A H Dorrah M Zamboni-rached and M Mojahedi Physical Review a 93 063864 (2016)

abril 17, 2022 Postar um comentário

Light Sci Appl. 2018; 7: 40.

Experimental demonstration of tunable refractometer based on orbital angular momentum of longitudinally structured calorie-free

Ahmed H. Dorrah

¹Edward Due south. Rogers Sr. Department of Electric and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4 Canada

Michel Zamboni-Rached

ⁱⁱSchool of Electrical and Computer Applied science, University of Campinas, Campinas – SP, 13083-852 Brazil

Mo Mojahedi

¹Edward S. Rogers Sr. Department of Electrical and Figurer Engineering, Academy of Toronto, Toronto, ON M5S 3G4 Canada

Received 2018 Feb six; Revised 2018 May six; Accepted 2018 May 13.

Abstract

The index of refraction plays a decisive office in the blueprint and classification of optical materials and devices; therefore, its proper and accurate decision is essential. In most refractive index (RI) sensing schemes, nonetheless, there is a trade-off between providing high-resolution measurements and roofing a broad range of RIs. We propose and experimentally demonstrate a novel mechanism for sensing the index of refraction of a medium by utilizing the orbital angular momentum (OAM) of structured light. Using a superposition of co-propagating monochromatic higher-order Bessel beams with equally spaced longitudinal wavenumbers, in a comb-similar setting, we generate non-diffracting rotating light structures in which the orientation of the beam's intensity profile is sensitive to the RI of the medium (here, a fluid). In principle, the sensitivity of this scheme can exceed ~2700°/RI unit of measurement (RIU) with a resolution of ~x^-v RIU. Furthermore, we bear witness how the unbounded degrees of liberty associated with OAM can exist deployed to offer a wide dynamic range by generating structured light that evolves into unlike patterns based on the change in RI. The rotating light structures are generated by a programmable spatial light modulator. This provides dynamic control over the sensitivity, which can be tuned to perform coarse or fine measurements of the RI in existent time. This, in turn, allows high sensitivity and resolution to be achieved simultaneously over a very wide dynamic range, which is a typical merchandise-off in all RI sensing schemes. We thus envision that this method volition open new directions in refractometry and remote sensing.

Introduction

The interactions of low-cal with a medium can be exploited to sense, mensurate, and study important properties of the medium^ane–xi. Ane important property is the index of refraction, which plays a crucial role in the design and classifications of nearly optical materials and devices. As such, modalities that tin can provide accurate, efficient, economic, and dynamic measurements of the index of refraction are always needed. In the by, various approaches to mensurate the alphabetize of refraction have been proposed and utilized. For instance, in laser-based refractive index (RI) sensing, the modify in RI in a medium can be inferred from the bending of refraction of the beam in the medium. This has been achieved in diverse means; it was originally washed past measuring the critical angle in the medium using a prism² or measuring the deportation of a beam that is obliquely incident on the sampleⁱⁱⁱ. More recently, other properties of light (east.g., diffraction) take been utilized to mensurate the index of refraction⁴. Although these previous techniques are relatively simple to implement, they lack reconfigurability, and their resolution is typically limited to ~10^-3 RI unit (RIU).

In some cases, the frequency response can exist more informative where a change in RI tin be linked to a shift in the transmission spectrum of a broadband source. This has been manifested by relating the RI change to a shift in the transmission spectrum of micro-ring resonators^12,13, micro-fiber resonators^14–sixteen, Mach–Zehnder interferometers^17,18, a shift in surface plasmon resonance^19,twenty, or detecting the shift in the reflection spectrum of Fabry–Perot resonators^21,22. Polarization is another holding of lite that has been used to diversify the RI measurements, thereby improving the sensing precision^23,24. Although these afterward techniques tin detect the index of refraction with high resolution (reaching ~10⁻⁶ RIU), they often endure from a lack of tunability and reconfigurability and accept a narrow dynamic range, high cost, sensitive interfacing and packaging requirements, and circuitous device fabrication processes.

To overcome these limitations, nosotros propose and demonstrate a novel RI sensing machinery that leverages on two important but unexplored degrees of freedom of light: low-cal'southward orbital angular momentum (OAM) and the ability to pattern calorie-free'due south intensity along its direction of propagation. We bear witness how these two degrees of liberty tin can be utilized to generate rotating intensity patterns (in the shape of flower petals), whose orientation and fashion profile are sensitive to the medium's index of refraction. The proposed scheme is light amplification by stimulated emission of radiation based: information technology only requires a laser source to be shaped past a spatial low-cal modulator (SLM) and only detected by a CCD camera. Hence, it does non require any sophisticated fabrication or packaging. Furthermore, because beam generation is performed using an SLM, which is addressed by a programmable computer-generated hologram (CGH), the proposed scheme is reconfigurable and provides a dynamic sensitivity that tin be tuned in real time. With such tunability, a coarse RI measurement can be performed first over a wide range, and the CGH tin can then be updated to obtain effectively measurements with higher resolution. This tunability can address the claiming of simultaneously achieving loftier sensitivity and a broad dynamic range, which is a typical trade-off in all RI sensing schemes.

Background

Low-cal beams with OAM possess helical phase-fronts due to their azimuthal phase dependency that follows due east^{iℓ ϕ} , where ℓ is the topological accuse or winding index of the helical phase-forepart. In fact, OAM differs fundamentally from the spin angular momentum (SAM) associated with polarization:^25–28 unlike SAM, which is limited to a value of ± ℏ per photon, OAM can larn unbounded values of ℓ ℏ per photon ( ℓ is an integer), thus offer additional degrees of liberty. These new degrees of freedom of light, namely, the OAM modes, take been utilized in imaging²⁶, optical trapping²⁹, cloth processing^thirty, data communications^31,32, and move sensing^9,10. Here, we show, for the first time (to the best of our cognition), the unexplored advantages of using light's OAM to mensurate the index of refraction of a given medium. Such a development can open new directions in refractometry and remote sensing using structured light.

Concept

OAM beams are characterized by twisted phase-fronts with a singularity in the beam's center. As such, they behave cypher intensity in the beam's heart, whereas their intensity is distributed over a cylindrical surface along the beam's axis. An OAM axle with topological charge ℓ possesses ℓ inter-twined helices in its phase-front²⁵. For the same value of ℓ , the amount of phase twist in each helix, over a finite distance, depends on the wavenumber and the RI of the medium. In essence, the helical phase tin can encounter dissimilar amounts of stretching (or pinch) if the same beam propagates in unlike media (with different RI), every bit shown beneath. Although characterized past a helical phase-forepart, when looking at the transverse intensity profiles of an OAM style, the intensity is distributed over a continuous ring. Hence, the amount of phase helicity is not readily detected past but looking at the beam'south intensity contour, and its detection typically requires a wavefront sensing appliance. Nonetheless, with a judicious superposition of 2 OAM beams of opposite helicities such that the two OAM beams carry topological charges with opposite signs, it becomes possible to directly map the helicity in the beam's stage-front end to a modulation in the beam'southward intensity profile, which can exist hands detected by a CCD camera. This occurs every bit a result of introducing singularities into the stage-front along the azimuthal direction ϕ , which in turn creates discontinuities in the axle'southward transverse intensity profile—often producing intensity patterns in the form of bloom petals^33,34.

