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18 pages, 6433 KiB  
Article
Dual-Wavelength Interferometric Detection Technology for Wind and Temperature Fields in the Martian Middle and Upper Atmosphere Based on LCTF
by Yanqiang Wang, Biyun Zhang, Chunmin Zhang, Shiping Guo, Tingyu Yan, Yifan He and William Ward
Remote Sens. 2024, 16(19), 3591; https://doi.org/10.3390/rs16193591 - 26 Sep 2024
Viewed by 627
Abstract
A dual-wavelength spaceborne Martian polarized wind imaging Michelson interferometer based on liquid crystal tunable filters (LCTF-MPWIMI) has been proposed for the remote sensing detection of dynamic parameters such as wind speed and temperature in the middle and upper atmosphere of Mars. Using the [...] Read more.
A dual-wavelength spaceborne Martian polarized wind imaging Michelson interferometer based on liquid crystal tunable filters (LCTF-MPWIMI) has been proposed for the remote sensing detection of dynamic parameters such as wind speed and temperature in the middle and upper atmosphere of Mars. Using the detected Martian oxygen atom emission lines at 557.7 nm and 630.0 nm as observation spectral lines, this technology extends the detection altitude range for Martian atmospheric wind speed and temperature to 60–180 km. By leveraging the different spectral line visibility of the interferograms at the two wavelengths, a novel method for measuring Martian atmospheric temperature is proposed: the dual-wavelength spectral line visibility product method. This new approach reduces the uncertainty of temperature detection compared to traditional single spectral line visibility methods, while maintaining the precision of wind speed measurements. The feasibility of the LCTF-MPWIMI for measuring wind and temperature fields in the Martian middle and upper atmosphere has been validated through theoretical modeling and computer simulations. The interferometer, as a key component of the system, has been designed and analyzed. The proposed LCTF-MPWIMI instrument is free of mechanical moving parts, offering flexible wavelength selection and facilitating miniaturization. The dual-wavelength temperature measurement method introduced in this work provides superior temperature measurement precision compared to any single spectral line when the signal-to-noise ratio (SNR) of the interferograms is comparable. Moreover, this method does not impose specific requirements on the atomic state of the spectral lines, making it broadly applicable to similar interferometric wind measurement instruments. These innovations offer advanced tools and methodologies for measuring wind speeds and temperatures in the atmospheres of Mars and other planets. Full article
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Figure 1
<p>Optical layout of the LCTF-MPWIMI. The LCTF serves both as an optical filter and as a linear polarizer, with the red arrow indicating that its transmission axis forms a 45° angle with the <span class="html-italic">x</span>-axis. P<sub>2</sub> refers to an array of linear polarizers, consisting of four sub-linear polarizers with their transmission axes at angles of 0°, 45°, 90°, and 135° relative to the <span class="html-italic">x</span>-axis, respectively.</p>
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<p>Mars atmospheric wind field detection with LCTF-MPWIMI: (<b>a</b>) single-field-of-view Limb; (<b>b</b>) dual-Field-of-view limb geometry.</p>
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<p>The limb profiles of [Oi] 557.7 nm dayglow intensity observed with UVIS on 28 April 2019 (green dots) [<a href="#B10-remotesensing-16-03591" class="html-bibr">10</a>].</p>
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<p>Wind velocity error variations with OPD and SNR.</p>
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<p>Emissions line visibility with OPD and atmospheric temperature.</p>
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<p>Simulated four normalized zero-wind interferograms of 557.7 nm airglow. The phase steps are 0°, 90°, 180°, and 270°, respectively. From the center to the edge of the images, the incident angle varies from 0° to 9.3°. The simulated atmospheric temperature is 150 K.</p>
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<p>Simulated four normalized zero wind interferograms of 630.0 nm airglow. The phase steps are 0°, 90°, 180°, and 270°, respectively. From the center to the edge of the images, the incident angle varies from 0° to 9.3°. The simulated atmospheric temperature is 150 K.</p>
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<p>Typical wind speed and temperature data from the Mars climate database [<a href="#B19-remotesensing-16-03591" class="html-bibr">19</a>]: (<b>a</b>) typical Martian atmospheric wind speed map; (<b>b</b>) typical Martian atmospheric temperature map.</p>
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<p>Four-phase stepping wind imaging interferograms for the 557.7 nm airglow spectral lines (<span class="html-italic">SNR</span> = 300).</p>
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<p>Four-phase stepping wind imaging interferograms for the 630.0 nm airglow spectral lines (<span class="html-italic">SNR</span> = 115).</p>
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<p>Retrieval of Martian atmospheric wind speed using 557.7 nm interferograms: (<b>a</b>) retrieved wind speed map; (<b>b</b>) wind speed error map.</p>
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<p>Retrieval of Martian atmospheric temperature using two spectral lines: (<b>a</b>) retrieved temperature map from single spectral line; (<b>b</b>) single spectral line temperature error map; (<b>c</b>) retrieved temperature map from dual-spectral line; (<b>d</b>) dual-spectral line temperature error map.</p>
Full article ">Figure 13
<p>Typical wind speed and temperature profiles from the Mars Climate Database [<a href="#B19-remotesensing-16-03591" class="html-bibr">19</a>], along with their gradients with respect to height, at a solar longitude of Ls = 250°, local time of 12:00, and a Martian longitude and latitude of 0°: (<b>a</b>) horizontal wind speed profile; (<b>b</b>) calculated horizontal wind speed gradient; (<b>c</b>) temperature profile; (<b>d</b>) calculated temperature gradient.</p>
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17 pages, 2151 KiB  
Review
Research Prospects for the Optimization of Magneto-Optical Trap Parameters for Cold Atom Interferometers
by Dongyi Li, Fangjun Qin, Rui Xu and An Li
Appl. Sci. 2024, 14(16), 7062; https://doi.org/10.3390/app14167062 - 12 Aug 2024
Viewed by 1096
Abstract
This study examines parameter optimization for magneto-optical traps (MOTs) to increase trapping efficiency and improve cold atom interferometer performance. Operational principles of MOTs, control parameters, and performance metrics such as volume, atomic loading time, and resonance frequency are discussed. This research also reviews [...] Read more.
This study examines parameter optimization for magneto-optical traps (MOTs) to increase trapping efficiency and improve cold atom interferometer performance. Operational principles of MOTs, control parameters, and performance metrics such as volume, atomic loading time, and resonance frequency are discussed. This research also reviews existing studies on the parameter optimization of MOTs, highlights challenges, and offers suggestions for future research. It proposes enhancing performance metrics, optimization techniques, and operational models to increase precision and practicality in parameter optimization for MOTs in cold atom interferometers. Full article
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Figure 1
<p>Diagram of a typical MOT.</p>
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<p>Schematic diagram of a one-dimensional MOT.</p>
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<p>Atomic motion in an MOT.</p>
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<p>Atomic gravimeter measurement sequence diagram.</p>
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<p>Atomic gravimeter structure diagram.</p>
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<p>The number of atoms at various vibration frequencies and laser intensities.</p>
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<p>Atom-loading process in the MOT.</p>
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28 pages, 10554 KiB  
Review
Classical and Atomic Gravimetry
by Jie Fang, Wenzhang Wang, Yang Zhou, Jinting Li, Danfang Zhang, Biao Tang, Jiaqi Zhong, Jiangong Hu, Feng Zhou, Xi Chen, Jin Wang and Mingsheng Zhan
Remote Sens. 2024, 16(14), 2634; https://doi.org/10.3390/rs16142634 - 18 Jul 2024
Cited by 1 | Viewed by 4086
Abstract
Gravity measurements have important applications in geophysics, resource exploration, geodesy, and inertial navigation. The range of classical gravimetry includes laser interferometer (LI)-based absolute gravimeters, spring relative gravimeters, superconducting gravimeters, airborne/marine gravimeters, micro-electromechanical-system (MEMS) gravimeters, as well as gravity satellites and satellite altimetry. Atomic [...] Read more.
Gravity measurements have important applications in geophysics, resource exploration, geodesy, and inertial navigation. The range of classical gravimetry includes laser interferometer (LI)-based absolute gravimeters, spring relative gravimeters, superconducting gravimeters, airborne/marine gravimeters, micro-electromechanical-system (MEMS) gravimeters, as well as gravity satellites and satellite altimetry. Atomic gravimetry is a new absolute gravity measurement technology based on atom interferometers (AIs) and features zero drift, long-term stability, long-term continuous measurements, and high precision. Atomic gravimetry has been used to measure static, marine, and airborne gravity; gravity gradient; as well as acceleration to test the weak equivalence principle at the China Space Station. In this paper, classical gravimetry is introduced, and the research progress on static and airborne/marine atomic gravimeters, space AIs, and atomic gravity gradiometers is reviewed. In addition, classical and atomic gravimetry are compared. Future atomic gravimetry development trends are also discussed with the aim of jointly promoting the further development of gravity measurement technologies alongside classical gravimetry. Full article
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Figure 1
<p>(<b>a</b>) Schematic of the working principle of the FG5 and A10 gravimeters [<a href="#B46-remotesensing-16-02634" class="html-bibr">46</a>,<a href="#B48-remotesensing-16-02634" class="html-bibr">48</a>]; (<b>b</b>) FG5 gravimeter [<a href="#B46-remotesensing-16-02634" class="html-bibr">46</a>]; (<b>c</b>) A10 gravimeter [<a href="#B48-remotesensing-16-02634" class="html-bibr">48</a>].</p>
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<p>(<b>a</b>) Schematic of zero-length spring suspension; (<b>b</b>) gPhoneX gravimeter [<a href="#B50-remotesensing-16-02634" class="html-bibr">50</a>]; (<b>c</b>) CG6 gravimeter [<a href="#B51-remotesensing-16-02634" class="html-bibr">51</a>].</p>
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<p>The iGrav SG transportable superconducting gravimeter [<a href="#B57-remotesensing-16-02634" class="html-bibr">57</a>].</p>
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<p>(<b>a</b>) GT-2A airborne gravimeter [<a href="#B64-remotesensing-16-02634" class="html-bibr">64</a>]; (<b>b</b>) SEA III marine gravimeter [<a href="#B65-remotesensing-16-02634" class="html-bibr">65</a>]; (<b>c</b>) SGA-WZ strapdown gravimeter [<a href="#B67-remotesensing-16-02634" class="html-bibr">67</a>].</p>
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<p>(<b>a</b>) Schematic of the mass–spring system of a MEMS gravimeter based on displacement sensing [<a href="#B68-remotesensing-16-02634" class="html-bibr">68</a>]; (<b>b</b>) the schematic of a MEMS accelerometer for measuring gravity based on resonant sensing [<a href="#B74-remotesensing-16-02634" class="html-bibr">74</a>].</p>
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<p>Schematic of satellite altimetry.</p>
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<p>Spatiotemporal diagram of AIs using SRTs. Three Raman pulses split, redirect, and recombine the atomic cloud. Atoms interfere after the third pulse.</p>
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<p>Typical transportable static atomic gravimeters. (<b>a</b>) LNE-SYRTE in France [<a href="#B114-remotesensing-16-02634" class="html-bibr">114</a>]; (<b>b</b>) Exail in France [<a href="#B101-remotesensing-16-02634" class="html-bibr">101</a>]; (<b>c</b>) University of California, Berkeley [<a href="#B102-remotesensing-16-02634" class="html-bibr">102</a>]; (<b>d</b>) Humboldt University in Germany [<a href="#B115-remotesensing-16-02634" class="html-bibr">115</a>]; (<b>e</b>) Zhejiang University and Zhejiang University of Technology [<a href="#B116-remotesensing-16-02634" class="html-bibr">116</a>]; (<b>f</b>) Huazhong University of Science and Technology e.g., [<a href="#B25-remotesensing-16-02634" class="html-bibr">25</a>]; (<b>g</b>) University of Science and Technology of China e.g., [<a href="#B26-remotesensing-16-02634" class="html-bibr">26</a>]; (<b>h</b>) WAG-C5-1 of APM [<a href="#B117-remotesensing-16-02634" class="html-bibr">117</a>].</p>
Full article ">Figure 8 Cont.
