Single Dip Research Articles

Realization and simulation of silicon-on-sapphire mid-infrared one-dimensional photonic crystal cavities

The mid-infrared (MIR) waveband is significant for chemical and biological sensing since it covers several atmospheric windows and molecular fingerprint regions. On-chip photonic integrated one-dimensional (1D) microcavities have great potential for high-performance mid-IR sensing because of their high sensitivity and compact structure. However, high-performance 1D microcavities based on the promising silicon-on-sapphire (SoS) MIR platform have not yet been designed or realized. Based on the photonic band structure induced by 1D photonic crystals (PhC), a high-performance Bragg reflector, an inward apodized Bragg grating, and a free spectral range (FSR)-free PhC microcavity integrated system operating in the MIR waveband were developed on the SoS platform. By carefully designing the period and penetration depth of the corrugation in the Bragg reflector, a stopband of 45 nm and an extinction ratio of −12 dB were achieved. The inward apodized Bragg grating was optimized by adjusting the apodization depth and the number of periods, resulting in a quality factor of 1043 at a wavelength of 3088.4 nm. Furthermore, introducing a Fabry–Pérot (F-P) cavity between two Bragg reflectors (with side-coupled light) and precisely tuning the stopband of the Bragg reflector and the FSR of the F-P cavity enabled the realization of an FSR-free PhC microcavity. This microcavity exhibited a single deep resonance dip with subnanometer bandwidth across a record-wide operational waveband from 3025 to 3200 nm, achieving a quality factor of approximately 5090. The MIR 1D PhC microcavities on the SoS platform hold great promise for high-performance gas detection and molecular sensing in future applications.

Kinematic source rupture on listric faults for the 2024 Noto Peninsula, Japan, earthquake (Mw 7.5) estimated from near-field strong-motion waveforms

On January 1, 2024, an Mw 7.5 reverse-fault earthquake occurred along the submarine active faults just offshore of the Noto Peninsula in central Japan. This earthquake is one of the largest inland crustal earthquakes that ever recorded in Japan. We performed inversions of near-field strong-motion waveforms (0.05–0.25 Hz) to investigate the kinematic rupture process of this earthquake. For the inversions, we assumed listric fault planes, where the dip angles are steeper near the seafloor and become gentler with increased depth, based on the location of the submarine fault offsets, the structure at shallow depths revealed by seismic reflection surveys, and the relocated aftershock distribution at deep depths. To constrain our kinematic source model, we also utilized geodetic information published by the Geospatial Information Authority of Japan. We identified that the rupture process had three phases. The first phase was the initial rupture with a small slip of ~ 2 m around the rupture initiation area. After several seconds, the second and third phases, which were the main ruptures, propagated in the southwest and northeast directions, respectively. During the second and third phases, five large slip areas with average slips of 5–6 m significantly contributed to the strong-motion waveforms. By forward-simulating the geodetic data, we found that the rupture at shallow depths included an oblique component close to a right-lateral component and that, in the eastern part of the source region, the rupture transferred from a southeastward-dipping fault to a northwestward-dipping fault. The listric fault with high dip angles at shallow depths is better than a planar fault with a single dip angle for accurately modeling the fault slip near the surface and the coseismic displacement close to the fault trace during the reverse-fault earthquake.Graphical abstract

Zero-crosstalk silicon photonic refractive indexsensor with subwavelength gratings.

Silicon photonic index sensors have received significant attention for label-free bio and gas-sensing applications, offering cost-effective and scalable solutions. Here, we introduce an ultra-compact silicon photonic refractive index sensor that leverages zero-crosstalk singularity responses enabled by subwavelength gratings. The subwavelength gratings are precisely engineered to achieve an anisotropic perturbation-led zero-crosstalk, resulting in a single transmission dip singularity in the spectrum that is independent of device length. The sensor is optimized for the transverse magnetic mode operation, where the subwavelength gratings are arranged perpendicular to the propagation direction to support a leaky-like mode and maximize the evanescent field interaction with the analyte space. Experimental results demonstrate a high wavelength sensitivity of - 410nm/RIU and an intensity sensitivity of 395dB/RIU, with a compact device footprint of approximately 82.8μm2. Distinct from other resonant and interferometric sensors, our approach provides an FSR-free single-dip spectral response on a small device footprint, overcoming common challenges faced by traditional sensors, such as signal/phase ambiguity, sensitivity fading, limited detection range, and the necessity for large device footprints. This makes our sensor ideal for simplified intensity interrogation. The proposed sensor holds promise for a range of on-chip refractive index sensing applications, from gas to biochemical detection, representing a significant step towards efficient and miniaturized photonic sensing solutions.