When the longitudinal wavenumbers of the superimposed OAM modes are slightly dissimilar, the beating betwixt the spatial frequency harmonics will outcome in a lite structure whose intensity contour tin rotate along its optical path. The rotation of light'due south intensity pattern along its propagation management has been previously reported in refs^35–39. The angular orientation of the rotating beam petals is a role of both its propagation altitude ( z ) and its optica l 50 ength. In other words, at a fixed detection plane along z, the beam orientation will besides vary if the RI of the medium is changed. This variation in the beam's angular orientation tin be interpreted based on the fact that the angular velocity of the rotating intensity pattern (petals) is directly linked to the corporeality of its phase helicity, which, in turn, depends on the RI. Every bit such, it is possible to develop a laser-based sensing scheme using OAM modes such that the modify in the RI tin exist linked to the change in the angular orientation of the beam'southward transverse intensity profile that, in turn, is easily detected past a CCD camera. In short, at a given transverse plane, by measuring the angular orientation of the rotating intensity pattern (petals) in an unknown medium with respect to its orientation in air (equally a reference), the RI of the unknown medium tin then be accurate l y measured.

Results

Theoretical framework

The rotating intensity design Ψ(ρ,ϕ,z,t) is equanimous of multiple OAM modes, where each OAM mode ψ _ℓ is a superposition of equal frequency co-propagating Bessel beams of different transverse and longitudinal wavenumbers. The resulting waveform, Ψ(ρ,ϕ,z,t), is thus given past³⁹

$Ψ (ρ, ϕ, z, t) = \sum_{ℓ = - \infty}^{\infty} ψ_{ℓ} = {eastward}^{- i ω t} \sum_{ℓ = - \infty}^{\infty} \sum_{chiliad = - North}^{N} A_{ℓ, m} J_{ℓ} (k_{ρ}^{ℓ, k} ρ) e^{i ℓ ϕ} {eastward}^{i k_{z}^{ℓ, m} z}$

one

Each OAM mode ψ _ℓ carries a specific topological accuse ℓ and is composed of iiN + 1 Bessel beams of equal order (l). For the m -th Bessel beam in ψ _ℓ , the transverse wavenumber ${one thousand}_{ρ}^{ℓ, 1000}$ is related to the longitudinal wavenumber $m_{z}^{ℓ, m}$ by the consistency relation ${thousand}_{ρ}^{ℓ, m} = \sqrt{1000^{two} - {({chiliad}_{z}^{ℓ, m})}^{ii}}$ . An important property of these OAM modes is that with the superposition of twoN + 1 Bessel beams with different spatial frequencies, the longitudinal intensity profile of the resultant axle can be modulated along the z-management (i.e., along the axle centrality) in a controlled manner^40,41. This is accomplished, in office, by the coefficients A _ℓ,yard in Eq. (1), which correspond different complex weighting factors for each Bessel axle in the superposition, calculated co-ordinate to

$A_{ℓ, chiliad} = \frac{1}{L} \int_{0}^{Fifty} F_{ℓ} (z) {east}^{- i (two π m ∕ Fifty) z} d z$

Function F _ℓ(z) in Eq. (two) is the desired longitudinal (centric) intensity profile. With the proper definition of the topological charges ℓ and the associated office F _ℓ(z), Eq. (1) can be deployed to generate a rotating OAM light construction with a predefined longitudinal extent that is independent of the transverse beam's dimensions. The longitudinal control over the beam's intensity profile is also an important belongings that will be utilized to extend the dynamic range of the proposed sensor, as discussed in section "Extending the Dynamic Range of Sensing." Finally, a summation of two (or more than) OAM modes with opposite signs for the topological accuse ℓ transforms the regular rings associated with the Bessel axle's transverse intensity contour into petal-like shapes whose rotation per RI change tin be detected by a CCD photographic camera³⁹.

The longitudinal wavenumbers of each OAM manner are equally spaced in the 1000-space around a constant parameter Q _ℓ in a comb-like setting. More specifically, $k_{z}^{ℓ, m} = Q_{ℓ} + two π m ∕ Fifty$ , where chiliad ∈ [ -N,N] and L is the distance over which the desired profiles are generated. Fig.1a depicts the spatial frequencies of two OAM modes with opposite helicities, that is, ψ _-1 and ψ _one , where the longitudinal wavenumbers are centered at slightly shifted constants, Q _-one and Q _ane , respectively. Fig.1a also depicts the respective weighting factors A _ℓ,m for each Bessel axle in the superposition of Eq. (i), every bit obtained with Eq. (two). The phase and aamplitude of the coefficients A _ℓ,m are evaluated such that the resulting axle extends for fifty cm (as divers past F(z)). This approach allows flexible control over the beam's range without altering its transverse localization.

An external file that holds a picture, illustration, etc. Object name is 41377_2018_34_Fig1_HTML.jpg

Schematic diagram illustrating the longitudinal wavenumbers of OAM modes ψ _-ane and

ψ ₁ . a Longitudinal wavenumbers in air ( n = ane). b Longitudinal wavenumbers in a medium with unknown refractive index n. Here, each OAM manner consists of nine Bessel beams whose longitudinal wavenumbers are equally spaced in a comb-like setting effectually a constant parameter Q _ℓ . Additionally, F _ℓ(z) = 1 for 0cm ≤z ≤ 50cm and $\tilde{Q} ≃ northward \times Q$ (see Supplementary Materials for derivation)

For this scenario, F _ℓ(z) was fix equal to unity for a distance of 0 cm ≤ z ≤ 50 cm ( ℓ = ±1) with propagation in air. As the effigy shows, the complex coefficients A _ℓ,m course a rummage-like structure in the one thousand-infinite. Analogous with optical frequency combs, in which the spectral range of frequencies is related to the light amplification by stimulated emission of radiation temporal pulse width^43,44, here, the span of the spatial frequency rummage is related to the radial extent of the beam (axle localization). As such, a more radially localized rotating light structure yields a k-space comb that spans a wider range of spatial frequencies.

Furthermore, in optical frequency combs, the spectral components are equally spaced in accordance to the laser repetition rate^43,44, whereas in the instance of our k-space comb, the teeth are equally spaced by a factor of iiπ.