<p>Typical transportable static atomic gravimeters. (<b>a</b>) LNE-SYRTE in France [<a href="#B114-remotesensing-16-02634" class="html-bibr">114</a>]; (<b>b</b>) Exail in France [<a href="#B101-remotesensing-16-02634" class="html-bibr">101</a>]; (<b>c</b>) University of California, Berkeley [<a href="#B102-remotesensing-16-02634" class="html-bibr">102</a>]; (<b>d</b>) Humboldt University in Germany [<a href="#B115-remotesensing-16-02634" class="html-bibr">115</a>]; (<b>e</b>) Zhejiang University and Zhejiang University of Technology [<a href="#B116-remotesensing-16-02634" class="html-bibr">116</a>]; (<b>f</b>) Huazhong University of Science and Technology e.g., [<a href="#B25-remotesensing-16-02634" class="html-bibr">25</a>]; (<b>g</b>) University of Science and Technology of China e.g., [<a href="#B26-remotesensing-16-02634" class="html-bibr">26</a>]; (<b>h</b>) WAG-C5-1 of APM [<a href="#B117-remotesensing-16-02634" class="html-bibr">117</a>].</p>
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<p>Comparison of the results of the gravimeters at the ICAG-2017 e.g., [<a href="#B49-remotesensing-16-02634" class="html-bibr">49</a>].</p>
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<p>Marine atomic gravimeters and their measurement results. (<b>a</b>) The marine atomic gravimeter developed by ONERA and (<b>b</b>) its measurement results along the calibration line in a marine environment e.g., [<a href="#B28-remotesensing-16-02634" class="html-bibr">28</a>]. (<b>c</b>) A marine atomic gravimeter developed by ZJUT [<a href="#B146-remotesensing-16-02634" class="html-bibr">146</a>] and (<b>d</b>) its measurement results from a gravity survey in a marine area [<a href="#B147-remotesensing-16-02634" class="html-bibr">147</a>]. (<b>e</b>) A marine atomic gravimeter jointly developed by APM and NUE, and (<b>f</b>) its measurement results from a repeated gravity survey in a marine area [<a href="#B148-remotesensing-16-02634" class="html-bibr">148</a>].</p>
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<p>(<b>a</b>) The airborne atomic gravimeter (GIRAFE) of ONERA on an aircraft, a classical LaCoste&amp;Romberg (L&amp;R) spring airborne gravimeter, and an iMAR strapdown airborne gravimeter for comparative measurements; (<b>b</b>) data measured by GIRAFE in an airborne survey e.g., [<a href="#B29-remotesensing-16-02634" class="html-bibr">29</a>].</p>
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<p>The space dual-species AI developed by APM at the China Space Station.</p>
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<p>(<b>a</b>) AGG of the University of Birmingham [<a href="#B162-remotesensing-16-02634" class="html-bibr">162</a>] and (<b>b</b>) AGG of APM e.g., [<a href="#B31-remotesensing-16-02634" class="html-bibr">31</a>].</p>
Full article ">
19 pages, 2622 KiB  
Article
Sensitivity of a Point-Source-Interferometry-Based Inertial Measurement Unit Employing Large Momentum Transfer and Launched Atoms
by Jinyang Li, Timothy Kovachy, Jason Bonacum and Selim M. Shahriar
Atoms 2024, 12(6), 32; https://doi.org/10.3390/atoms12060032 - 11 Jun 2024
Viewed by 1451
Abstract
We analyze theoretically the sensitivity of accelerometry and rotation sensing with a point source interferometer employing large momentum transfer (LMT) and present a design of an inertial measurement unit (IMU) that can measure rotation around and acceleration along each of the three axes. [...] Read more.
We analyze theoretically the sensitivity of accelerometry and rotation sensing with a point source interferometer employing large momentum transfer (LMT) and present a design of an inertial measurement unit (IMU) that can measure rotation around and acceleration along each of the three axes. In this design, the launching technique is used to realize the LMT process without the need to physically change directions of the Raman pulses, thus significantly simplifying the apparatus. We also describe an explicit scheme for such an IMU. Full article
(This article belongs to the Special Issue Advances in and Prospects for Matter Wave Interferometry)
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Figure 1
<p>Illustration of the pulse sequence for the LMT protocol for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>. For <span class="html-italic">n</span>th order LMT, 4<span class="html-italic">n</span> <math display="inline"><semantics> <mi>π</mi> </semantics></math> pulses are added. A pulse with an odd (even) subsript has an effective propagation direction opposite (identical) to that of the three original pulses.</p>
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<p>Accelerometry sensitivity of a PSI as a function of <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mo>Ω</mo> </msub> <msub> <mi>σ</mi> <mi mathvariant="normal">f</mi> </msub> </mrow> </semantics></math> for Gaussian <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> when (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>a</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>a</mi> </msub> <mo>=</mo> <mrow> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </mrow> </semantics></math>. When <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mo>Ω</mo> </msub> <msub> <mi>σ</mi> <mi mathvariant="normal">f</mi> </msub> </mrow> </semantics></math> is close to zero, the sensitivity depends on <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>. As <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mo>Ω</mo> </msub> <msub> <mi>σ</mi> <mi mathvariant="normal">f</mi> </msub> </mrow> </semantics></math> increases to about <math display="inline"><semantics> <mi>π</mi> </semantics></math>, the sensitivity converges to the analytical expressions.</p>
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<p>(<b>a</b>) Optimal order of LMT as a function the Raman beam intensity. For <span class="html-italic">n</span>th order of LMT, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi mathvariant="normal">t</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msub> <mi>k</mi> <mrow> <mi>eff</mi> </mrow> </msub> </mrow> </semantics></math>. For the Raman transition <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>m</mi> <mi>F</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>→</mo> <mi>F</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>m</mi> <mi>F</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mi mathvariant="normal">s</mi> </msub> <mo>=</mo> <mn>5.01</mn> <mrow> <mtext> </mtext> <mi>mW</mi> </mrow> <mo>⋅</mo> <msup> <mrow> <mi>cm</mi> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>b</b>) LMT-induced sensitivity enhancement factor as a function of the light intensity.</p>
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<p>Ratio of the actual sensitivity to the ideal sensitivity for <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.2</mn> <mrow> <mtext> </mtext> <mi>mm</mi> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mo>Ω</mo> </msub> <mo>=</mo> <mrow> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> <msup> <mrow> <mrow> <mtext> </mtext> <mi>cm</mi> </mrow> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, when the finite initial size of the atomic cloud is considered. The temperature of the atoms is set at 6 µK.</p>
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<p>Schematic illustration of a three-dimensional PSI apparatus. The diagram on the top shows a cross-sectional view in the <span class="html-italic">y</span>-<span class="html-italic">z</span> plane. The red arrows reprenst the Raman beams and the blue ones the imaging beams.</p>
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<p>Coupling between <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>m</mi> <mi>F</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>m</mi> <mi>F</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> Zeeman ground states of <sup>87</sup>Rb induced by copropagating and counter-propagating Raman beams along the <span class="html-italic">z</span> axis. Here, <math display="inline"><semantics> <mi>v</mi> </semantics></math> is the amplitude of the atomic offset velocity in the <span class="html-italic">z</span> direction. The offset velocity is assumed to be in the <span class="html-italic">z</span> direction. <math display="inline"><semantics> <mrow> <msubsup> <mi>R</mi> <mrow> <mn>1</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>±</mo> <mi>z</mi> </mrow> </msubsup> </mrow> </semantics></math> represents the higher (lower)-frequency Raman beam in the <math display="inline"><semantics> <mrow> <mo>±</mo> <mi>z</mi> </mrow> </semantics></math> direction. The wavenumbers of both Raman beams are denoted as <math display="inline"><semantics> <mi>k</mi> </semantics></math>, and the small wavenumber difference between them is neglected. In the designations of the quantum states, the last element indicates the momentum in the z-direction. The transitions driven by three pairs of Raman beams have three different resonant frequencies in the presence of a offset velocity. (<b>a</b>) Coupling induced by copropagating Raman beams. The transition frequency is simply the hyperfine splitting. (<b>b</b>) Coupling induced by <math display="inline"><semantics> <mrow> <msubsup> <mi>R</mi> <mn>1</mn> <mrow> <mo>+</mo> <mi>z</mi> </mrow> </msubsup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>R</mi> <mn>2</mn> <mrow> <mo>−</mo> <mi>z</mi> </mrow> </msubsup> </mrow> </semantics></math>. The transition frequency is shifted up by <math display="inline"><semantics> <mrow> <mn>2</mn> <mi>k</mi> <mi>v</mi> <mo>+</mo> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>ℏ</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>/</mo> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mrow> </mrow> </semantics></math>, where <math display="inline"><semantics> <mrow> <mn>2</mn> <mi>k</mi> <mi>v</mi> </mrow> </semantics></math> is Doppler shift and <math display="inline"><semantics> <mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>ℏ</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>/</mo> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mrow> </mrow> </semantics></math> is the recoil frequency. (<b>c</b>) Coupling induced by <math display="inline"><semantics> <mrow> <msubsup> <mi>R</mi> <mn>1</mn> <mrow> <mo>−</mo> <mi>z</mi> </mrow> </msubsup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>R</mi> <mn>2</mn> <mrow> <mo>+</mo> <mi>z</mi> </mrow> </msubsup> </mrow> </semantics></math>. The transition frequency is shifted down by <math display="inline"><semantics> <mrow> <mn>2</mn> <mi>k</mi> <mi>v</mi> <mo>+</mo> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>ℏ</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>/</mo> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mrow> </mrow> </semantics></math>.</p>
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<p>(<b>a</b>) Illustration of the pulse sequence for the LMT protocol for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>. (<b>b</b>) Timing of the VCO frequency for this protocol. The VCO determines the relative frequency between the two Raman beams. When the VCO is tuned to <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>HFS</mi> </mrow> </msub> <mo>+</mo> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>ℏ</mo> <msub> <mi>k</mi> <mrow> <mi>eff</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>/</mo> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mrow> <mo>+</mo> <mi>k</mi> <mi>v</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>R</mi> <mn>1</mn> <mrow> <mo>+</mo> <mi>z</mi> </mrow> </msubsup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>R</mi> <mn>2</mn> <mrow> <mo>−</mo> <mi>z</mi> </mrow> </msubsup> </mrow> </semantics></math> take effect. When the VCO is tuned to <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>HFS</mi> </mrow> </msub> <mo>−</mo> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>ℏ</mo> <msub> <mi>k</mi> <mrow> <mi>eff</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>/</mo> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mrow> <mo>−</mo> <mi>k</mi> <mi>v</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>R</mi> <mn>1</mn> <mrow> <mo>−</mo> <mi>z</mi> </mrow> </msubsup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>R</mi> <mn>2</mn> <mrow> <mo>+</mo> <mi>z</mi> </mrow> </msubsup> </mrow> </semantics></math> take effect. See text for details.</p>