A Retrospective Study: Are the Multi-Dips in the THz Spectrum during Laser Filamentation Caused by THz–Plasma Interactions?

During the process of terahertz (THz) wave generation via femtosecond laser filamentation in air, as well as through the mixing of THz waves with externally injected plasma filaments, THz waves engage in interactions with the plasma. A characteristic feature of this interaction is the modulation of the THz radiation spectrum by the plasma, which includes the generation of THz spectral dips. This information is essential for understanding the underlying mechanisms of THz–plasma interactions or for inferring plasma parameters. However, a current debate exists on the number of THz spectral dips observed after the interaction, with different opinions of single versus multiple dips, thus leaving the interaction mechanisms still ambiguous. In this work, we retrospectively analyzed the experimental appearance of multiple dips in the THz spectrum and found that the current observations of such dips are predominantly a result of the water vapor absorption with a low spectral resolution. Additionally, we observed that altering the acquisition width of the temporal THz signal also influenced the dips’ number. Hence, in future research, simultaneous attention should be paid to the following two aspects of THz–plasma interactions: (1) It is necessary to ensure a sufficiently wide time-domain window to accurately represent the spectral dip characteristics. (2) The spectral dips should be carefully distinguished from the water absorption lines before being further studied. On the other hand, for the case of a single dip in the THz spectrum, we also put forward a new viewpoint of the resonance between surface plasmon waves and THz waves, which should also be taken into consideration in future studies.

High linearity temperature-compensated SPR fiber sensor for the detection of glucose solution concentrations

In the realm of optical fiber sensing, conventional surface plasmon resonance (SPR) sensors often face challenges due to their low linearity, broad resonance, and susceptibility to external temperature changes. To that end, we evaluate a groundbreaking approach with a temperature-compensating SPR biosensor based on the Mach-Zehnder interferometer (MZI), featuring a novel tapered multimode fiber-single mode fiber-multimode fiber (MMF-SMF-MMF) architecture. Specifically, the SPR is excited on a silver film modified by reduced graphene oxide (rGO) and π-π stacked pyrene-1-boric acid providing specific glucose binding ability. The MZI effect compensating for ambient temperature influence on the SPR is realized with the MMF-SMF-MMF structure. The study also explores the use of Fast Fourier Transform (FFT) for separate, granular analysis of the MZI and SPR signals and improves the linearity of the sensor using a novel centroid-fitting technique. This method utilizes the shift of calculated centroid coordinates rather than the single dip resonance wavelength from SPR, enhancing the overall performance of the sensor. The experimental results demonstrate excellent glucose sensitivity of 2.819 nm/mM, a linear range of 0–10 mmol/L, a LOD of 0.12 mM/L and concurrent temperature compensation. Compact and easy to fabricate, the proposed SPR sensor provides a novel solution for accurate glucose concentration detection in human blood.

Numerical Investigation of Double Transonic Dip Behaviors in Supercritical Airfoil Flutter

This numerical study examines the behavior of the double transonic dip with a supercritical airfoil. A two-dimensional model of a wing section with two degrees of freedom was used for the investigation. The flowfield was modeled by solving the unsteady Reynolds-averaged compressible Navier–Stokes equations using the Spalart–Allmaras turbulence model, and the equations of motion were solved to determine the structural dynamics. The present model successfully captured the behavior of the double transonic dip on the flutter boundary of the supercritical airfoil, which contrasts with the well-known behavior of the single transonic dip exhibited by conventional symmetric airfoils. Although the mechanism of the first dip at a lower Mach number corresponded to that of the well-known conventional transonic dip, the second dip at a higher Mach number was uniquely observed for the supercritical airfoil. The analysis established that, for the supercritical airfoil, the motion of the shock wave over the upper surface was significantly affected by the behavior of the boundary layer around the highly cambered aft region of the lower surface during flutter. The behavior of the boundary layer involving the separation and reattachment over the lower surface caused the unusual shock wave motion over the upper surface under the Mach number condition at the bottom of the second dip. This motion exerted negative damping forces on the motion of the airfoil, thereby becoming the primary contributor to generating the second dip experienced by the supercritical airfoil.