A powerful property of frequency combs is their ability to link the precision of optical frequencies with microwave frequencies⁴⁴, thus providing an accurate and precise spectral ruler that can be interfaced (accessed) with electronic circuitry. Here, by interfering two g-space combs, we provide a tool that tin map the shift in the spatial frequencies of an OAM mode in a given medium, which is also linked to the helicity of the phase-front in that medium, into a rotation in the transverse intensity contour that tin be easily detected by a CCD photographic camera. In general, the orientation of the rotating lite patterns (petals) resulting from the superposition of two OAM modes, ψ _{ℓ _i} and ψ _{ℓ ₂} , is given by³⁹

where ΔQ =Q _{ℓ _i} -Q _{ℓ ₂} and z is the detection plane. Fig.1a shows the example when the beam is propagating in air (n = 1) and where the longitudinal wavenumbers $k_{z}^{ℓ = ane, chiliad}$ and $k_{z}^{ℓ = - 1, m}$ , associated with ψ _ane and ψ _-1 , are centered around Q _ane and Q _-1 , respectively, and are equally spaced past a gene of 2π. Interestingly, when the aforementioned axle is allowed to propagate in a different medium with an unknown RI ( n > 1), the longitudinal wavenumbers ${\tilde{m}}_{z}^{ℓ, m}$ are shifted and are centered effectually larger values, ${\tilde{Q}}_{i}$ and ${\tilde{Q}}_{- i}$ , as depicted in Fig.1b. This is a upshot of the consistency relation ${\tilde{k}}_{z}^{ℓ, m} = \sqrt{k^{2} - {(1000_{ρ}^{ℓ, g})}^{2}}$ , in which $k = ω \frac{due north}{c}$ , whereas the values of $k_{ρ}^{ℓ, m}$ are preserved at the boundary. In a medium with index of refraction n, the wavenumbers are still equally spaced just have a smaller spacing, equal to 2 π ∕n . As Fig.1b shows, for this case, the spacing between the centers of the ii spatial frequency combs becomes $Δ \tilde{Q} = {\tilde{Q}}_{1} - {\tilde{Q}}_{- 1} ≃ Δ Q ∕ n$ (see Supplementary Materials for derivation).

According to Eq. (three), the change in ΔQ , which is a part of RI, modifies the angular orientation of the beam's petals (Φ). Therefore, in any unknown medium, the rotating intensity pattern exhibits a specific athwart orientation Φ that depends on the RI of the medium. Past comparing the angular orientation of the intensity pattern in a medium with an unknown alphabetize of refraction (n), denoted equally $Φ_{ℓ_{1}, ℓ_{2}} (Δ \tilde{Q})$ , with respect to its orientation in air as a reference ( n = 1), denoted equally Φ_{ℓ _i,ℓ ₂}(ΔQ), at the same propagation distance, i tin then authentic fifty y determine the RI of the unknown medium.

Experimental measurements

To examine the performance of the proposed sensing scheme, we generated and tested multiple scenarios in which the rotating structured beams are composed of the superposition of OAM modes ψ ₁ and ψ _-1 , that is, Ψ(ρ,ϕ,z = 0,t) =ψ ₁ +ψ _-1 . The beams were generated by an SLM and function F _ℓ(z) was chosen such that

$F_{ℓ} (z) = \{\begin{matrix} F_{1} = F_{- i} = one, 0 c thousand \leq z \leq 50 c one thousand \\ F_{1} = F_{- 1} = 0, e l s eastward w h e r e \end{matrix}$

Using Eq. (3) and the fact that $Δ \tilde{Q} ≃ Δ Q ∕ n$ , information technology follows that the differential angular orientation of the axle in a given medium with an unknown index of refraction ( n ) with respect to its orientation in air ( n = ane), denoted as θ (for ℓ = 1, - 1), is given past

$θ = Φ_{1, - 1} (Δ Q) - Φ_{one, - one} (Δ \tilde{Q}) = \frac{Δ Q (1 - \frac{1}{north}) z}{ii}$

five

The waveforms were transmitted in air as a reference ( north = one), in improver to water ( northward = one.335), vegetable oil ( n = 1.475), and cinnamon oil ( northward = 1.57), and so detected at the propagation altitude z = 22 cm inside the medium. The selection of these media was made to offer a broad range of RIs. Fig.2 illustrates the measured and calculated variations of θ as a function of the RI of the medium. In each medium, the differential angle θ represents the orientation of the rotating lite petal in that medium with respect to its orientation in air. The measured values of θ are obtained after identifying the centroids of the detected petal-like structures based on locating the local maxima. From Eq. (v), it tin exist observed that the sensitivity of this scheme (∂θ∕∂north ) is direct proportional toΔQ . This is confirmed in Fig.2a–d, which correspond to cases in which ΔQ = 23.62, 47.24, 53.xiv, and 59.02 m⁻ⁱ, respectively. In each case, the unknown index of refraction ( n ) is evaluated with

An external file that holds a picture, illustration, etc. Object name is 41377_2018_34_Fig2_HTML.jpg

Measured and fake transverse beam profiles of the rotating beam petals in air, h2o, vegetable oil, and cinnamon oil at the propagation distance z = 22 cm for different cases of ΔQ .

a ΔQ = 23.62m^-1 , b ΔQ = 47.24m^-i , c ΔQ = 53.14m^-ane , and d ΔQ = 59.02m^-i . The green arrows designate the orientation of the petals in air, and the blueish arrows show the orientation of the petals in the medium. The corresponding theoretical and measured quantities ( θ and northward ) are listed for each example

Fig.3a displays the differential orientation angle θ equally a function of the RI for various values of ΔQ . The markers on the effigy correspond to the measured θ in water (blue), vegetable oil (greenish), and cinnamon oil (cherry-red), where the detection aeroplane is withal at z = 22 cm. The gradient of each bend corresponds to the sensitivity (∂θ∕∂northward ). From the figure, information technology is evident that cases with larger ΔQ possess larger slopes and thus exhibit higher sensitivity, in agreement with Eq. (5). Furthermore, the measured RIs for each ΔQ scenario, as evaluated using Eq. (half dozen), are shown in Fig.3b. Each marker represents an boilerplate of at least v contained measurements. The measured RI values averaged over all ΔQ scenarios in each medium are also listed in the effigy, and the nominal values (from the vendor) are shown in the dashed lines. The averaged measured RIs are: 1.331, ane.476, and 1.570 for water, vegetable oil, and cinnamon oil, respectively. The respective standard deviations in estimating θ are σ _θ = 1.9^∘ , 1.474^∘ , and 0.644^∘ , respectively.