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<p>Layout of a PSI-based IMU using <sup>87</sup>Rb atoms. (<b>a</b>) Optical system. Each of the six MOT/molasses beams is produced and controlled by an AOM optimized at a modulation frequency of 100 MHz. The modulation frequency is ramped down to 60 MHz during the process of optical molasses cooling. These beams also pass through an EOM to generate the repumping frequency component. The Raman beams (red) are produced by a 500 MHz AOM and an EOM. The imaging beam (blue) is produced by an AOM tuned to 92 MHz. The Raman beams are directed to each of the three fiber ports with PBSs (black boxes with a slash inside) and variable retarders VR1 and VR 2. The imaging beam is directed to each of the three fiber ports with variable retarders VR 3 and VR 4. (<b>b</b>) Frequencies of the beams used in (<b>a</b>).</p>
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14 pages, 12042 KiB  
Article
Tightly Trapped Atom Interferometer inside a Hollow-Core Fiber
by Yitong Song, Wei Li, Xiaobin Xu, Rui Han, Chengchun Gao, Cheng Dai and Ningfang Song
Photonics 2024, 11(5), 428; https://doi.org/10.3390/photonics11050428 - 3 May 2024
Cited by 2 | Viewed by 1645
Abstract
We demonstrate a fiber-guided atom interferometer in a far-off-resonant trap (FORT) of 100 μK. The differential light shift (DLS) introduced by the FORT leads to the inhomogeneous dephasing of the tightly trapped atoms inside a hollow-core fiber. The DLS-induced dephasing is greatly suppressed [...] Read more.
We demonstrate a fiber-guided atom interferometer in a far-off-resonant trap (FORT) of 100 μK. The differential light shift (DLS) introduced by the FORT leads to the inhomogeneous dephasing of the tightly trapped atoms inside a hollow-core fiber. The DLS-induced dephasing is greatly suppressed in π/2-π-π/2 Doppler-insensitive interferometry. The spin coherence time is extended to 13.4 ms by optimizing the coupling of the trapping laser beam into a quasi-single-mode hollow-core anti-resonant fiber. The Doppler-sensitive interferometry shows a much shorter coherence time, indicating that the main limits to our fiber-guided atom interferometer are the wide axial velocity distribution and the irregular modes of the Raman laser beams inside the fiber. This work paves the way for portable and miniaturized quantum devices, which have advantages for inertial sensing at arbitrary orientations and in dynamic environments. Full article
(This article belongs to the Special Issue The Integration of Quantum Communication and Quantum Sensors)
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<p>Experiment details: (<b>a</b>) diagram of the experimental system. PM-SMF, polarization-maintaining single-mode fiber; CCD, charge-coupled device camera; PBS, polarizing beam splitter; NPBS, non-polarizing beam splitter; MOT, magneto-optical trap; SDM, shortpass dichroic mirrors; APD, avalanche photodetector. The APD is used to detect the transmission power of the probe laser pulses. The blue and yellow arrows correspond to <math display="inline"><semantics> <mrow> <mn>780</mn> <mo> </mo> <mi>nm</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>852</mn> <mo> </mo> <mi>nm</mi> </mrow> </semantics></math> laser beams, respectively; (<b>b</b>) <sup>87</sup>Rb D2 transition hyperfine structure and frequency configurations of the laser beams.</p>
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<p>Time sequence: (<b>a</b>) in the interferometry experiment, the laser-cooled atoms inside the HC-ARF are trapped radially by the the <math display="inline"><semantics> <mrow> <mn>852</mn> <mo> </mo> <mi>nm</mi> </mrow> </semantics></math> laser beam; (<b>b</b>) in the experiment of measuring the radial temperature of the atomic cloud, the <math display="inline"><semantics> <mrow> <mn>852</mn> <mo> </mo> <mi>nm</mi> </mrow> </semantics></math> laser beam needs to be turned off before detecting the optical depth of different expansion times <math display="inline"><semantics> <mi>τ</mi> </semantics></math>.</p>
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<p>A 7-core hollow-core anti-resonant fiber: (<b>a</b>) microscopic view of the cross-section; (<b>b</b>) the near-field mode profile with <math display="inline"><semantics> <mrow> <mn>40</mn> <mo>×</mo> </mrow> </semantics></math> magnification. In the pseudo-color image, red and blue colors represent the strongest and weakest optical intensities, respectively.</p>
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<p>Far-field mode profiles: (<b>a</b>) the setup of the HC-ARF coupling scheme; (<b>b</b>) the far-field mode profile during the optimization; (<b>c</b>) optimized far-field mode profile captured by the CCD.</p>
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<p>Loading of atoms into the HC-ARF: (<b>a</b>) absorption images of the atomic cloud above the HC-ARF at different loading times <span class="html-italic">t</span>. A coordinate system is established, in which the tip of the HC-ARF is considered as the origin; (<b>b</b>) resonant optical depth <math display="inline"><semantics> <msub> <mi>OD</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>s</mi> </mrow> </msub> </semantics></math> versus expansion time <math display="inline"><semantics> <mi>τ</mi> </semantics></math>. Each data point is an average of five repetitions, and the curve is a fit with Equation (<a href="#FD4-photonics-11-00428" class="html-disp-formula">4</a>).</p>
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<p>Optical pumping: (<b>a</b>) Zeeman sub-levels of <sup>87</sup>Rb and the transition rules. Red cross indicates that the <sup>87</sup>Rb atoms in the sub-level <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>m</mi> <mi>F</mi> </msub> <mrow> <mo>=</mo> <mn>0</mn> <mo>〉</mo> </mrow> </mrow> </semantics></math> no longer interact with the pump laser pulse; (<b>b</b>) microwave Raman spectra with and without the pump laser pulse. The measured data points are fitted to the sinc functions.</p>
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<p>Configuration of Raman laser pulses: (<b>a</b>) co-propagating. The polarization is configured as <math display="inline"><semantics> <mrow> <msup> <mi>σ</mi> <mo>+</mo> </msup> <mtext>-</mtext> <msup> <mi>σ</mi> <mo>+</mo> </msup> </mrow> </semantics></math>; (<b>b</b>) counter-propagating. Two pairs of lin ⊥ lin polarized Raman laser pulses (diagonal and antidiagonal).</p>
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<p>Doppler-insensitive Raman transition: (<b>a</b>) normalized population versus Raman detunings; (<b>b</b>) normalized population versus pulse durations.</p>
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<p>Doppler-insensitive interferometry inside the HC-ARF: (<b>a</b>) interference fringes with interrogation times <math display="inline"><semantics> <mrow> <mn>2</mn> <mi>T</mi> </mrow> </semantics></math> of <math display="inline"><semantics> <mrow> <mn>2</mn> <mo> </mo> <mi>ms</mi> </mrow> </semantics></math>. The red squares and blue triangles correspond to the far-field mode profiles of the trapping laser beam during and after the optimization, respectively. Each data point is an average of three repetitions with the standard deviation; (<b>b</b>) contrast versus interrogation times <math display="inline"><semantics> <mrow> <mn>2</mn> <mi>T</mi> </mrow> </semantics></math> under the condition of optimal coupling. Each data point corresponds to an average of three fitting results from the interference fringes, and the curve is a fit with an exponential decay function [see Equation (<a href="#FD8-photonics-11-00428" class="html-disp-formula">8</a>)].</p>
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<p>Doppler broadening of Raman transition due to the axial velocity distribution. The error bars are the standard deviation of the average of three experimental runs.</p>
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<p>Doppler-sensitive interference fringes: (<b>a</b>) with different <math display="inline"><semantics> <mi>π</mi> </semantics></math> pulse widths <math display="inline"><semantics> <msub> <mi>τ</mi> <mi>π</mi> </msub> </semantics></math>, and interrogation times <math display="inline"><semantics> <mrow> <mn>2</mn> <mi>T</mi> <mo>=</mo> <mn>60</mn> <mo> </mo> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>. The error bars are the standard deviation of the average of three repetitions; (<b>b</b>) at different loading times <span class="html-italic">t</span>. The interrogation times are <math display="inline"><semantics> <mrow> <mn>60</mn> <mo> </mo> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>, and the trap depth is <math display="inline"><semantics> <mrow> <mn>100</mn> <mo> </mo> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">K</mi> </mrow> </semantics></math>. The error bars represent the standard deviation of three experimental runs; (<b>c</b>) with different depths <span class="html-italic">U</span>. The interrogation times are <math display="inline"><semantics> <mrow> <mn>60</mn> <mo> </mo> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>, and the loading time is <math display="inline"><semantics> <mrow> <mn>50</mn> <mo> </mo> <mi>ms</mi> </mrow> </semantics></math>. Each data point is an average of three repetitions with the standard deviation.</p>
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<p>Doppler-sensitive interferometry inside the HC-ARF: (<b>a</b>) contrast versus interrogation times <math display="inline"><semantics> <mrow> <mn>2</mn> <mi>T</mi> </mrow> </semantics></math>. Each data point corresponds to an average of three fitting results from the interference fringes, and the damped curve is a fit with Equation (<a href="#FD8-photonics-11-00428" class="html-disp-formula">8</a>); (<b>b</b>) interference fringes with different Doppler shifts. The contrasts of the blue triangles, red squares, and green dots are <math display="inline"><semantics> <mrow> <mn>0.08</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>0.39</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mn>0.79</mn> </mrow> </semantics></math>, respectively.</p>
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<p>The irregular far-field mode profiles of Raman laser pulses: (<b>a</b>) coupling into the HC-ARF from the bottom; (<b>b</b>) re-coupling from the top to bottom.</p>
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16 pages, 3398 KiB  
Article
Enhancing Pure Inertial Navigation Accuracy through a Redundant High-Precision Accelerometer-Based Method Utilizing Neural Networks
by Qinyuan He, Huapeng Yu, Dalei Liang and Xiaozhuo Yang
Sensors 2024, 24(8), 2566; https://doi.org/10.3390/s24082566 - 17 Apr 2024
Cited by 2 | Viewed by 1597
Abstract
The pure inertial navigation system, crucial for autonomous navigation in GPS-denied environments, faces challenges of error accumulation over time, impacting its effectiveness for prolonged missions. Traditional methods to enhance accuracy have focused on improving instrumentation and algorithms but face limitations due to complexity [...] Read more.