Lamb Dip of a Quadrupole Transition in H_{2}.

The saturated absorption spectrum of the hyperfineless S(0) quadrupole line in the (2-0) band of H_{2} is measured at λ=1189 nm, using the NICE-OHMS technique under cryogenic conditions (72K). It is the first time that a Lamb dip of a molecular quadrupole transition has been recorded. At low (150-200W) saturation powers a single narrow Lamb dip is observed, ruling out an underlying recoil doublet of 140kHz. Studies of Doppler-detuned resonances show that the redshifted recoil component can be made visible for low pressures and powers, and prove that the narrow Lamb dip must be interpreted as the blue recoil component. A transition frequency of 252 016 361 164 (8)kHz is extracted, which is off by -2.6 (1.6)MHz from molecular quantum electrodynamical calculations therewith providing a challenge to theory.

ERYTHROMYCIN-DOPED NANOFIBRE COATING ENHANCES OSSEOINTEGRATION AND INHIBITS STAPHYLOCOCCUS AUREUS-INDUCED OSTEOLYSIS

Failure of osseointegration and periprosthetic joint infection (PJI) are the two main reasons of implant failure after total joint replacement (TJR). Nanofiber (NF) implant surface coating represents an alternative local drug eluting device that improves osseointegration and decreases the risk of PJI. The purpose of this study was to investigate the therapeutic efficacies of erythromycin (EM)-loaded coaxial PLGA/PCL-PVA NF coating in a rat S. aureus-infected tibia model.NF coatings with 100mg and 1000mg EM were prepared. NF without EM was included as positive control. 56 Sprague Dawley rats were divided into 4 groups. A titanium pin (1.0-mm x 8 mm) was placed into the tibia through the intercondylar notch. S. aureus (SA) was introduced by both direct injection of 10 μl broth (1 × 104 CFU) into the medullary cavity and single dip of Ti pins into a similar solution prior to insertion. Rats were sacrificed at 8 and 16 weeks after surgery. The outcome measurements include μCT based quantitative osteolysis evaluation and hard tissue histology.Results: EM-NF coating (EM100 and EM1000) reduced osteolysis at 8 and 16 weeks, compared to EM0 and negative control. The effective infection control by EM-NFs was further confirmed by hard tissue section analysis. The Bone implant contact (BIC) and bone area fraction Occupancy (BAFO) within 200 µm of the surface of the pins were used to evaluate the osseointegration and new bone formation around the implants. At 16 weeks, the bone implant contact (BIC) of EM 100 (35.08%) was higher than that of negative control (3.43%) and EM0 (0%). The bone area fraction occupancy within 200 µm (BAFO) of EM100 (0.63 mm2) was higher than that of negative control (0.390 mm2) and EM0 (0.0 mm2). The BAFO of EM100 was also higher than that of EM1000 (0.3mm2).There was much less osteolysis observed with EM100 and EM1000 NF coatings at 16 weeks, as compared to EM0 positive control, p=0.08 and p=0.1, respectively. Osseointegration and periprosthetic bone formation was enhanced by EM-NFs, especially EM100. Data from this pilot study is promising for improving implant surface fabrication strategies.

Theory of Floquet-driven dissipative cavity magnonics

Coherent magnon-photon coupling under the Floquet drive has been demonstrated in recent experimental measurements and theoretical calculations. In this work, we studied the Floquet-driven dissipative cavity magnonics theoretically. Our results show that the Floquet states ${d}^{(n)}$ ($n$ is integer) with dissipative coupling give rise to many level attractions in the transmission spectrum. As driving amplitude increases or driving frequency decreases, the coupling for the $n=0$ mode becomes weak while for the sideband $(n=\ifmmode\pm\else\textpm\fi{}1)$ becomes strong due to renormalized coupling strength. When the coherent and the dissipative couplings are simultaneously present, a series of sharp dips originating from the interference of two couplings occur, in contrast to a single dip with zero damping in previous work. The modified zero-damping condition is derived for the Floquet system. Moreover, we give a generalized selection rule for coupling between Floquet states by considering the multiphoton process that has not yet been studied in previous work. Our results open up promising roads for exploring the dissipative magnon-photon coupling with the Floquet driving technology.