An external file that holds a picture, illustration, etc. Object name is 41377_2018_34_Fig3_HTML.jpg

Performance of the proposed sensing scheme at z = 22 cm.

a Differential orientation (θ) as a function of the RI for different values of at z = 27 cm, ΔQ has the units of m⁻¹. The vertical dash lines correspond to the refractive indices of water (1.335), vegetable oil (ane.475), and cinnamon oil (1.57). b Measured refractive indices as a role of ΔQ . The markers correspond the measured RI values, and the dashed lines represent the nominal values from the vendor. The averaged measured values for the refractive indices are 1.331, ane.476, and ane.570 for water, vegetable oil, and cinnamon oil, respectively. c Resolution of the proposed scheme as a function of ΔQ under different scenarios of the standard difference ( σ _θ ) in estimating θ . d Sensitivity and dynamic range of the proposed scheme equally a function of ΔQ at iii different detection planes: z = 18, 22, and 27 cm

Fig.3c shows the resolution of the proposed sensing mechanism as a function of ΔQ . The resolution of a sensor is the smallest change in the measurand (here, RI) that can be detected. The resolution ( σ _r ) is inversely proportional to the sensor'south sensitivity and direct proportional to the standard deviation of the output variable ( σ _θ ), according to σ _r =σ _θ∕(∂θ∕∂n). For the current setup, the sensor's resolution is on the order of 10^-three RIU, corresponding to a sensitivity of 270°/RIU. As Fig.3c indicates, by further improving the optical setup (e.g., a better camera and laser with less noise) and, hence, reducing σ _θ or by increasing the value of ΔQ , the sensor'southward resolution tin can exist improved further (reaching ≃ 10^-five RIU). We also notation that a larger separation in ΔQ typica l 50 y implies that longitudinal wavenumbers $k_{z}^{ℓ, m}$ are also more widely separated in the spatial frequency domain. Through the consistency relation $g_{ρ}^{ℓ, m} = \sqrt{k^{two} - {({yard}_{z}^{ℓ, grand})}^{2}}$ , a larger separation in ${grand}_{z}^{ℓ, g}$ implies a wider bridge in the spatial frequency ${chiliad}_{ρ}^{ℓ, m}$ and, hence, a more localized beam (if ${thousand}_{ρ}^{ℓ, m}$ acquires large values).

The ability to generate highly localized structured light depends on the SLM's pixel pitch Δ10 , where $k_{ρ}^{ℓ, m}$ and ΔQ are inversely proportional to Δx . Past using commercially available SLMs with a pixel pitch of approximately four μm (the SLM used in our experiment has a pixel pitch of 36 μm), it is possible to achieveΔQ ≈ 600 m^-i . Therefore, by using currently available SLM technology, it is possible to reach resolutions in the social club of 10^-v RIU. This tin can be further improved by an social club of magnitude ( σ _r ≈ x^-six ) when using metasurfaces or phase masks to generate beams with higher values of ΔQ .

From Eq. (5), it is clear that in addition to an increasing ΔQ , the sensitivity of our scheme depends on the altitude z (i.east., increasing z yields higher sensitivity). This is depicted in the green curves of Fig.3d. In principle, the sensitivity of our scheme tin exceed 2600°/RIU at the plane of detection z = 27 cm for ΔQ = 600 m^-1 (see section "Improving the sensor'southward performance by increasing the interaction length"). In this case, college sensitivity is achieved at the expense of reducing the sensor's dynamic range. For the profiles displayed in Fig.ii, the upper bound on the dynamic range is dictated past the value of n associated with θ = 180^∘ , after which the rotating pattern reproduces itself (i.e., becomes degenerate). The dynamic range is evaluated past solving 1∕[one - 2π∕(zΔQ)] and is plotted as blue curves in Fig.3d as a function of ΔQ at three dissimilar detection planes forth z . It is observed that in contrast to the sensitivity, the dynamic range decreases at larger values of ΔQ (and distance z). Therefore, in that location is a merchandise-off betwixt simultaneously achieving high sensitivity (and high resolution) and maintaining a large dynamic range for sensing. In section "Extending the dynamic range of sensing," we testify how higher OAM modes can be exploited to address this problem.

In all the previous cases, nosotros accept presented scenarios in which the rotating beam structures are detected at a stock-still plane z = 22 cm. In the adjacent department, we show the effect that the detection plane z has on the sensitivity and resolution of the proposed scheme.

Improving the sensor's performance by increasing the interaction length

In addition to the dependence on the spatial frequency separation ΔQ , the sensitivity of the proposed sensing scheme depends on the location of the detection plane forth z. In the previous cases, nosotros considered a detection aeroplane that was stock-still at z = 22 cm. Here, we discuss how higher sensitivities tin can be achieved by setting the detection plane at further distances. Fig.4a depicts the sensor response ( θ ) equally a function of RI when the detection airplane is fix at z = 27 cm. The markers represent to the measured θ in water (bluish), vegetable oil (dark-green), and cinnamon oil (red). It is observed that at z = 27 cm, the differential bending θ acquires larger values for the same RI change. Hence, the sensitivity of the scheme tin can be improved past setting the detection plane at a longer distance along z , in agreement with Eq. (5).

An external file that holds a picture, illustration, etc. Object name is 41377_2018_34_Fig4_HTML.jpg

Performance of the proposed sensing scheme at z = 27 cm.

a Differential orientation (θ) as a function of the RI for different values of at z = 27 cm, ΔQ has the units of yard⁻¹.The vertical dash lines represent to the refractive indices of water (one.335), vegetable oil (i.475), and cinnamon oil (1.57). A degeneracy in detection appears at due north = 1.489 for ΔQ = seventy.86m^-ane (red dashed curve). The solid blackness bend depicts the example of z = 22 cm for comparing. b Measured refractive indices as a function of ΔQ . The markers represent the measured RIs and the dashed lines represent the nominal values from the vendor. The averaged measured values are ane.342, i.480, and 1.576 for water, vegetable oil, and cinnamon oil, respectively. c Resolution of the proposed scheme at z = 27 cm as a function of ΔQ nether different scenarios of the standard deviation σ _θ. d Accurateness of the proposed sensor at z = 27 cm as a office of ΔQ under different scenarios of the mean accented error in θ ( ν _θ )

The measured RIs at z = 27 cm, evaluated from Eq. (6), are plotted in Fig.4b for each ΔQ scenario. Each mark represents an boilerplate of at to the lowest degree 5 unlike measurements. The measured RI values, averaged over all ΔQ scenarios in each medium, are listed in the effigy and denoted past ñ , whereas the nominal values (from the vendor) are shown in the dashed lines. The measured RI at z = 27 cm, averaged over all scenarios of ΔQ , are 1.342, 1.480, and one.576 for h2o, vegetable oil, and cinnamon oil. The standard deviations in estimating the angular orientation θ in water, vegetable oil, and cinnamon oil are σ _θ = 1.24^∘ , 1.91^∘ , and one.17^∘ , respectively. Once again, this uncertainty represents the main limiting factor for the resolution of the proposed scheme.