The pure inertial navigation system, crucial for autonomous navigation in GPS-denied environments, faces challenges of error accumulation over time, impacting its effectiveness for prolonged missions. Traditional methods to enhance accuracy have focused on improving instrumentation and algorithms but face limitations due to complexity and costs. This study introduces a novel device-level redundant inertial navigation framework using high-precision accelerometers combined with a neural network-based method to refine navigation accuracy. Experimental validation confirms that this integration significantly boosts navigational precision, outperforming conventional system-level redundancy approaches. The proposed method utilizes the advanced capabilities of high-precision accelerometers and deep learning to achieve superior predictive accuracy and error reduction. This research paves the way for the future integration of cutting-edge technologies like high-precision optomechanical and atom interferometer accelerometers, offering new directions for advanced inertial navigation systems and enhancing their application scope in challenging environments. Full article
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<p>Accelerometer axis diagram.</p>
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<p>Redundant high-precision accelerometer inertial navigation system.</p>
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<p>System hardware structure design diagram.</p>
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<p>Displacement error prediction network structure diagram.</p>
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<p>AUV diagram for experiment.</p>
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<p>Full-course predicted trajectory diagram for Experiment 1.</p>
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<p>Error comparison of each method in Experiment 1. (<b>a</b>) Represents the global error and (<b>b</b>) represents the CDF of the RTE for the method.</p>
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<p>Full-course predicted trajectory diagram for Experiment 2.</p>
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<p>Error comparison of each method in Experiment 2. (<b>a</b>) Represents the global error and (<b>b</b>) represents the CDF of the RTE for the method.</p>
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15 pages, 9880 KiB  
Article
Effect of Polymer Adjuvant Type and Concentration on Atomization Characteristics of Nozzle
by Peng Hu, Ruirui Zhang, Liping Chen, Longlong Li, Qing Tang, Wenlong Yan and Jiajun Yang
Agriculture 2024, 14(3), 404; https://doi.org/10.3390/agriculture14030404 - 1 Mar 2024
Viewed by 1193
Abstract
(1) Background: Various types of adjuvants are added during application to enhance the spraying effect, but most adjuvant formulations are proprietary products, and their exact compositions have not been disclosed. (2) Methods: The spatial distributions of droplet sizes and velocities generated by the [...] Read more.
(1) Background: Various types of adjuvants are added during application to enhance the spraying effect, but most adjuvant formulations are proprietary products, and their exact compositions have not been disclosed. (2) Methods: The spatial distributions of droplet sizes and velocities generated by the atomization of different polymer adjuvants were measured based on the phase Doppler interferometer (PDI) measurement method. The sub-area statistical method was used to quantitatively analyze the droplet characteristics at various points below the nozzle. (3) Results: The polymer (polyethylene oxide (PEO))/associative surfactant (sodium dodecyl sulfate (SDS)) can increase the droplet size and velocity of the solution after atomization by increasing the amount of the polymer/associative surfactant to reduce the equilibrium surface tension of the solution and increase the viscosity. Different concentrations of polymer/associative surfactant atomization produced larger droplet sizes and better uniformity than polymer/non-associative surfactant (polysorbate-20 (Tween20)). At the same position, the relationship between droplet velocities for the three adjuvant combinations was roughly as follows: PEO/SDS solution had the highest velocity, followed by PEO solution, and PEO/Tween20 solution had the lowest velocity. (4) Conclusions: The optimal of droplet size distribution can be achieved by adding appropriate amounts of associative surfactants to polymers. Full article
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<p>LU120-03 nozzle structure.</p>
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<p>Droplet parameter measurement platform. 1. Power supply 2. Data management computer 3. AIMS 2.3.0.1 system software 4. Signal processor 1 channel 5. Signal processor 2 channel 6. Signal processor 3 channel 7. 3-axis positioning and moving system 8. Optical transmitter 9. Optical receiver 10. Nozzle.</p>
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<p>Distribution schematic of fan nozzle spray area measurement sub-areas.</p>
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<p>Viscosity and equilibrium surface tension distribution of solutions with different concentrations adjuvants. (<b>a</b>) Viscosity distribution graph. (<b>b</b>) Equilibrium surface tension distribution graph. (Note: The graphs a–g are arranged according to the size of all means, with a representing the largest mean, and where there is a letter with the same labeling, the difference is not significant, and where there is a letter with a different labeling, the difference is significant.).</p>
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<p>Droplet size distribution in vertical direction at 250 kPa pressure. (<b>a</b>) <span class="html-italic">z</span>-axis cross-section. (<b>b</b>) Vertical direction. (Note: The graphs a–g are arranged according to the size of all means, with a representing the largest mean, and where there is a letter with the same labeling, the difference is not significant, and where there is a letter with a different labeling, the difference is significant.).</p>
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<p>Droplet size distribution at z = 0.6 m under different spray pressures. (<b>a</b>) 150 kPa; (<b>b</b>) 250 kPa; (<b>c</b>) 350 kPa.</p>
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<p>Droplet size distribution at spray pressure of 250 kPa. (<b>a</b>) z = 0.1 m; (<b>b</b>) z = 0.3 m; (<b>c</b>) z = 0.5 m.</p>
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<p>RS distribution of droplet size at different spray pressures. (<b>a</b>) 150 kPa; (<b>b</b>) 250 kPa; (<b>c</b>) 350 kPa.</p>
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<p>Distribution of droplet velocity at Z = 0.6 m under different pressures. (<b>a</b>) 150 kPa; (<b>b</b>) 200 kPa; (<b>c</b>) 250 kPa; (<b>d</b>) 300 kPa; (<b>e</b>) 350 kPa.</p>
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<p>Droplet velocity distribution of different adjuvant combinations in the vertical direction below the spray nozzle. (Note: The graphs a,b,c,d are arranged according to the size of all means, with a representing the largest mean, and where there is a letter with the same labeling, the difference is not significant, and where there is a letter with a different labeling, the difference is significant.).</p>
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13 pages, 3906 KiB  
Article
High-Precision Atom Interferometer-Based Dynamic Gravimeter Measurement by Eliminating the Cross-Coupling Effect
by Yang Zhou, Wenzhang Wang, Guiguo Ge, Jinting Li, Danfang Zhang, Meng He, Biao Tang, Jiaqi Zhong, Lin Zhou, Runbing Li, Ning Mao, Hao Che, Leiyuan Qian, Yang Li, Fangjun Qin, Jie Fang, Xi Chen, Jin Wang and Mingsheng Zhan
Sensors 2024, 24(3), 1016; https://doi.org/10.3390/s24031016 - 4 Feb 2024
Cited by 4 | Viewed by 1896
Abstract
A dynamic gravimeter with an atomic interferometer (AI) can perform absolute gravity measurements with high precision. AI-based dynamic gravity measurement is a type of joint measurement that uses an AI sensor and a classical accelerometer. The coupling of the two sensors may degrade [...] Read more.
A dynamic gravimeter with an atomic interferometer (AI) can perform absolute gravity measurements with high precision. AI-based dynamic gravity measurement is a type of joint measurement that uses an AI sensor and a classical accelerometer. The coupling of the two sensors may degrade the measurement precision. In this study, we analyzed the cross-coupling effect and introduced a recovery vector to suppress this effect. We improved the phase noise of the interference fringe by a factor of 1.9 by performing marine gravity measurements using an AI-based gravimeter and optimizing the recovery vector. Marine gravity measurements were performed, and high gravity measurement precision was achieved. The external and inner coincidence accuracies of the gravity measurement were ±0.42 mGal and ±0.46 mGal after optimizing the cross-coupling effect, which was improved by factors of 4.18 and 4.21 compared to the cases without optimization. Full article
(This article belongs to the Collection Inertial Sensors and Applications)
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<p>Principle of the joint gravity measurement and the introduction of the cross-coupling effect.</p>
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<p>(Color online). AI-based dynamic gravimeter for the marine gravity measurement.</p>
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<p>(Color online.) Allan standard deviation of the measured gravity value at the National Geodetic Observatory in Wuhan for 2T = 30 ms.</p>
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<p>(Color online). Gravity comparison with a shore-based gravity reference site under mooring state.</p>
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<p>(Color online.) (<b>a</b>) The trajectory of the survey line during the marine gravity measurement. (<b>b</b>) The power spectral density amplitude of the measured acceleration in the z direction under the mooring state (black dashed line) and sailing state (red solid line).</p>
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<p>(Color online). (<b>a</b>) The calibration curves for the recovery vector components during the gravity survey measurement, the z component of recovery vector is subtracted by 0.99 for the convenience of display. (<b>b</b>) The recovered fringe when the recovery vector is set to {0, 0, 1}. (<b>c</b>) The recovered fringe when the recovery vector is set to its optimized value {0.0060, −0.0034, 0.9860}.</p>
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<p>(Color online). Data processing process of the gravity anomaly. The red solid line in (<b>a</b>–<b>c</b>) represents data of the three survey lines, while the blue dashed line represents other data during the gravity survey. (<b>a</b>) The recovery acceleration <span class="html-italic">a<sub>rec</sub></span><sub>,<span class="html-italic">z</span></sub>(<span class="html-italic">t</span>). (<b>b</b>) The recovery acceleration <span class="html-italic">a<sub>rec</sub></span><sub>,<span class="html-italic">z</span></sub>(<span class="html-italic">t</span>) after the low-pass filter. (<b>c</b>) The calculated motion acceleration <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi mathvariant="italic">mot</mi> <mo>,</mo> <mi>z</mi> </mrow> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> after the low-pass filter. (<b>d</b>) The measured gravity anomaly of the AI-based gravimeter (red solid line) and the classical shipborne strapdown gravimeter (black dotted line).</p>
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<p>(Color online). Comparing the gravity anomaly measurement during the three survey lines. (<b>a</b>) Before deducting the sea surface height-induced gravity. (<b>b</b>) After deducting the sea surface height-induced gravity. The inset figure is the measured water depth during the three survey lines.</p>
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12 pages, 8742 KiB  
Article
Quantum Applications of an Atomic Ensemble Inside a Laser Cavity
by Andrei Ben Amar Baranga, Gennady A. Koganov, David Levron, Gabriel Bialolenker and Reuben Shuker
Photonics 2024, 11(1), 46; https://doi.org/10.3390/photonics11010046 - 2 Jan 2024
Viewed by 1607
Abstract
Many quantum device signals are proportional to the number of the participating atoms that take part in the detection devices. Among these are optical magnetometers, atomic clocks, quantum communications and atom interferometers. One way to enhance the signal-to-noise ratio is to introduce atom [...] Read more.