Fabrication and optimization of LSM infiltrated cathode electrode for anode supported microtubular solid oxide fuel cells

In this study, anode supported microtubular solid oxide fuel cells (SOFCs) with LSM (lanthanum strontium manganite) catalyst infiltrated LSM-YSZ (yttria stabilized zirconia) cathodes are developed to increase the density of triple/three phase boundaries (TPBs) in the cathode, thereby to improve the cell performance. For this purpose, two different porous YSZ layers are formed on the dense YSZ electrolyte, i.e., one is with co-sintering while the other one is not. Incorporation of LSM into these porous YSZ layers is achieved via dip coating of a sol-gel based infiltration solution. The effects of the fabrication method for porous YSZ, LSM solution dwelling time and the thickness of the porous YSZ layer on the cell performance are experimentally investigated and optimized in the given order. A reference cell having a conventional dip coated cathode prepared by mixing the commercial LSM and YSZ powders is also fabricated for comparison. The results show that among the cases considered, the highest peak power density of 0.828 W/cm2 can be obtained from the cell, whose single dip coated porous electrolyte layer co-sintered with the dense electrolyte is impregnated with LSM for a dwelling time of 45 min. On the other hand, the peak power density of the reference cell is measured as only 0.558 W/cm2. These results reveal that ∼50% increase in the maximum cell performance compared to that of the reference cell can be achieved by LSM infiltration after the optimizations.

Analysis of transmission spectra and cladding mode evolution of etched double-cladding fiber long-period grating

We demonstrate the inscription of long-period gratings (LPGs) in a fluorine-doped double-cladding fiber (DCF) with a low refractive index trench. The grating has a single resonance dip in the wide wavelength range from 1200 nm to 2400 nm. The transmission spectra and cladding mode evolution of the DCF-LPG during the cladding etching process are investigated theoretically and experimentally. The mode order of the cladding modes decreases with one order of every 20μm diameter etching, while the resonance wavelength of the DCF-LPG barely shifts when the fiber diameter decreases gradually. Only LP11 mode can be excited in the etched DCF-LPG when the fiber diameter is less than 40μm. The characteristics of the DCF-LPGs are investigated experimentally. The torsion sensitivity was measured to be −0.154 nm/(rad/m) and −0.271 nm/(rad/m) at 1.55μm and 2.0μm wavebands, respectively, which are one order of magnitude higher than that of the conventional LPG. The proposed DCF-LPG has potential application as a tunable band-rejection filter in broad wavebands.

A sol-gel based bioactive glass coating on laser textured 316L stainless steel substrate for enhanced biocompatability and anti-corrosion properties

In this study, we have synthesized a new family of bioactive glass (BAG) comprising of SiO2, Na2O, CaO, K2O, P2O5 and MgO via sol-gel route. The composition of these oxides has been selected in such a way that the BAG has a coefficient of thermal expansion close to the 316L stainless steel (SS). In vitro test by soaking the BAG in simulated body fluid (SBF) showed good bioactivity due to ion exchange between alkali ions from the BAG and the ions present in the SBF. The 316L SS substrates were textured using laser before application of BAG using sol-gel dip coating. Pull off test showed that the adhesion strength in case of textured and non-textured surfaces were 4.2 MPa and 2.46 MPa, respectively. This implies that texturing of the surface is helpful in increasing the adhesion strength of the coating. The bioactive glass-coated textured surface of 316L SS substrate showed better corrosion resistance in SBF than the bare substrate for which the respective corrosion current densities were 35 nA/cm2 and 353 nA/cm2, thereby indicating that the single layer sol-gel dip coating of bioactive glass improved the corrosion resistance of the 316L SS implants. The analysis of scanning electron microscopy (SEM) images of the coating after immersion in SBF shows the formation and growth of the apatite layer that will further inhibit the corrosion of the substrate. These results demonstrate the potential of the synthesized BAG as a coating material for 316L SS implants.

Observation of single photon interference and Hong-Ou-Mandel dip

We have built a cost effective Hong-Ou-Mandel (HOM) interferometer by using a conventional cheap multimode laser and a home-made coincidence counting circuit and we have observed the single photon interference and HOM dip making us to measure the bandpass filter. We expect it could be a good model to the graduates trying a variety of quantum optics experiments.