Fig.4c depicts the resolution equally a function of ΔQ for different scenarios of σ _θ. Resolution is calculated as σ _r =σ _θ∕(∂θ∕∂n), from which the minimum detectable change in the index of refraction ( n ) is obtained. With ΔQ = 50 thou^-ane and steps of σ _θ = 2^∘ , the resolution is seven.5 × 10^-3 RIU, which is amend than the example of z = 22 cm; however, with σ _θ = 0.1^∘ , the resolution can reach 3.7 × 10^-4 . Furthermore, the resolution can be improved by an society of magnitude by generating more localized beams in which ΔQ is ten times larger, every bit shown in Fig.4c. For example, the proposed scheme tin can be utilized to measure RI with a resolution on the order of 10^-v , when ΔQ is set up to 600 yard^-1 , which is feasible when using SLMs with a 4 μm pixel pitch.

Accurateness is another of import metric in RI sensing. In our proposed scheme, accurateness is determined past the reliable detection of the centroids' maxima in the rotating petal-similar structures. Hence, the error in determining θ represents the primary limiting cistron for the accuracy of the proposed scheme. Past taking the derivative of Eq. (6) with respect to θ , the accurateness tin be expressed every bit

where ν _θ denotes the mean absolute error in θ . In our experiments, ν _θ was ~ii° at z = 27 cm. Fig.4d depicts the sensor'southward accurateness every bit a office of ΔQ under unlike scenarios of ν _θ . Similar to the resolution and sensitivity, the sensor'due south accuracy can be dramatically improved by using beams with larger values of ΔQ . We also note that because beams with larger values of ΔQ are more localized, they lead to lower values of ν _θ . Equation (vii) as well characterizes the sensor'southward precision (i.east., the repeatability of RI measurements over time). This is readily performed by replacing ν _θ with the standard deviation ( σ _θ ). A more than detailed analysis on the sensor's tolerance to the deviations in θ , z , and ΔQ tin can exist plant in the Supplementary Materials.

Extending the dynamic range of sensing

In the previous department, nosotros showed that the sensitivity and resolution can be improved past extending the length over which the beam interacts with the medium. This improvement is achieved at the expense of limiting the dynamic range of the sensing scheme. For example, consider Fig.5a, which depicts the instance when the rotating light blueprint, with ΔQ = 70.86m^-1 , is detected at z = 27 cm. It is observed that for vegetable oil, θ =175°. This implies that the rotating light pattern is very close to acquiring a degenerate orientation compared to a axle that propagates in air. Furthermore, in the case of cinnamon oil, θ exceeds 180°; hence, mapping the orientation to the index of refraction is no longer unique. In other words, a measured value of θ = 203^∘ is degenerate with θ = (203^∘ - 180^∘) = 23^∘ . Such degeneracy can as well be verified from Fig.4a, as marked by the dashed scarlet curve. It is thus not clear, in this instance, if the RI value should exist mapped to 1.59 or 1.044 (corresponding to the measured orientations θ = 203^∘ and θ = 23^∘ , respectively). As previously mentioned, the dynamic range is constrained when θ reaches 180^∘ , where it tin readily be calculated from 1∕[i - 2π∕(z ΔQ)]. This suggests that at ΔQ = lxx.86 grand^-1 and z = 27 cm, the dynamic range of this sensing scheme only spans the range from n = 1 to northward~1.489.

An external file that holds a picture, illustration, etc. Object name is 41377_2018_34_Fig5_HTML.jpg

Using higher gild OAM modes to extend the dynamic range of sensing.

a Measured and imitation transverse beam profiles in h2o, vegetable oil, and cinnamon oil at z = 27 cm for ΔQ = 70.86m^-1 . b Schematic diagram showing the evolution of the generated beam from three to two petals with propagation in the different media. c Measured and false athwart orientation of the rotating pattern at propagation distance z = 27 cm. The rotating beam is designed to evolve into three petals (instead of ii, as in a) when the intensity profile is degenerate with the case of air, thus extending the dynamic range of the sensing scheme

Consequently, there is a articulate trade-off between achieving high sensitivity and maintaining a wide dynamic range for RI measurement. Notwithstanding, past incorporating larger topological charges ( ℓ > 1) in the superposition of Eq. (1), it becomes possible to mitigate this merchandise-off. OAM modes with larger ℓ accept ℓ -helices in their phase-front. When these OAM modes—with contrary signs of ℓ —are superimposed, the rotating pattern will possess more than 2 rotating petals equally a result of introducing boosted phase singularities in the azimuthal direction ( ϕ )³⁹. In this case, it is possible to generate rotating beams that can change their number of petals (evolving from ii to three petals, for instance) as the optical length (index of refraction) is increased. This, in plough, can exist utilized to break the degeneracy of the ii-petal patterns and extend the dynamic range of the sensor while maintaining loftier sensitivity and resolution.

For case, consider the waveform Ψ(ρ,ϕ,z = 0,t) =ψ _-1 +ψ _i +ψ ₂ , in which the part F _ℓ(z) is defined equally

$F_{ℓ} (z) = \{\begin{matrix} F_{- 1} = 1, 0 c thou \leq z \leq 50 c m \\ F_{2} = 1.5, 0 c chiliad \leq z \leq 18 c yard \\ F_{1} = 1, xviii c m < z \leq 50 c m \\ F_{- 1} = F_{1} = F_{2} = 0, due east fifty s due east w h e r east \end{matrix}$

eight

Here, the beam is composed of OAM modes with ℓ = 1, −1, and 2. As such, the propagating beam (in air) is designed to possess three petals over the range 0cm ≤z ≤ 18cm before it reduces to ii petals over the range 18cm <z ≤ 50cm . Note that this is the behavior of the beam when propagating in air. The ability to control the topological accuse of structured low-cal along the beam centrality is depicted in Fig.5b, which illustrates the evolution of the axle from three petals to two petals with propagation. This is made possible by the careful effective and destructive interference among the nine co-propagating Bessel beams in the superposition. With Eq. (8), only OAM modes with non-zero intensity will contribute to the axle center, while the contributions of the other OAM modes are distributed over the outer rings of the beam³⁹. Those contributions that are stored in the outer rings can and then exist restored to the beam'due south center at the prescribed locations defined by Eq. (viii). In principle, one tin discern this event as a practical manifestation of the Hilbert's hotel paradox concept^45,46, in which the spatial redistribution of the local topological accuse varies with propagation while the total charge is globally conserved. As the beam is immune to propagate in different media, the altitude over which the beam carries two or 3 petals changes, depending on the RI (from the relation $Δ \tilde{Q} = Δ Q ∕ n$ and Eq. (3)). Figure5c represents the measured and simulated transverse intensity profiles of the rotating light structure (with ΔQ = 70.86m^-1 ) in air, water, vegetable oil, and cinnamon oil at a propagation altitude of z = 27 cm. Here, once the sensing scheme approaches the limit of its dynamic range (i.east., θ approaching 180°), the same beam evolves into a new intensity profile with three petals instead of two. We note that this evolution occurs without changing the initial hologram on the SLM. In this case, the dynamic range is no longer constrained by the condition θ = 180^∘ . In other words, there is a one-to-ane mapping between the beam orientation and index of refraction of the medium. Accordingly, the dynamic range is now extended from RI = i to ii.91, as opposed to the previous case (department "Improving the sensor's operation by increasing the interaction length"), where the span of dynamic range is from RI = 1 to 1.489. Interestingly, the dynamic range of the beam can be extended even further by incorporating higher OAM modes in the superposition of Eq. (1).