Many quantum device signals are proportional to the number of the participating atoms that take part in the detection devices. Among these are optical magnetometers, atomic clocks, quantum communications and atom interferometers. One way to enhance the signal-to-noise ratio is to introduce atom entanglement that increases the signal in a super-radiant-like effect. A coherent em field inside a laser cavity is suggested to achieve atoms’ correlation/entanglement. This may also play an important role in the basic quantum arena of many-body physics. An initial novel experiment to test the realization of atoms’ correlation is described here. A Cs optical magnetometer is used as a tool to test the operation of a cell-in-cavity laser and its characteristics. A vapor cell is inserted into an elongated external cavity of the pump laser in Littrow configuration. Higher atom polarization and reduced laser linewidth are obtained leading to better magnetometer sensitivity and signal-to-noise ratio. The Larmor frequency changes of the Free Induction Decay of optically pumped Cs atomic polarization in the ambient earth magnetic field at room temperature is measured. Temporal changes in the magnetic field of less than 10 pT/√Hz are measured. The first-order dependence of the magnetic field on temperature and temperature gradients is eliminated, important in many practical applications. Single and gradiometric magnetometer configurations are presented. Full article
(This article belongs to the Special Issue Quantum Optics: Science and Applications)
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Figure 1

Figure 1
<p>Absorption coefficient <span class="html-italic">α</span> as a function of the number of atoms N for no correlation Ω<span class="html-italic"><sub>c</sub></span> = 0 (black dashed line), in presence of correlation Ω<span class="html-italic"><sub>c</sub></span> = 0.0001 (red dash-dotted line) and Ω<span class="html-italic"><sub>c</sub></span> = 0.001 (blue solid line).</p>
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<p>Diode laser with a Cs vapor cell inside the elongated external cavity. Shown is the fluorescence through the cell at resonant D2 line.</p>
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<p>Pulsed magnetometer with the vapor cell inside the pump laser cavity. The probe laser is perpendicular to the pump.</p>
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<p>Jones matrix presentation for laser beam polarization inside the Littrow external cavity.</p>
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<p>ASD of the pulsed magnetometer with the vapor cell inside the pump laser cavity and outside the cavity. The noise floor is lowered by introducing the cell inside the cavity without affecting the applied field intensity measurement as emphasized in the insert. The insert shows the field at 11.3 Hz and 13 Hz.</p>
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<p>(<b>A</b>) Small baseline gradiometer, single vapor cell. (<b>B</b>) Large baseline gradiometer with two vapor cells inside the pump laser cavity.</p>
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<p>ASD of a pulsed magnetometer in gradiometric configuration with two beams at 1.5 mm distance inside a single cell illuminated by an external cavity diode laser.</p>
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<p>Pulsed magnetometer with co-linear laser beams and vapor cell inside the pump laser cavity.</p>
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<p>The emission spectrum of the Toptica AR-coated diode laser LD-0860-0080-AR-1 measured by a 30 cm spectrometer and recorded on a digital oscilloscope.</p>
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<p>The emission spectrum of the Toptica AR-coated diode laser LD-0860-0080-AR-1 narrowed and tuned by a Littrow configuration external cavity. The linewidth is significantly smaller than the spectrometer’s resolution and was separately measured by a wavemeter.</p>
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15 pages, 16135 KiB  
Article
Designing an Automatic Frequency Stabilization System for an External Cavity Diode Laser Using a Data Acquisition Card in the LabVIEW Platform
by Yueyang Wu, Fangjun Qin, Yang Li, Zhichao Ding and Rui Xu
Appl. Sci. 2024, 14(1), 308; https://doi.org/10.3390/app14010308 - 29 Dec 2023
Cited by 2 | Viewed by 1628
Abstract
The frequency stability of free-running lasers is susceptible to the influence of environmental factors, which cannot meet the long-term frequency stabilization requirements for atom interferometry precision measurements. To obtain a frequency-stabilized 780 nm laser beam, an automatic frequency stabilization system for an external [...] Read more.
The frequency stability of free-running lasers is susceptible to the influence of environmental factors, which cannot meet the long-term frequency stabilization requirements for atom interferometry precision measurements. To obtain a frequency-stabilized 780 nm laser beam, an automatic frequency stabilization system for an external cavity diode laser (ECDL) based on rubidium (Rb) atomic saturated absorption spectrum was designed using a commercial data acquisition (DAQ) card. The signals acquired by the A/D terminal are processed and analyzed by LabVIEW, which can automatically identify all the locking points and output the piezoelectric ceramic transducer (PZT) scan and digital feedback through the D/A terminal. The experimental results show that the system can lock to six different frequencies separately and realize automatic relocking within 3.5 s after unlocking. The system has a stability of 1.68 × 10−10@1 s and 4.77 × 10−12@1000 s, which meets the laboratory’s requirements for atomic interference experiments. Full article
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Figure 1
<p>Hyperfine energy level structure of the Rb atom.</p>
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<p>SAS of the Rb D2 line transition.</p>
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<p>(<b>a</b>) Primary, (<b>b</b>) secondary, and (<b>c</b>) tertiary differential curves of Rb atomic SAS.</p>
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<p>(<b>a</b>) Block diagram of the frequency stabilization system structure. (<b>b</b>) The physical diagram of the frequency stabilization system. (<b>c</b>) The interactive interface of the frequency stabilization system.</p>
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<p>Obtaining reference voltage and error fragment program.</p>
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<p>(<b>a</b>) The original error signal and (<b>b</b>) the smoothed 40 points of the error signal. (<b>c</b>) The scanning signal with the corresponding SAS signal and error signal.</p>
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<p>6 error fragments of the error signal.</p>
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<p>Fitting curve of locking point scanning voltage with corresponding frequency and wavelength.</p>
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<p>Incremental PI control program.</p>
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<p>Automatic relock program.</p>
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<p>Wavelength changes after frequency locking.</p>
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<p>Measurement of (<b>a</b>) linewidth of SAS and (<b>b</b>) frequency discrimination sensitivity.</p>
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<p>(<b>a</b>) Variation and (<b>b</b>) Allan variance of the error signal when the frequency is locked.</p>
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<p>Automatic locking process of the frequency stabilization system.</p>
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<p>Input/output synchronization of the DAQ card program.</p>
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<p>Differentiation to obtain error signal program.</p>
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<p>Peak position extraction of the error signal program.</p>
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<p>Error signal over-zero position and error fragment extraction program.</p>
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<p>Reference voltage extraction of the PZT.</p>
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34 pages, 1758 KiB  
Article
Towards the Simplest Model of Quantum Supremacy: Atomic Boson Sampling in a Box Trap
by Vitaly V. Kocharovsky, Vladimir V. Kocharovsky, William D. Shannon and Sergey V. Tarasov
Entropy 2023, 25(12), 1584; https://doi.org/10.3390/e25121584 - 25 Nov 2023
Cited by 2 | Viewed by 1144
Abstract
We describe boson sampling of interacting atoms from the noncondensed fraction of Bose–Einstein-condensed (BEC) gas confined in a box trap as a new platform for studying computational ♯P-hardness and quantum supremacy of many-body systems. We calculate the characteristic function and statistics of atom [...] Read more.