Simultaneous Measurement of High Refractive Index and Temperature Based on SSRS-FBG

In this study, a novel and simple optical fiber sensor is proposed, simulated, and experimentally demonstrated for simultaneous measurement of high refractive index (RI) and temperature. The sensor consists of a section of single-mode-silica rod-single-mode (SSRS) fiber structure cascaded to fiber Bragg grating (FBG) (SSRS-FBG). The simultaneous measurement of high RI and temperature was realized in this study by monitoring the output power level and wavelength shift of the SSRS-FBG single dip transmission spectrum corresponding to different surrounding high RI and ambient temperature values. The experimental results showed that the cascaded sensor had a sensitivity of 108.07 dBm/RIU for high RI ranging at 1.45-1.531, while the temperature sensitivity was 9.31 pm/°C from 35 °C to 85 °C. This sensor is most suitable for high RI applications, such as for monitoring oil quality stored in terminal and power transformers.

Free-spectral-range-free filters with ultrawide tunability across the S + C + L band

Optical filters are essential parts of advanced optical communication and sensing systems. Among them, the ones with an ultrawide free spectral range (FSR) are especially critical. They are promising to provide access to numerous wavelength channels highly desired for large-capacity optical transmission and multipoint multiparameter sensing. Present schemes for wide-FSR filters either suffer from limited cavity length or poor fabrication tolerance or impose an additional active-tuning control requirement. We theoretically and experimentally demonstrate a filter that features FSR-free operation capability, subnanometer optical bandwidth, and acceptable fabrication tolerance. Only one single deep dip within a record-large waveband ( S + C + L band) is observed by appropriately designing a side-coupled Bragg-grating-assisted Fabry–Perot filter, which has been applied as the basic sensing unit for both the refractive index and temperature measurement. Five such basic units are also cascaded in series to demonstrate a multichannel filter. This work provides a new insight to design FSR-free filters and opens up a possibility of flexible large-capacity integration using more wavelength channels, which will greatly advance integrated photonics in optical communication and sensing.

Ultrawide FSR microring racetrack resonator with an integrated Fabry–Perot cavity for refractive index sensing

We propose a simple route to obtain an ultrawide free spectral range (FSR) using a composite racetrack microring resonator (RMRR) with Fabry–Perot cavities. The presence of two Fabry–Perot cavities in the straight arm of the RMRR suppresses other peaks while yielding a single resonance dip with a high extinction ratio. A comprehensive, detailed mathematical description and finite difference time domain (FDTD) simulations investigate the proposed device performances. The modified RMRR yields an FSR value greater than 150 nm while retaining the unmodified ring resonator’s Q -factor. Other than its application in communications, the device also will be highly useful for sensing. FDTD simulations show a sensitivity of 185 nm/RIU, which is 2.6 times larger than the unmodified RMRR. The proposed device also enjoys compactness and a wide tunability range and has less fabrication complexity. The proposed sensor could be a strong candidate in optical communications, optical switching, and many other applications.

Reflectionless dual standing-wave microcavity resonator units for photonic integrated circuits

We propose a novel photonic circuit element configuration that emulates the through-port response of a bus coupled traveling-wave resonator using two standing-wave resonant cavities. In this "reflectionless resonator unit", the two constituent cavities, here photonic crystal (PhC) nanobeams, exhibit opposite mode symmetries and may otherwise belong to a single design family. They are coupled evanescently to the bus waveguide without mutual coupling. We show theoretically, and verify using FDTD simulations, that reflection is eliminated when the two cavities are wavelength aligned. This occurs due to symmetry-induced destructive interference at the bus coupling region in the proposed photonic circuit topology. The transmission is equivalent to that of a bus-coupled traveling-wave (e.g. microring) resonator for all coupling conditions. We experimentally demonstrate an implementation fabricated in a new 45 nm silicon-on-insulator complementary metal-oxide semiconductor (SOI CMOS) electronic-photonic process. Both PhC nanobeam cavities have a full-width half-maximum (FWHM) mode length of 4.28 μm and measured intrinsic Q's in excess of 200,000. When the resonances are tuned to degeneracy and coalesce, transmission dips of the over-coupled PhC nanobeam cavities of -16 dB and -17 dB nearly disappear showing a remaining single dip of -4.2 dB, while reflection peaks are simultaneously reduced by 10 dB, demonstrating the quasi-traveling-wave behavior. This photonic circuit topology paves the way for realizing low-energy active devices such as modulators and detectors that can be cascaded to form wavelength-division multiplexed links with smaller power consumption and footprint than traveling wave, ring resonator based implementations.