Word

Nosotros proposed and experimentally demonstrated a novel tunable RI sensing scheme based on the OAM of lite. By calculation OAM modes with reverse helicities, nosotros created petal-like light structures that rotate along their optical length. The angular orientation of the rotating petal-like construction depends on the RI of the medium. The rotation in the transverse intensity contour can be measured easily with respect to a reference (air) using a CCD camera, and the differential measurement tin can and then exist utilized to accurately identify the RI change. Hence, the proposed sensing scheme is based on a uncomplicated setup that but requires an SLM for beam generation and a CCD camera for detection. The sensitivity is but limited by the available SLM bandwidth and can in principle exceed 2700°/RIU with a resolution of 10^-five RIU using SLMs with a 4 μm pixel pitch, which are widely available commercially. We too proposed a novel mechanism to expand the dynamic range of sensing by incorporating higher OAM modes in the transmitted beam, thus scanning the range from RI = 1 to over two.91. The programmability of the SLM allows the sensitivity, resolution, and dynamic range of the sensor to be reconfigured on demand, thus providing a tunable sensing mechanism that can provide coarse and fine RI measurements in real time.

Future considerations and outlook

In this work, we showed how structured light with OAM can be utilized to measure the real part of the RI. Hereafter considerations include the following: First, the proposed scheme can be deployed to gauge the imaginary part of the RI associated with propagation losses. This can exist performed by detecting the intensity level of the measured images. By quantifying the corporeality of attenuation in the intensity blueprint with respect to the reference medium (air) at a given detection altitude z , the imaginary role of the RI can then exist estimated past applying Beer's law⁴⁷. Since the adult waveform allows control over the longitudinal intensity profile, it is possible to generate rotating light structures that are immune to the propagation losses effects^48,49. 2nd, the sensitivity and resolution can be dramatically enhanced by deploying OAM modes with faster rotation rates. This is readily accomplished in various ways, such as using accelerated OAM modes⁵⁰ or generating rotating beams with larger values of ΔQ , as explained earlier. Another arroyo is to replace the 2nd lens in the iv-f imaging organisation of Fig.half-dozen past a lens with much shorter focal length. This scheme compresses the beam'due south longitudinal extent and thus acts equally a photonic gear that amplifies the axle's rotation charge per unit forth the z-direction and, hence, boosts the sensitivity and resolution of the entire scheme. Third, similar to OAM, polarization can be exploited to extend the dynamic range of sensing⁵¹. Finally, it is likewise interesting to extend the current sensor to characterize the RI of non-homogeneous media. This will be the subject field of futurity work.

An external file that holds a picture, illustration, etc. Object name is 41377_2018_34_Fig6_HTML.jpg

Experimental setup used to generate and detect the structured light used for sensing.

A computer-generated hologram (CGH) was sent to a transmissive SLM that encodes the desired blueprint on a 532 nm collimated laser beam. The SLM was sandwiched in a polarizer–analyzer configuration as it operates with maximum efficiency on the vertically polarized incident low-cal. The generated pattern was then filtered and imaged using a 4-f imaging organization. Finally, the beam evolution was recorded inside the fluid using a CCD camera on a translation stage where the z = 0 aeroplane lay to the correct of Lens 2 (at its focal plane)

Materials and methods

To exam the proposed sensing mechanism, we performed the post-obit experimental procedure: first, the Bessel beams superposition in Eq. (1) was computed and transformed into second transmission CGHs. The holograms were designed for the case of air ( n = 1) and were sent to a transmissive SLM (Holoeye LC2012). The manual part at the SLM plane is given past

H(x,y) = ½{β(x,y) +α(x,y)cos[Θ(x,y) - 2π(u ₀ x +v ₀ y)]}

Hither, α(x,y) and Θ(ten,y) represent the amplitude and phase of Ψ(ρ,ϕ,z = 0,t), respectively, and β(x,y) is a bias function chosen as a soft envelope for the amplitude α(x,y) according to β(x,y) = [1 +α ²(10,y)]∕ii ⁵². The design is as well interfered with a aeroplane wave exp[2π i(u ₀ x +v ₀ y)]. This superposition shifts the encoded pattern off-axis (in the Fourier plane) to the spatial frequencies ( u ₀,v ₀ ), thus making it easier to filter out the shifted pattern from the undesired on-axis noise with an iris. In our experiment, u ₀ and v ₀ were set to 1∕(4Δx), where Δx is the SLM pixel pitch (Δx = 36 μm in our case).

Given the twisted nematic nature of our SLM, which makes information technology operate with maximum efficiency on linearly polarized light, we used a polarizer–analyzer combination oriented at (0^∘ ) and (xc^∘ ) with respect to the SLM axis, as depicted in Fig.vi. The CGH was used to encode the desired blueprint on an expanded and collimated green laser beam (532 nm). The generated waveform was then imaged and filtered using a iv-f optical system and an iris, which blocked the undesired diffraction orders and on-axis noise. Then, the generated waveform was transmitted into a glass tank at normal incidence. The tank was placed in the focal plane of the four-f imaging organisation ( z = 0 plane), as shown in Fig.half-dozen. Finally, the generated waveform was recorded within the fluid using a CCD camera at a stock-still detection airplane along z. To establish the broad dynamic range capability of our sensing machinery, we considered the following fluids: (a) deionized h2o ( n = 1.335), (b) vegetable oil ( n = ane.475), and (c) cinnamon oil ( n = 1.57).

Electronic supplementary material

Acknowledgements

We would like to acknowledge the support from the Natural Sciences and Engineering Research Council in Canada, the Ontario Graduate Scholarship, FAPESP (Grant No. 2015/26444-viii), and CNPq (Grant No. 304718/2016-five). The authors are also thankful for Prof. Stewart Aitchison and Prof. John Sipe for their stimulating discussions.