We describe boson sampling of interacting atoms from the noncondensed fraction of Bose–Einstein-condensed (BEC) gas confined in a box trap as a new platform for studying computational ♯P-hardness and quantum supremacy of many-body systems. We calculate the characteristic function and statistics of atom numbers via the newly found Hafnian master theorem. Using Bloch–Messiah reduction, we find that interatomic interactions give rise to two equally important entities—eigen-squeeze modes and eigen-energy quasiparticles—whose interplay with sampling atom states determines the behavior of the BEC gas. We infer that two necessary ingredients of ♯P-hardness, squeezing and interference, are self-generated in the gas and, contrary to Gaussian boson sampling in linear interferometers, external sources of squeezed bosons are not required. Full article
(This article belongs to the Special Issue Selected Featured Papers from Entropy Editorial Board Members)
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Figure 1
<p>Schematic diagram of the Bloch–Messiah reduction in Equation (<a href="#FD26-entropy-25-01584" class="html-disp-formula">26</a>): Three irreducible steps of the Bogoliubov transformation (<a href="#FD26-entropy-25-01584" class="html-disp-formula">26</a>) of the creation/annihilation operators and wave functions from the observational bare-atom basis to the quasiparticle basis.</p>
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<p>An average quasiparticle occupation, <math display="inline"><semantics> <msub> <mover accent="true"> <mi>N</mi> <mo stretchy="false">˜</mo> </mover> <mi mathvariant="bold">k</mi> </msub> </semantics></math>, (red solid curves) and the single-mode squeezing parameter, <math display="inline"><semantics> <msub> <mi>r</mi> <mi mathvariant="bold">k</mi> </msub> </semantics></math>, (blue dashed curves) vs the absolute value of the anomalous correlator <math display="inline"><semantics> <mrow> <mo>|</mo> <mo>〈</mo> <msub> <mover accent="true"> <mi>s</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="bold">k</mi> </msub> <msub> <mover accent="true"> <mi>s</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="bold">k</mi> </msub> <mo>〉</mo> <mo>|</mo> <mo>≡</mo> <mo>|</mo> <mi>α</mi> <mo>|</mo> </mrow> </semantics></math> for different fixed values of the normal correlator <math display="inline"><semantics> <mrow> <mo>〈</mo> <msubsup> <mover accent="true"> <mi>s</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi mathvariant="bold">k</mi> </mrow> <mo>†</mo> </msubsup> <msub> <mover accent="true"> <mi>s</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="bold">k</mi> </msub> <mo>〉</mo> <mo>≡</mo> <mi>η</mi> </mrow> </semantics></math>. Red solid curves representing the average quasiparticle occupation are, in fact, the arcs of a circle with a radius <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> centered at <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>.</p>
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<p>Probability <math display="inline"><semantics> <msub> <mi>ρ</mi> <mi>n</mi> </msub> </semantics></math> for sampling <span class="html-italic">n</span> atoms in a single eigen-squeeze mode (<a href="#FD44-entropy-25-01584" class="html-disp-formula">44</a>): Orange points correspond to the moderate values of the anomalous correlator <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> <mo>=</mo> <mn>9.5</mn> </mrow> </semantics></math> (<b>left</b> panel) or <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> <mo>=</mo> <mn>10.4</mn> </mrow> </semantics></math> (<b>central</b> panel); blue and green points on the left and central panels show probabilities <math display="inline"><semantics> <msub> <mi>ρ</mi> <mi>n</mi> </msub> </semantics></math> in the case of zero (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>) and maximal (<math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <mi>η</mi> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>η</mi> <mo>)</mo> </mrow> </msqrt> <mo>≃</mo> <mn>10.5</mn> </mrow> </semantics></math>) squeezing, respectively. The <b>right</b> panel exemplifies a dependence of the probabilities <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>0</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ρ</mi> <mn>5</mn> </msub> </mrow> </semantics></math> on the anomalous correlator <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> </mrow> </semantics></math>. The normal correlator for all panels has the same value <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>.</p>
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<p>Typical dependence of the joint probability distribution <math display="inline"><semantics> <msub> <mi>ρ</mi> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </msub> </semantics></math> for sampling <math display="inline"><semantics> <msub> <mi>n</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>n</mi> <mn>2</mn> </msub> </semantics></math> atoms in the two counter-propagating plane waves on the absolute value of the anomalous correlator, <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> </mrow> </semantics></math>. The normal correlator <math display="inline"><semantics> <mi>η</mi> </semantics></math> (that is, the average number of atoms per one mode) is set to be <math display="inline"><semantics> <mi>η</mi> </semantics></math> = 10. The anomalous correlator values are (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (which corresponds to an ideal, non-interacting gas), (<b>b</b>) <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> <mo>=</mo> <mn>9.7</mn> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> <mo>=</mo> <mn>10.4</mn> </mrow> </semantics></math>, and (<b>d</b>) <math display="inline"><semantics> <mrow> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mo>|</mo> <mi>α</mi> <mo>|</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <mi>η</mi> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>η</mi> <mo>)</mo> </mrow> </msqrt> </mrow> </semantics></math> (which corresponds to the two-mode squeezed vacuum state). Increasing <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> </mrow> </semantics></math> leads to growing up correlations between the random numbers <math display="inline"><semantics> <msub> <mi>n</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>n</mi> <mn>2</mn> </msub> </semantics></math>. Note, that the scale of the values <math display="inline"><semantics> <msub> <mi>ρ</mi> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </msub> </semantics></math> changes significantly from (<b>a</b>–<b>d</b>).</p>
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<p>The joint probability distribution <math display="inline"><semantics> <msub> <mi>ρ</mi> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </msub> </semantics></math> for atomic boson sampling from two excited atom states formed by a general-case unitary mixing of two eigen-squeeze modes (<a href="#FD44-entropy-25-01584" class="html-disp-formula">44</a>) with the same wave vector <math display="inline"><semantics> <mi mathvariant="bold">k</mi> </semantics></math>. The normal and anomalous correlators in Equation (<a href="#FD51-entropy-25-01584" class="html-disp-formula">51</a>) are <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> <mo>=</mo> <mn>10.4</mn> </mrow> </semantics></math>, respectively, which amounts to the mean quasiparticle occupation <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>N</mi> <mo stretchy="false">˜</mo> </mover> <mi mathvariant="bold">k</mi> </msub> <mo>≃</mo> <mn>0.95</mn> </mrow> </semantics></math> and the single-mode squeezing parameter <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi mathvariant="bold">k</mi> </msub> <mo>≃</mo> <mn>1.34</mn> </mrow> </semantics></math>. The panels show the evolution of the joint probability distribution by adjusting the matrix <span class="html-italic">V</span> of the unitary mixing in Equation (<a href="#FD26-entropy-25-01584" class="html-disp-formula">26</a>): (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>=</mo> <mfenced open="[" close="]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math> (the sampling states coincide with the eigen-squeeze modes and the probability distribution factorizes into the product of two single mode distributions given in Equation (<a href="#FD59-entropy-25-01584" class="html-disp-formula">59</a>)), (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mstyle> <mfenced open="[" close="]"> <mtable> <mtr> <mtd> <mrow> <msqrt> <mn>3</mn> </msqrt> <mo>+</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> <mo>)</mo> <mi>i</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> <mo>)</mo> <mi>i</mi> </mrow> </mtd> <mtd> <mrow> <msqrt> <mn>3</mn> </msqrt> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <mfenced open="[" close="]"> <mtable> <mtr> <mtd> <msqrt> <mn>3</mn> </msqrt> </mtd> <mtd> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </mtd> <mtd> <msqrt> <mn>3</mn> </msqrt> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mstyle> <mfenced open="[" close="]"> <mtable> <mtr> <mtd> <mrow> <mo>+</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </mtd> <mtd> <mrow> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math> (the sampling states coincide with the two counter-propagating plane waves and the probability distribution is given by the hypergeometric function in Equation (<a href="#FD77-entropy-25-01584" class="html-disp-formula">77</a>)). The scaling on all panels is the same because the most probable outcome <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> represents a Fock state invariant under unitary transformations.</p>
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<p>The joint probability distribution <math display="inline"><semantics> <msub> <mi>ρ</mi> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </msub> </semantics></math> for atomic boson sampling from two excited atom states formed by a unitary mixing of two eigen-squeeze modes (<a href="#FD44-entropy-25-01584" class="html-disp-formula">44</a>) with the same wave vector <math display="inline"><semantics> <mi mathvariant="bold">k</mi> </semantics></math>. The unitary matrix is <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <mfenced open="[" close="]"> <mtable> <mtr> <mtd> <msqrt> <mn>3</mn> </msqrt> </mtd> <mtd> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </mtd> <mtd> <msqrt> <mn>3</mn> </msqrt> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math>, which coincides with the matrix corresponding to the panel (c) in <a href="#entropy-25-01584-f005" class="html-fig">Figure 5</a>. The system of excited atoms is in a squeezed vacuum state with zero quasiparticle occupations, <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>N</mi> <mo stretchy="false">˜</mo> </mover> <mi mathvariant="bold">k</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The single-mode squeezing parameter, normal and anomalous correlators in Equation (<a href="#FD51-entropy-25-01584" class="html-disp-formula">51</a>) are <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi mathvariant="bold">k</mi> </msub> <mo>≃</mo> <mn>1.87</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mo>|</mo> <mi>α</mi> <mo>|</mo> </mrow> <mo>=</mo> <msqrt> <mn>110</mn> </msqrt> </mrow> </semantics></math>, respectively.</p>
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14 pages, 16445 KiB  
Article
Direct Laser Writing of Diffractive Structures on Bi-Layer Si/Ti Films Coated on Fused Silica Substrates
by Dmitrij A. Belousov, Roman I. Kuts, Konstantin A. Okotrub and Victor P. Korolkov
Photonics 2023, 10(7), 771; https://doi.org/10.3390/photonics10070771 - 4 Jul 2023
Cited by 3 | Viewed by 1330
Abstract
This paper presents the results of an investigation of direct laser writing on a titanium film with an antireflection capping silicon coating. Bi-layer films were deposited on fused silica substrates using an e-beam evaporation system. Modeling predicted that optical absorption for a bi-layer [...] Read more.
This paper presents the results of an investigation of direct laser writing on a titanium film with an antireflection capping silicon coating. Bi-layer films were deposited on fused silica substrates using an e-beam evaporation system. Modeling predicted that optical absorption for a bi-layer Si/Ti material can be increased by a factor of ~2 compared to a single-layer Ti film at 532 nm laser writing beam wavelength. It is experimentally proved that rate of thermochemical laser writing on Si/Ti films is at least 3 times higher than that on a single-layer Ti film with comparable thickness. The silicon layer was found to participate in the thermochemical reaction (silicide formation) under laser beam heating, which allows one to obtain sufficient position-dependent phase change (PDPC) of light reflected from exposed and unexposed areas. This results in much larger profile depth measured with a white light interferometer (up to 150 nm) than with an atomic force microscope (up to 25 nm). During direct laser writing on Si/Ti films, there is a broad range of writing beam power within which the PDPC and reflection coefficient for the exposed areas change insignificantly. The possibility of selective development of a thermochemically written pattern on a Ti film by removing the capping silicon layer on unexposed areas in a hot KOH solution is shown. Full article
(This article belongs to the Special Issue Direct Laser Writing for Photonic Applications)
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<p>Optimization of the antireflection Si layer thickness deposited on the Ti film: (<b>a</b>) Optimum Si layer thickness versus Ti layer thickness; (<b>b</b>) Examples of reflectance spectra (thickness Ti = 50 nm) for silicon layers of various thicknesses.</p>
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<p>Absorption coefficient characteristics: (<b>a</b>) Comparison of A(Si/Ti) and A(Ti); (<b>b</b>) Absorption coefficient ratio A(Si/Ti)/A(Ti).</p>
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<p>Reflectance spectra of the deposited films.</p>
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<p>Test structures written on a bi-layer Si/Ti film: (<b>a</b>) Scheme of laser writing of test structures; (<b>b</b>) Microimages (in reflected light) of test structure writings at different powers during direct laser writing on a bi-layer Si/Ti film.</p>
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<p>Change in reflectance (at 532 nm wavelength) from the exposed areas of the Si/Ti film.</p>
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<p>Raman spectra for Si/Ti film areas (right side) exposed at different laser beam power levels (V = 150 mm/s).</p>
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<p>Optically measured profile depth of structures fabricated on Si/Ti films.</p>
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<p>AFM profilogram of the formed structures (V = 75 mm/s).</p>
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<p>Image of single tracks on a Si/Ti film depending on the power of the laser writing beam (V = 300 mm/s).</p>
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<p>Microimages (in reflection) of test writing (V = 600 mm/s) before and after etching of the capping silicon layer.</p>
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<p>Change of reflectance at a wavelength of 532 nm from exposed areas before and after etching of the capping silicon layer: (<b>a</b>) V = 300 mm/s; (<b>b</b>) V = 600 mm/s.</p>
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<p>Optically measured profile depth of structures written on a bi-layer Si/Ti film before and after etching of the silicon capping layer.</p>
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10 pages, 3011 KiB  
Communication
Enhanced Readout from Spatial Interference Fringes in a Point-Source Cold Atom Inertial Sensor
by Jing Wang, Junze Tong, Wenbin Xie, Ziqian Wang, Yafei Feng and Xiaolong Wang
Sensors 2023, 23(11), 5071; https://doi.org/10.3390/s23115071 - 25 May 2023
Cited by 2 | Viewed by 1608
Abstract
When the initial size of an atom cloud in a cold atom interferometer is negligible compared to its size after free expansion, the interferometer is approximated to a point-source interferometer and is sensitive to rotational movements by introducing an additional phase shear in [...] Read more.