Coherent perfect absorption in unpatterned thin films of intrinsic semiconductor

We analyze the coherent perfect absorption (CPA) in unpatterned thin films in intrinsic semiconductor using the admittance matching method. The CPA is obtained at the phase and irradiance matching conditions. Controlling the modulation of relative input irradiances, a sharply single dip of CPA transforms to the broadband perfect absorption. The CPA parameters of admittance matching method are well agreement with the transfer matrix method. We analyze also coherently controlling the phase mechanism that is applicable for optical switching and modulation. This method provides the precision in design at visible regime. The method can enhance the design of future photonics devices for sensors and modulators.

Has the modern design of Attune total knee replacement improved outcome in patients with isolated patellofemoral arthritis?

Modern knee replacements aim to improve patient function in arthritis affecting different compartments of the knee. This study evaluates the Patient Reported Outcome Measure (PROM) and functional outcome of a modern total knee replacement (Attune, DePuy) in patients with isolated patellofemoral arthritis. A total of 50 consecutive patients with isolated unilateral patellofemoral arthritis having had Attune total knee replacements at a single institution between 2010 and 2016 were prospectively studied. Five patients who developed symptoms on the opposite side during the study and two patients lost to follow-up were excluded. One patient needed early revision for loosening, leaving a total of 42 patients to be followed up over a period of 4 years. The Oxford Knee score (OKS) and Knee Injury and Osteoarthritis Outcome Score (KOOS) recorded pre-operatively and at follow-up was compared. A Functional assessment at around 8 months after operation was undertaken. At average follow-up of 24 months the mean OKS score improved by 15 points and the KOOS score improved by 20 points. Final KOOS sub-score for Pain was 80, Symptom 80, and ADL 82, Sports & Recreation 32 and QOL 60. Functional assessment at mean 8 months showed that a significant number of patients were able to Kneel (50%); Sit cross legged (23%); sit on their heel (23%) and were able do a single leg dip test (86%). This unique study of a modern design total knee replacement (Attune) in patients with isolated unilateral patellofemoral arthritis shows good PROM scores at 2 years and good functional assessment results at 8 months. The PROM scores are marginally better than the published results with Attune's predecessor, in a similar cohort of patients, but falls short of the published results of patellofemoral replacement implants. Large randomised comparative studies between traditional and the modern implant design is recommended to answer the question if design modification has influenced clinical outcome in patients with patellofemoral arthritis.

Characterization of optical light curves of extreme variability quasars over a ∼16-yr baseline

ABSTRACT We study the optical light curves – primarily probing the variable emission from the accretion disc – of ∼900 extreme variability quasars (EVQs, with maximum flux variations more than 1 mag) over an observed-frame baseline of ∼16 yr using public data from the SDSS Stripe 82, PanSTARRS-1 and the Dark Energy Survey. We classify the multiyear long-term light curves of EVQs into three categories roughly in the order of decreasing smoothness: monotonic decreasing or increasing (3.7 per cent), single broad peak and dip (56.8 per cent), and more complex patterns (39.5 per cent). The rareness of monotonic cases suggests that the major mechanisms driving the extreme optical variability do not operate over time-scales much longer than a few years. Simulated light curves with a damped random walk model generally under-predict the first two categories with smoother long-term trends. Despite the different long-term behaviours of these EVQs, there is little dependence of the long-term trend on the physical properties of quasars, such as their luminosity, BH mass, and Eddington ratio. The large dynamic range of optical flux variability over multiyear time-scales of these EVQs allows us to explore the ensemble correlation between the short-term (≲6 months) variability and the seasonal-average flux across the decade-long baseline (the rms-mean flux relation). We find that unlike the results for X-ray variability studies, the linear short-term flux variations do not scale with the seasonal-average flux, indicating different mechanisms that drive the short-term flickering and long-term extreme variability of accretion disc emission. Finally, we present a sample of 16 EVQs, where the approximately bell-shaped large amplitude variation in the light curve can be reasonably well fit by a simple microlensing model.