Notes

Conflict of interest

The authors declare that they have no conflict of interest.

Footnotes

Commodity accustomed preview

Accepted article preview online: 18 May 2018

Publisher'southward note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Supplementary information is available for this paper at 10.1038/s41377-018-0034-9.

References

i. Fan XD, et al. Sensitive optical biosensors for unlabeled targets: a review. Anal. Chim. Acta. 2008;620:viii–26. doi: 10.1016/j.aca.2008.05.022. [PubMed] [CrossRef] [Google Scholar]

2. Moreels Eastward, de Greef C, Finsy R. Light amplification by stimulated emission of radiation low-cal refractometer. Appl. Opt. 1984;23:3010–3013. doi: 10.1364/AO.23.003010. [PubMed] [CrossRef] [Google Scholar]

3. Nemoto S. Measurement of the refractive index of liquid using laser axle displacement. Appl. Opt. 1992;31:6690–6694. doi: 10.1364/AO.31.006690. [PubMed] [CrossRef] [Google Scholar]

4. Makdisi Y, Zaidi AA, Bhatia KS. Laser refractometry of liquids with a diffraction grating. Opt. Commun. 1989;72:148–152. doi: x.1016/0030-4018(89)90385-iv. [CrossRef] [Google Scholar]

5. Daimon Thousand, Masumura A. Measurement of the refractive index of distilled water from the almost-infrared region to the ultraviolet region. Appl. Opt. 2007;46:3811–3820. doi: 10.1364/AO.46.003811. [PubMed] [CrossRef] [Google Scholar]

6. Zhang J, Wei D, Xie CD, Peng KC. Characteristics of assimilation and dispersion for rubidium D_two lines with the modulation transfer spectrum. Opt. Express. 2003;xi:1338–1344. doi: 10.1364/OE.11.001338. [PubMed] [CrossRef] [Google Scholar]

seven. Ottonelli Due south, et al. A compact three degrees-of-freedom motion sensor based on the light amplification by stimulated emission of radiation-self-mixing consequence. IEEE Photon Technol. Lett. 2008;20:1360–1362. doi: x.1109/LPT.2008.926569. [CrossRef] [Google Scholar]

8. Gagliardi Chiliad, Salza Grand, Avino Due south, Ferraro P, De Natale P. Probing the ultimate limit of fiber-optic strain sensing. Scientific discipline. 2010;330:1081–1084. doi: 10.1126/science.1195818. [PubMed] [CrossRef] [Google Scholar]

9. Cvijetic N, Milione G, Ip E, Wang T. Detecting lateral motion using light's orbital angular momentum. Sci. Rep. 2015;5:15422. doi: 10.1038/srep15422. [PMC gratuitous article] [PubMed] [CrossRef] [Google Scholar]

10. Lavery MPJ, Speirits FC, Barnett SM, Padgett MJ. Detection of a spinning object using light's orbital angular momentum. Science. 2013;341:537–540. doi: 10.1126/science.1239936. [PubMed] [CrossRef] [Google Scholar]

11. Brullot W, Vanbel MK, Swusten T, Verbiest T. Resolving enantiomers using the optical angular momentum of twisted light. Sci. Adv. 2016;2:e1501349. doi: 10.1126/sciadv.1501349. [PMC costless article] [PubMed] [CrossRef] [Google Scholar]

12. Zhang WW, Serna S, Le Roux Ten, Vivien L, Cassan East. Highly sensitive refractive index sensing past fast detuning the critical coupling condition of slot waveguide ring resonators. Opt. Lett. 2016;41:532–535. doi: 10.1364/OL.41.000532. [PubMed] [CrossRef] [Google Scholar]

xiii. Tsigaridas GN. A study on refractive index sensors based on optical micro-ring resonators. Photon Sens. 2017;7:217–225. doi: x.1007/s13320-017-0418-0. [CrossRef] [Google Scholar]

14. Xu ZL, Luo YY, Liu DM, Shum PP, Sun QZ. Sensitivity-controllable refractive index sensor based on cogitating θ-shaped microfiber resonator cooperated with Vernier outcome. Sci. Rep. 2017;7:9620. doi: 10.1038/s41598-017-10163-x. [PMC free commodity] [PubMed] [CrossRef] [Google Scholar]

xv. Xu F, Horak P, Brambilla G. Optical microfiber ringlet resonator refractometric sensor. Opt. Express. 2007;15:7888–7893. doi: x.1364/OE.fifteen.007888. [PubMed] [CrossRef] [Google Scholar]

16. Guo X, Tong LM. Supported microfiber loops for optical sensing. Opt. Express. 2008;16:14429–14434. doi: 10.1364/OE.16.014429. [PubMed] [CrossRef] [Google Scholar]

17. Wo JH, et al. Refractive index sensor using microfiber-based Mach–Zehnder interferometer. Opt. Lett. 2012;37:67–69. doi: 10.1364/OL.37.000067. [PubMed] [CrossRef] [Google Scholar]

xviii. Sun X, Dai DX, Thylén L, Wosinski L. High-sensitivity liquid refractive-index sensor based on a Mach–Zehnder interferometer with a double-slot hybrid plasmonic waveguide. Opt. Express. 2015;23:25688–25699. doi: ten.1364/OE.23.025688. [PubMed] [CrossRef] [Google Scholar]

nineteen. Monzón-Hernández D, Villatoro J. Loftier-resolution refractive alphabetize sensing by means of a multiple-peak surface plasmon resonance optical fiber sensor. Sens Actuators B. 2006;115:227–231. doi: 10.1016/j.snb.2005.09.006. [CrossRef] [Google Scholar]

xx. Shalabney A, Abdulhalim I. Sensitivity-enhancement methods for surface plasmon sensors. Laser Photon Rev. 2011;5:571–606. doi: ten.1002/lpor.201000009. [CrossRef] [Google Scholar]

21. Zhang JJ, et al. Microfiber Fabry–Perot interferometer made by taper-drawing technique and its application as a radio frequency interrogated refractive alphabetize sensor. Opt. Lett. 2012;37:2925–2927. doi: 10.1364/OL.37.002925. [PubMed] [CrossRef] [Google Scholar]

22. Zhang JJ, et al. Microfiber Fabry–Perot interferometer for dual-parameter sensing. J. Light Technol. 2013;31:1608–1615. doi: 10.1109/JLT.2013.2252145. [CrossRef] [Google Scholar]

23. Kim HJ, Kwon OJ, Lee SB, Han YG. Polarization-dependent refractometer for discrimination of temperature and ambient refractive index. Opt. Lett. 2012;37:1802–1804. doi: 10.1364/OL.37.001802. [PubMed] [CrossRef] [Google Scholar]