When the initial size of an atom cloud in a cold atom interferometer is negligible compared to its size after free expansion, the interferometer is approximated to a point-source interferometer and is sensitive to rotational movements by introducing an additional phase shear in the interference sequence. This sensitivity on rotation enables a vertical atom-fountain interferometer to measure angular velocity in addition to gravitational acceleration, which it is conventionally used to measure. The accuracy and precision of the angular velocity measurement depends on proper extraction of frequency and phase from spatial interference patterns detected via the imaging of the atom cloud, which is usually affected by various systematic biases and noise. To improve the measurement, a pre-fitting process based on principal component analysis is applied to the recorded raw images. The contrast of interference patterns are enhanced by 7–12 dB when the processing is present, which leads to an enhancement in the precision of angular velocity measurements from 6.3 μrad/s to 3.3 μrad/s. This technique is applicable in various instruments that involve precise extraction of frequency and phase from a spatial interference pattern. Full article
(This article belongs to the Special Issue Quantum Sensors and Quantum Sensing)
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<p>Schematic diagram of phase shears based on interference sequence of <math display="inline"><semantics> <mrow> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>−</mo> <mi>π</mi> <mo>−</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> Raman pulses. The two separate branches induce horizontal beam-tilt phase shear <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Φ</mi> </mrow> <mrow> <mi>H</mi> </mrow> </msub> </mrow> </semantics></math> and vertical timing-asymmetry phase shear <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Φ</mi> </mrow> <mrow> <mi>V</mi> </mrow> </msub> </mrow> </semantics></math>. Red and blue spheres represent the atom cloud in a ground state <math display="inline"><semantics> <mrow> <mo>|</mo> <mfenced open="" close="&#x27E9;" separators="|"> <mrow> <mi>g</mi> </mrow> </mfenced> </mrow> </semantics></math> and an excited state <math display="inline"><semantics> <mrow> <mo>|</mo> <mfenced open="" close="&#x27E9;" separators="|"> <mrow> <mi>e</mi> </mrow> </mfenced> </mrow> </semantics></math>, respectively.</p>
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<p>Schematic diagram of the apparatus. (<b>a</b>) The schematic diagram of phase shear in the atomic interferometer; (<b>b</b>) top view of the Earth rotation compensation stage. (<b>c</b>) The sequence of realizing phase shear in the atomic interferometer. The launching of atoms is marked as 0 ms.</p>
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<p>Apparatus of the fountain-type inertia sensor in APSI setup.</p>
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<p>Spatial interference fringes of the atom cloud captured on x-axis camera. (<b>a</b>) Horizontal spatial fringes caused by beam-tilt phase shear; (<b>b</b>) vertical spatial fringes caused by asymmetrical Raman pulse intervals.</p>
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<p>Normalized images of the first 6 principal components obtained from beam-tilt phase shear of <math display="inline"><semantics> <mrow> <mi>δ</mi> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>60</mn> <mo> </mo> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </semantics></math>.</p>
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<p>Direct and PCA fitting of spatial fringes. (<b>a</b>) and (<b>e</b>) are averaged raw images of horizontal and vertical fringe detections; (<b>b</b>) and (<b>f</b>) are 2nd highest ranked PC of (<b>a</b>) and (<b>e</b>); lower row of graphs(<b>c</b>,<b>d</b>,<b>g</b>) and (<b>h</b>) are integrations of the corresponding image in the upper row, along the direction parallel to fringes. Red lines depict the fitted curve from each data set.</p>
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<p>Derived spatial frequency from phase shears. Red solid lines represent theoretical predictions. Red dashed lines represent linear fitting of measured spatial frequencies. (<b>a</b>) and (<b>b</b>) are results from angular velocity measurements without and with PCA-based pre-fitting processing; (<b>c</b>) and (<b>d</b>) are results from asymmetrical timing phase shears. Top chart shows the residuals calculated by subtracting the predicted linear values from the measured data.</p>
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9 pages, 5254 KiB  
Communication
Tilt Measurement at the Quantum Cramer–Rao Bound Using a Higher-Order Hermite–Gaussian Mode
by Zhi Li, Yijian Wang, Hengxin Sun, Kui Liu and Jiangrui Gao
Photonics 2023, 10(5), 584; https://doi.org/10.3390/photonics10050584 - 17 May 2023
Cited by 2 | Viewed by 1436
Abstract
The quantum Cramer–Rao bound (QCRB) provides an ultimate precision limit in parameter estimation. The sensitivity of spatial measurements can be improved by using the higher-order Hermite–Gaussian mode. However, to date, the QCRB-saturating tilt measurement has not been realized. Here, we experimentally demonstrate tilt [...] Read more.
The quantum Cramer–Rao bound (QCRB) provides an ultimate precision limit in parameter estimation. The sensitivity of spatial measurements can be improved by using the higher-order Hermite–Gaussian mode. However, to date, the QCRB-saturating tilt measurement has not been realized. Here, we experimentally demonstrate tilt measurements using a higher-order HG40 mode as the probe beam. Using the balanced homodyne detection with an optimal local beam, which involves the superposition of high-order HG30 and HG50 modes, we demonstrate the precision of the tilt measurement approaching the QCRB. The signal-to-noise ratio of the tilt measurement is enhanced by 9.2 dB compared with the traditional method using HG00 as the probe beam. This scheme is more practical and robust to losses, which has potential applications in areas such as laser interferometer gravitational-wave observatories and high-sensitivity atomic force microscopes. Full article
(This article belongs to the Special Issue Optical Measurement Systems)
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<p>The tilt of the high-order Hermite–Gaussian beam. A tilted <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mn>40</mn> </msub> </mrow> </semantics></math> mode beam is decomposed in the <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mn>40</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mn>30</mn> </msub> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mn>50</mn> </msub> </mrow> </semantics></math> mode components. The tilt information <math display="inline"><semantics> <mi>θ</mi> </semantics></math> is carried by the combination of <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mn>30</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mn>50</mn> </msub> </mrow> </semantics></math> modes. The <math display="inline"><semantics> <msub> <mi>ω</mi> <mn>0</mn> </msub> </semantics></math> is the beam waist of the incident <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mn>00</mn> </msub> </mrow> </semantics></math> mode.</p>
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<p>Mode−order dependence of the normalized SNRs for the optimal (red circle dots) and intermediate (black square dots) balanced homodyne detection.</p>
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<p>Experimental setup for tilt measurements with different high-order HG modes. Mode converter (MC), detector(D), beam splitter (BS), signal generator (SG), balanced homodyne detection (BHD), 50/50 beam splitter (BS), digital storage oscilloscope (DSO), and split detector (SD).</p>
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<p>The phase monitoring from the digital storage oscilloscope (DSO) for the demodulated tilt signal. Trace1 represents the demodulated signals by the spectrum analyzer (SA). Trace2 represents the interference signals from one half of the split detector.</p>
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<p>The different high−order HG modes as the probe beam vs. the total power of the tilt signal. (<b>a</b>) The <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mrow> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> mode as the probe and the <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> mode as the local oscillator. (<b>b</b>–<b>e</b>) The <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mrow> <mn>4</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> modes as the probes, with the intermediate high-order modes as the local oscillators. (<b>f</b>–<b>i</b>) The <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mi>H</mi> <msub> <mi>G</mi> <mrow> <mn>4</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> modes as the probes, and the optimal high-order modes as local oscillators. Trace1 (black line) is the shot noise level with the signal beam being blocked. Trace2 (blue line) corresponds to the total power of the tilt signal with the scanning of the phase of the local beam.</p>
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<p>The total power of SNR with different probe beams. The circle dots and square dots, respectively, represent the experimental results with an optimal BHD and intermediate BHD. The red lines are the theoretical fittings.</p>
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28 pages, 21547 KiB  
Review
Crystal-Based X-ray Interferometry and Its Application to Phase-Contrast X-ray Imaging, Zeff Imaging, and X-ray Thermography
by Akio Yoneyama, Daiko Takamatsu, Thet-Thet Lwin, Shigehito Yamada, Tetsuya Takakuwa, Kazuyuki Hyodo, Keiichi Hirano and Satoshi Takeya
Appl. Sci. 2023, 13(9), 5424; https://doi.org/10.3390/app13095424 - 26 Apr 2023
Cited by 6 | Viewed by 2743
Abstract
Crystal-based X-ray interferometry (CXI) detects X-ray phase shifts by using the superposition of waves, and its sensitivity is the highest among the other X-ray phase-detecting methods. Therefore, phase-contrast X-ray imaging (PCXI) using CXI has the highest density resolution among the PCXI methods and [...] Read more.
