The optical phase
We demonstrate a Bell state analyzer that operates directly on frequency mismatch. Based on electrooptic modulators and Fouriertransform pulse shapers, our quantum frequency processor design implements interleaved Hadamard gates in discrete frequency modes. Experimental tests on entangledphoton inputs reveal fidelities of
 Award ID(s):
 2034019
 NSFPAR ID:
 10369358
 Publisher / Repository:
 Optical Society of America
 Date Published:
 Journal Name:
 Optica
 Volume:
 9
 Issue:
 3
 ISSN:
 23342536
 Page Range / eLocation ID:
 Article No. 280
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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$\mathrm{\varphi <\#comment/>}$ is a key quantity in the physics of light propagating through a turbulent medium. In certain respects, however, the statistics of the phasefactor ,$\mathrm{\psi <\#comment/>}=\mathrm{exp}<\#comment/>(i\mathrm{\varphi <\#comment/>})$ , are more relevant than the statistics of the phase itself. Here, we present a theoretical analysis of the 2D phasefactor spectrum${F}_{\mathrm{\psi <\#comment/>}}(\mathit{\kappa <\#comment/>})$ of a random phase screen. We apply the theory to four types of phase screens, each characterized by a powerlaw phase structure function,${D}_{\mathrm{\varphi <\#comment/>}}(r)=(r/{r}_{c}{)}^{\mathrm{\gamma <\#comment/>}}$ (where${r}_{c}$ is the phase coherence length defined by${D}_{\mathrm{\varphi <\#comment/>}}({r}_{c})=1\phantom{\rule{thickmathspace}{0ex}}{\mathrm{r}\mathrm{a}\mathrm{d}}^{2}$ ), and a probability density function${p}_{\mathrm{\alpha <\#comment/>}}(\mathrm{\alpha <\#comment/>})$ of the phase increments for a given spatial lag. We analyze phase screens with turbulent ($\mathrm{\gamma <\#comment/>}=5/3$ ) and quadratic ($\mathrm{\gamma <\#comment/>}=2$ ) phase structure functions and with normally distributed (i.e., Gaussian) versus Laplacian phase increments. We find that there is a pronounced bump in each of the four phasefactor spectra${F}_{\mathrm{\psi <\#comment/>}}(\mathrm{\kappa <\#comment/>})$ . The precise location and shape of the bump are different for the four phasescreen types, but in each case it occurs at$\mathrm{\kappa <\#comment/>}\sim <\#comment/>1/{r}_{c}$ . The bump is unrelated to the wellknown “Hill bump” and is not caused by diffraction effects. It is solely a characteristic of the refractiveindex statistics represented by the respective phase screen. We show that the secondorder$\mathrm{\psi <\#comment/>}$ statistics (covariance function, structure function, and spectrum) characterize a random phase screen more completely than the secondorder$\mathrm{\varphi <\#comment/>}$ counterparts. 
We study the relationship between the input phase delays and the output mode orders when using a pixelarray structure fed by multiple singlemode waveguides for tunable orbitalangularmomentum (OAM) beam generation. As an emitter of a freespace OAM beam, the designed structure introduces a transformation function that shapes and coherently combines multiple (e.g., four) equalamplitude inputs, with the
$k$ th input carrying a phase delay of$(k<\#comment/>1)\mathrm{\Delta <\#comment/>}\mathrm{\phi <\#comment/>}$ . The simulation results show that (1) the generated OAM order ℓ is dependent on the relative phase delay$\mathrm{\Delta <\#comment/>}\mathrm{\phi <\#comment/>}$ ; (2) the transformation function can be tailored by engineering the structure to support different tunable ranges (e.g.,$l=\{<\#comment/>1\},\{<\#comment/>1,+1\},\{<\#comment/>1,0,+1\}$ , or$\{<\#comment/>2,<\#comment/>1,+1,+2\}$ ); and (3) multiple independent coaxial OAM beams can be generated by simultaneously feeding the structure with multiple independent beams, such that each beam has its own$\mathrm{\Delta <\#comment/>}\mathrm{\phi <\#comment/>}$ value for the four inputs. Moreover, there is a tradeoff between the tunable range and the mode purity, bandwidth, and crosstalk, such that the increase of the tunable range leads to (a) decreased mode purity (from 91% to 75% for$l=<\#comment/>1$ ), (b) decreased 3 dB bandwidth of emission efficiency (from 285 nm for$l=\{<\#comment/>1\}$ to 122 nm for$l=\{<\#comment/>2,\phantom{\rule{thickmathspace}{0ex}}<\#comment/>1,\phantom{\rule{thickmathspace}{0ex}}+1,\phantom{\rule{thickmathspace}{0ex}}+2\}$ ), and (c) increased crosstalk within the Cband (from$<\#comment/>23.7$ to$<\#comment/>13.2\phantom{\rule{thickmathspace}{0ex}}\mathrm{d}\mathrm{B}$ when the tunable range increases from 2 to 4). 
