran/9433  

   Conlon, David; Lim, Jeck (July 2025, Transactions of the American Mathematical Society)

   We show that if ${\mathcal{L}}_1$and ${\mathcal{L}}_2$are linear transformations from ${\mathbb{Z}}^d$to ${\mathbb{Z}}^d$satisfying certain mild conditions, then, for any finite subset $A$of ${\mathbb{Z}}^d$, $|{\mathcal{L}}_{1}A+{\mathcal{L}}_{2}A|\ge <\#comment/>{(|det({\mathcal{L}}_1){|}^{1/d}+|det({\mathcal{L}}_2\left){|}^{1/d}\right)}^d|A|-<\#comment/>o\left(|A|\right).$This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of ${\mathcal{L}}_1$and ${\mathcal{L}}_2$. As an application, we prove a lower bound for $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|$when $A$is a finite set of real numbers and $\mathrm{\lambda <\#comment/>}$is an algebraic number. In particular, when $\mathrm{\lambda <\#comment/>}$is of the form $(p/q{)}^{1/d}$for some $p,q,d\in <\#comment/>\mathbb{N}$, each taken as small as possible for such a representation, we show that $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|\ge <\#comment/>(p^{1/d}+q^{1/d}{)}^d|A|-<\#comment/>o(|A|).$This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $\mathrm{\lambda <\#comment/>}=\sqrt{2}$. 

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4. Random walk on group extensions

   https://doi.org/10.1090/tran/9309  

   Golsefidy, S Alireza; Srinivas, Srivatsa (January 2025, Transactions of the American Mathematical Society)

   We study random walks on various group extensions. Under certain bounded generation and bounded scaled conditions, we estimate the spectral gap of a random walk on a quasi-random-by-nilpotent group in terms of the spectral gap of its projection to the quasi-random part. We also estimate the spectral gap of a random-walk on a product of two quasi-random groups in terms of the spectral gap of its projections to the given factors. Based on these results, we estimate the spectral gap of a random walk on the ${\mathbb{F}}_q$-points of a perfect algebraic group $\mathbb{G}$in terms of the spectral gap of its projections to the almost simple factors of the semisimple quotient of $\mathbb{G}$. These results extend a work of Lindenstrauss and Varjú and an earlier work of the authors. Moreover, using a result of Breuillard and Gamburd, we show that there is an infinite set $\mathcal{P}$of primes of density one such that, if $k$is a positive integer and $\mathbb{G}=\mathbb{U}⋊<\#comment/>({\mathrm{SL}}_2{)}_{\mathbb{Q}}^m$is a perfect group and $\mathbb{U}$is a unipotent group, then the family of all the Cayley graphs of $\mathbb{G}\left(\mathbb{Z}/\underset{i=1}{\overset{k}{\prod <\#comment/>}}{p}_i\mathbb{Z}\right)$, $p_i\in <\#comment/>\mathcal{P}$, is a family of expanders. 

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5. Spectrally simplified approach for leveraging legacy geostationary oceanic observations

   https://doi.org/10.1364/AO.465491  

   Houskeeper, Henry F.; Hooker, Stanford B.; Cavanaugh, Kyle C. (January 2022, Applied Optics)

   The use of multispectral geostationary satellites to study aquatic ecosystems improves the temporal frequency of observations and mitigates cloud obstruction, but no operational capability presently exists for the coastal and inland waters of the United States. The Advanced Baseline Imager (ABI) on the current iteration of the Geostationary Operational Environmental Satellites, termed the $R$Series (GOES-R), however, provides sub-hourly imagery and the opportunity to overcome this deficit and to leverage a large repository of existing GOES-R aquatic observations. The fulfillment of this opportunity is assessed herein using a spectrally simplified, two-channel aquatic algorithm consistent with ABI wave bands to estimate the diffuse attenuation coefficient for photosynthetically available radiation, $K_d\left(\mathrm{P}\mathrm{A}\mathrm{R}\right)$. First, anin situABI dataset was synthesized using a globally representative dataset of above- and in-water radiometric data products. Values of $K_d\left(\mathrm{P}\mathrm{A}\mathrm{R}\right)$were estimated by fitting the ratio of the shortest and longest visible wave bands from thein situABI dataset to coincident,in situ $K_d\left(\mathrm{P}\mathrm{A}\mathrm{R}\right)$data products. The algorithm was evaluated based on an iterative cross-validation analysis in which 80% of the dataset was randomly partitioned for fitting and the remaining 20% was used for validation. The iteration producing the median coefficient of determination ( $R^2$) value (0.88) resulted in a root mean square difference of $0.319{\mathrm{m}}^{-<\#comment/>1}$, or 8.5% of the range in the validation dataset. Second, coincident mid-day images of central and southern California from ABI and from the Moderate Resolution Imaging Spectroradiometer (MODIS) were compared using Google Earth Engine (GEE). GEE default ABI reflectance values were adjusted based on a near infrared signal. Matchups between the ABI and MODIS imagery indicated similar spatial variability ( $R^2=0.60$) between ABI adjusted blue-to-red reflectance ratio values and MODIS default diffuse attenuation coefficient for spectral downward irradiance at 490 nm, $K_d\left(490\right)$, values. This work demonstrates that if an operational capability to provide ABI aquatic data products was realized, the spectral configuration of ABI would potentially support a sub-hourly, visible aquatic data product that is applicable to water-mass tracing and physical oceanography research. 

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