Crystal-based X-ray interferometry (CXI) detects X-ray phase shifts by using the superposition of waves, and its sensitivity is the highest among the other X-ray phase-detecting methods. Therefore, phase-contrast X-ray imaging (PCXI) using CXI has the highest density resolution among the PCXI methods and enables fine, non-destructive observation with a density resolution below sub-mg/cm3. It has thus been applied in a wide range of fields, including biology, medicine, geology, and industry, such as visualization of the testis and brains of aged rats with tumors, human embryos at each Carnegie stage, air hydrates in old Antarctic ice, and ion distribution in electrolytes. Novel imaging methods have also been developed to take advantage of its high sensitivity, such as visualization of the effective atomic number (Zeff) and the three-dimensional temperature of samples. This article reviews the principles and history of PCXI and crystal-based X-ray interferometers, as well as a CXI system using synchrotron radiation and its potential applications from biomedical to industrial. Full article
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<p>Principle of phase-contrast X-ray imaging: (<b>a</b>) schematic view of absorption and phase shift of X-rays and (<b>b</b>) sensitivity ratio of phase- and absorption-contrast imaging against atomic number.</p>
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<p>Various types of crystal-based X-ray interferometers.</p>
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<p>Schematic view of STXI; <span class="html-italic">dθ</span> must be stabilized to within a few tens of prad for STXI operation.</p>
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<p>Schematic view (<b>left</b>) and photo (<b>right</b>) of ST-CXI system at beamline BL-14C of Photon Factory of KEK in Japan.</p>
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<p>Interference patterns obtained using 17.8-keV SR (<b>left</b>) and 35-keV SR (<b>right</b>).</p>
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<p>Sectional (<b>a</b>,<b>b</b>) and three-dimensional (<b>c</b>) volume rendering images of mouse brain. Reprinted with permission from Ref. [<a href="#B32-applsci-13-05424" class="html-bibr">32</a>]. Copyright 2013, IOP Publishing.</p>
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<p>Tumor-bearing rat testis: (<b>A</b>) sectional CT image obtained by CXI; (<b>B</b>) corresponding histopathological (HE-stained) gray-scale image and (<b>C</b>) color image. Periphery of testis (brown arrow); cystic areas (red arrow); solid areas (green arrows); fibrous septa (blue arrow); artificially torn region in HE images (black arrow) (1, 2, 3: solid tumor areas, 4: fibrous septa, 5: cyst) [<a href="#B42-applsci-13-05424" class="html-bibr">42</a>].</p>
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<p>Three-dimensional image of tumor-bearing rat testis obtained by CXI. Serial cut sections of tumor: (<b>A</b>) surface area, (<b>B</b>) internal area, (<b>C</b>) enlargement of boxed region in (<b>B</b>); a: blood vessel, b: seminiferous tubules, c: tunica albuginea, d: tumor, e: different densities apparent in solid area of tumor, f: cystic area of tumor, g: blood clot, h: fibrous septa [<a href="#B42-applsci-13-05424" class="html-bibr">42</a>].</p>
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<p>Three-dimensional reconstruction of tumor in rat brain: (<b>a</b>) overview of brain and tumor microvasculature; (<b>b</b>) magnified image of boxed region shown in (<b>a</b>); (<b>c</b>) tumor microvasculature viewed from different angle. Characteristics of tumor microvasculature are well depicted; e.g., tortuous, saccular (arrow), and dilated blood vessels are visible and exhibit haphazard interconnection patterns [<a href="#B40-applsci-13-05424" class="html-bibr">40</a>].</p>
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<p>Preparation for human embryo imaging. (<b>a</b>): A gel cylinder containing the human embryo sample is fixed to an acrylic plank; (<b>b</b>): the sample from (<b>a</b>) is placed in an acrylic cell ready for imaging. Samples mounted in the gel were different between photo (<b>a</b>) and photo (<b>b</b>); (<b>c</b>): the gel cylinder containing the human embryo sample is fixed to an acrylic plank. Kapton polyimide film is molded into a cup and fixed to an acrylic plank; (<b>d</b>): the gel and sample in (<b>c</b>) are placed in an acrylic cell and ready for imaging.</p>
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<p>Representative studies using CXI images acquired from the Kyoto collection. CT planes and reconstructed images were shown. (<b>a</b>): Coronal plane of the face and reconstruction of the secondary palate immediately before fusion (Carnegie stage (CS) 23) [<a href="#B50-applsci-13-05424" class="html-bibr">50</a>]. (<b>b</b>): Sagittal plane of the eyeball and reconstruction of the eyeball using the optic nerve (CS17 and CS19) [<a href="#B51-applsci-13-05424" class="html-bibr">51</a>]. (<b>c</b>): Transverse plane of the temporal bone and reconstruction of the inner ear with an enlarged brain and inner ear (CS21) [<a href="#B52-applsci-13-05424" class="html-bibr">52</a>]. (<b>d</b>): Sagittal plane of the thigh and reconstruction of the femur (CS19, CS21) [<a href="#B53-applsci-13-05424" class="html-bibr">53</a>]. (<b>e</b>): Transverse plane of the thorax, reconstruction, and median line of the bronchial tree (CS22) [<a href="#B54-applsci-13-05424" class="html-bibr">54</a>]. (<b>f</b>): Transverse plane of the thorax, reconstruction of the entire embryo and thoracic cage, and analysis of the reconstructed image (CS22) [<a href="#B55-applsci-13-05424" class="html-bibr">55</a>]. (<b>g</b>): Transverse plane of the thorax and abdomen and reconstruction of the thoracic and abdominal organs (CS21) [<a href="#B56-applsci-13-05424" class="html-bibr">56</a>]. In this sample, the liver is defective, and the positions of the stomach, pancreas, and heart have been changed.</p>
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<p>Schematic of experimental setup for CXI under temperature-controlled conditions: (<b>a</b>) top view of STXI and (<b>b</b>) schematics of experimental setup for cryochamber.</p>
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<p>(<b>a</b>) 3D image of cylindrical ice core sample including air hydrates visualized using a CXI system. The dark gray circles in the image correspond to air hydrate crystals [<a href="#B61-applsci-13-05424" class="html-bibr">61</a>]. (<b>b</b>) Air hydrates in an ice core sample were visualized using an optical microscope for comparison. (<b>c</b>) Density difference between air hydrate crystals and surrounding ice crystals plotted against size of air hydrate crystals. The point considerably higher than the others is possibly not an air hydrate but another type of hydrate, such as a sulphate hydrate, CO<sub>2</sub> hydrate, or Ar hydrate. (Reprinted with permission from [<a href="#B57-applsci-13-05424" class="html-bibr">57</a>]. Copyright 2006, AIP Publishing).</p>
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<p>X-ray images of THF hydrates: (<b>a</b>) THF hydrate with ice at 253 K and (<b>b</b>) THF hydrate with water at 276 K at interparticle pore spaces obtained by CXI [<a href="#B63-applsci-13-05424" class="html-bibr">63</a>]. Densities of ice, CH4, and air hydrates are the values at 233 K, and those of THF hydrate and water are the values at 273 K. HDPE denotes high-density polyethylene. (<b>c</b>) Absorption contrast image of THF hydrate and ice in a polypropylene (PP) tube at 123 K measured by monochromatic X-rays of 8 keV using synchrotron radiation, where the two cannot be distinguished [<a href="#B64-applsci-13-05424" class="html-bibr">64</a>].</p>
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<p>Sectional images obtained by CXI of (<b>a</b>) CH<sub>4</sub> hydrate and (<b>b</b>) natural gas hydrate with ice (white part) at 193 K. Reprinted with permission from [<a href="#B60-applsci-13-05424" class="html-bibr">60</a>,<a href="#B61-applsci-13-05424" class="html-bibr">61</a>]. Copyright 2011 and 2012, American Chemical Society.</p>
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<p>(<b>a</b>) Schematic of experimental setup of CXI system for in situ observation of LIB behavior. (<b>b</b>) Schematic of LIB cell, where WE is LiFePO<sub>4</sub> composite electrode, CE is Li electrode, and electrolyte is 1 M LiClO<sub>4</sub> dissolved in carbonate solvent (EC:DEC = 1:2 <span class="html-italic">v</span>/<span class="html-italic">v</span>). Continuous phase map of electrolyte in region indicated by red dashed line was analyzed [<a href="#B70-applsci-13-05424" class="html-bibr">70</a>]. Reproduced with permission from ref [<a href="#B70-applsci-13-05424" class="html-bibr">70</a>]. Copyright 2022 Springer Nature.</p>
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<p>(<b>a</b>) Constant-current (0.06 mA, C/2 rate) charging profile and (<b>b</b>) corresponding extracted phase maps of electrolyte. Since Δ<span class="html-italic">p</span> is estimated as change from initial image of t = 0 s (without an applied current), change in Δ<span class="html-italic">p</span> reflects change in electrolyte density from the initial state caused by charging. (<b>c</b>) Δ<span class="html-italic">p</span> profiles of x-axis averaged over y-axis in (<b>b</b>) [<a href="#B71-applsci-13-05424" class="html-bibr">71</a>]. (Reproduced with permission from ref [<a href="#B71-applsci-13-05424" class="html-bibr">71</a>]. Copyright 2018 American Chemical Society.) SOC: state of charge.</p>
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<p>(<b>a</b>) Fast charge/discharge cycle (0.36 mA, 3C rate, five cycles) and potential profiles of first cycle, (<b>b</b>) corresponding extracted phase maps of electrolyte, and (<b>c</b>) Δ<span class="html-italic">p</span> profile of x-axis direction charge (points A to F) and discharge (points G to L) [<a href="#B71-applsci-13-05424" class="html-bibr">71</a>]. Reproduced with permission from ref [<a href="#B71-applsci-13-05424" class="html-bibr">71</a>]. Copyright 2018 American Chemical Society.</p>
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<p>(<b>a</b>) Phase map of LAB cell obtained before current was applied. (<b>b</b>) Charge-discharge curve at C/2-rate (50 mA) and corresponding phase maps acquired at A–H [<a href="#B72-applsci-13-05424" class="html-bibr">72</a>]. (Reproduced with permission from ref [<a href="#B72-applsci-13-05424" class="html-bibr">72</a>]. Copyright 2020 Royal Society of Chemistry).</p>
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<p>Images of mouse leg obtained using (<b>a</b>) degree of coherence contrast, (<b>b</b>) phase contrast, and (<b>c</b>) absorption contrast. Reproduced from ref. [<a href="#B73-applsci-13-05424" class="html-bibr">73</a>] with the permission of AIP Publishing.</p>
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<p>Photos of metal foils and corresponding phase maps, absorption maps, and Z<sub>eff</sub> images. Average Zeff values were 16.4, 25.4, 27.9, and 28.8 for Al, Fe, Ni, and Cu foils, respectively [<a href="#B74-applsci-13-05424" class="html-bibr">74</a>]. Reproduced from ref. [<a href="#B74-applsci-13-05424" class="html-bibr">74</a>] with the permission of AIP Publishing.</p>
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<p>(<b>a</b>) Z<sub>eff</sub> image and (<b>b</b>) photo of rusted iron plate. Orange circles indicate rusted areas [<a href="#B75-applsci-13-05424" class="html-bibr">75</a>].</p>
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<p>Time-resolved projection images of thermal flow in heated water (<b>left</b>) and images calculated using finite element method (<b>right</b>) [<a href="#B76-applsci-13-05424" class="html-bibr">76</a>].</p>
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<p>Schematic of sample cell (<b>left</b>) and three-dimensional temperature distribution of water heated using ceramic heater (<b>right</b>) [<a href="#B76-applsci-13-05424" class="html-bibr">76</a>].</p>
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