We report on spectroscopic measurements on the
$4{f}^{7}6{s}^{2}{\phantom{\rule{thickmathspace}{0ex}}}^{8}{\phantom{\rule{negativethinmathspace}{0ex}}S}_{7/2}^{\circ <\#comment/>}\to <\#comment/>4{f}^{7}{(}^{8}\phantom{\rule{negativethinmathspace}{0ex}}{S}^{\circ <\#comment/>})6s6p{(}^{1}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{P}^{\circ <\#comment/>}){\phantom{\rule{thinmathspace}{0ex}}}^{8}\phantom{\rule{negativethinmathspace}{0ex}}{P}_{9/2}$ transition in neutral europium151 and europium153 at 459.4 nm. The center of gravity frequencies for the 151 and 153 isotopes, reported for the first time in this paper, to our knowledge, were found to be 652,389,757.16(34) MHz and 652,386,593.2(5) MHz, respectively. The hyperfine coefficients for the$6s6p{(}^{1}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{P}^{\circ <\#comment/>}){\phantom{\rule{thinmathspace}{0ex}}}^{8}\phantom{\rule{negativethinmathspace}{0ex}}{P}_{9/2}$ state were found to be$\mathrm{A}(151)=<\#comment/>228.84(2)\phantom{\rule{thickmathspace}{0ex}}\mathrm{M}\mathrm{H}\mathrm{z}$ ,$\mathrm{B}(151)=226.9(5)\phantom{\rule{thickmathspace}{0ex}}\mathrm{M}\mathrm{H}\mathrm{z}$ and$\mathrm{A}(153)=<\#comment/>101.87(6)\phantom{\rule{thickmathspace}{0ex}}\mathrm{M}\mathrm{H}\mathrm{z}$ ,$\mathrm{B}(153)=575.4(1.5)\phantom{\rule{thickmathspace}{0ex}}\mathrm{M}\mathrm{H}\mathrm{z}$ , which all agree with previously published results except for A(153), which shows a small discrepancy. The isotope shift is found to be 3163.8(6) MHz, which also has a discrepancy with previously published results. 
By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1Wasserstein metric over the planar grid
$\{0,1,\dots , n\}^2$ has$L_1$ distortion bounded below by a constant multiple of$\sqrt {\log n}$ . We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if$\{G_n\}_{n=1}^\infty$ is a sequence of graphs whose isoperimetric dimension and Lipschitzspectral dimension equal a common number$\delta \in [2,\infty )$ , then the 1Wasserstein metric over$G_n$ has$L_1$ distortion bounded below by a constant multiple of$(\log G_n)^{\frac {1}{\delta }}$ . We proceed to compute these dimensions for$\oslash$ powers of certain graphs. In particular, we get that the sequence of diamond graphs$\{\mathsf {D}_n\}_{n=1}^\infty$ has isoperimetric dimension and Lipschitzspectral dimension equal to 2, obtaining as a corollary that the 1Wasserstein metric over$\mathsf {D}_n$ has$L_1$ distortion bounded below by a constant multiple of$\sqrt {\log  \mathsf {D}_n}$ . This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of$L_1$ embeddable graphs whose sequence of 1Wasserstein metrics is not$L_1$ embeddable. 
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$R$ Series (GOESR), however, provides subhourly imagery and the opportunity to overcome this deficit and to leverage a large repository of existing GOESR aquatic observations. The fulfillment of this opportunity is assessed herein using a spectrally simplified, twochannel aquatic algorithm consistent with ABI wave bands to estimate the diffuse attenuation coefficient for photosynthetically available radiation,${K}_{d}(\mathrm{P}\mathrm{A}\mathrm{R})$ . First, anin situ ABI dataset was synthesized using a globally representative dataset of above and inwater radiometric data products. Values of${K}_{d}(\mathrm{P}\mathrm{A}\mathrm{R})$ were estimated by fitting the ratio of the shortest and longest visible wave bands from thein situ ABI dataset to coincident,in situ ${K}_{d}(\mathrm{P}\mathrm{A}\mathrm{R})$ data products. The algorithm was evaluated based on an iterative crossvalidation 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$0.319\phantom{\rule{thinmathspace}{0ex}}{\mathrm{m}}^{<\#comment/>1}$ , or 8.5% of the range in the validation dataset. Second, coincident midday 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 bluetored reflectance ratio values and MODIS default diffuse attenuation coefficient for spectral downward irradiance at 490 nm,${K}_{d}(490)$ , 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 subhourly, visible aquatic data product that is applicable to watermass tracing and physical oceanography research.