24. Bahrami F, Maisonneuve K, Meunier M, Aitchison JS, Mojahedi M. An improved refractive alphabetize sensor based on genetic optimization of plasmon waveguide resonance. Opt. Express. 2013;21:20863–20872. doi: x.1364/OE.21.020863. [PubMed] [CrossRef] [Google Scholar]

25. Mandel L, Wolf East. Optical Coherence and Breakthrough Optics. Cambridge: Cambridge University Press; 1995. [Google Scholar]

26. Yao AM, Padgett MJ. Orbital angular momentum: origins, beliefs and applications. Adv. Opt. Photon. 2011;3:161–204. doi: 10.1364/AOP.3.000161. [CrossRef] [Google Scholar]

27. Allen L, Beijersbergen MW, Spreeuw RJC, Woerdman JP. Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes. Phys. Rev. A. 1992;45:8185–8189. doi: 10.1103/PhysRevA.45.8185. [PubMed] [CrossRef] [Google Scholar]

28. Molina-Terriza Chiliad, Torres JP, Torner Fifty. Twisted photons. Nat. Phys. 2007;3:305–310. doi: 10.1038/nphys607. [CrossRef] [Google Scholar]

29. Padgett M, Bowman R. Tweezers with a twist. Nat. Photonics. 2011;v:343–348. doi: 10.1038/nphoton.2011.81. [CrossRef] [Google Scholar]

30. Duocastella Thou, Arnold CB. Bessel and annular beams for materials processing. Laser Photon Rev. 2012;6:607–621. doi: x.1002/lpor.201100031. [CrossRef] [Google Scholar]

31. Bozinovic Northward, et al. Terabit-scale orbital angular momentum manner division multiplexing in fibers. Science. 2013;340:1545–1548. doi: x.1126/science.1237861. [PubMed] [CrossRef] [Google Scholar]

32. Willner AE, et al. Optical communications using orbital angular momentum beams. Adv. Opt. Photon. 2015;vii:66–106. doi: 10.1364/AOP.7.000066. [CrossRef] [Google Scholar]

33. Chávez-Cerda Southward, McDonald GS, New GHC. Nondiffracting beams: travelling, standing, rotating and screw waves. Opt. Commun. 1996;123:225–233. doi: 10.1016/0030-4018(95)00538-two. [CrossRef] [Google Scholar]

34. Tervo J, Turunen JP. Rotating scale-invariant electromagnetic fields. Opt. Express. 2001;ix:ix–fifteen. doi: ten.1364/OE.9.000009. [PubMed] [CrossRef] [Google Scholar]

35. Abramochkin E, Losevsky N, Volostnikov 5. Generation of screw-type laser beams. Opt. Commun. 1997;141:59–64. doi: 10.1016/S0030-4018(97)00215-0. [CrossRef] [Google Scholar]

36. Pääkkönen P, et al. Rotating optical fields: experimental demonstration with diffractive optics. J. Mod. Opt. 1998;45:2355–2369. doi: 10.1080/09500349808231245. [CrossRef] [Google Scholar]

37. Schechner YY, Piestun R, Shamir J. Wave propagation with rotating intensity distributions. Phys. Rev. E. 1996;54:R50–R53. doi: x.1103/PhysRevE.54.R50. [PubMed] [CrossRef] [Google Scholar]

38. Daria VR, Palima DZ, Glückstad J. Optical twists in stage and amplitude. Opt. Limited. 2011;19:476–481. doi: x.1364/OE.nineteen.000476. [PubMed] [CrossRef] [Google Scholar]

39. Dorrah AH, Zamboni-Rached 1000, Mojahedi M. Controlling the topological charge of twisted low-cal beams with propagation. Phys. Rev. A. 2016;93:063864. doi: ten.1103/PhysRevA.93.063864. [CrossRef] [Google Scholar]

40. Zamboni-Rached M. Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: frozen waves. Opt. Express. 2004;12:4001–4006. doi: 10.1364/OPEX.12.004001. [PubMed] [CrossRef] [Google Scholar]

41. Zamboni-Rached M, Recami E, Hernández-Figueroa HE. Theory of "frozen waves": modeling the shape of stationary wave fields. J. Opt. Soc. Am. A. Opt. Paradigm Sci. Vis. 2005;22:2465–2475. doi: 10.1364/JOSAA.22.002465. [PubMed] [CrossRef] [Google Scholar]

42. McGloin D, Dholakia K. Bessel beams: diffraction in a new light. Contemp. Phys. 2005;46:15–28. doi: 10.1080/0010751042000275259. [CrossRef] [Google Scholar]

43. Holzwarth R, et al. Optical frequency synthesizer for precision spectroscopy. Phys. Rev. Lett. 2000;85:2264–2267. doi: x.1103/PhysRevLett.85.2264. [PubMed] [CrossRef] [Google Scholar]

44. Diddams SA, et al. Direct link between microwave and optical frequencies with a 300 THz femtosecond light amplification by stimulated emission of radiation comb. Phys. Rev. Lett. 2000;84:5102–5105. doi: ten.1103/PhysRevLett.84.5102. [PubMed] [CrossRef] [Google Scholar]

45. Gbur G. Fractional vortex Hilbert's hotel. Optica. 2016;3:222–225. doi: 10.1364/OPTICA.3.000222. [CrossRef] [Google Scholar]

46. Potoček V, et al. Breakthrough Hilbert hotel. Phys. Rev. Lett. 2015;115:160505. doi: 10.1103/PhysRevLett.115.160505. [PubMed] [CrossRef] [Google Scholar]

47. Swinehart DF. The Beer–Lambert police force. J. Chem. Educ. 1962;39:333. doi: 10.1021/ed039p333. [CrossRef] [Google Scholar]

48. Zamboni-Rached Chiliad. Diffraction-attenuation resistant beams in absorbing media. Opt. Express. 2006;14:1804–1809. doi: x.1364/OE.14.001804. [PubMed] [CrossRef] [Google Scholar]

49. Dorrah AH, Zamboni-Rached M, Mojahedi M. Generating attenuation-resistant frozen waves in absorbing fluid. Opt. Lett. 2016;41:3702–3705. doi: x.1364/OL.41.003702. [PubMed] [CrossRef] [Google Scholar]

50. Schulze C, et al. Accelerated rotation with orbital angular momentum modes. Phys. Rev. A. 2015;91:043821. doi: x.1103/PhysRevA.91.043821. [CrossRef] [Google Scholar]

51. Corato-Zanarella M, Dorrah AH, Zamboni-Rached M, Mojahedi M. Arbitrary control of polarization and intensity profiles of diffraction-attenuation-resistant beams along the propagation management. Phys. Rev. Appl. 2018;ix:024013. doi: ten.1103/PhysRevApplied.9.024013. [CrossRef] [Google Scholar]

52. Arrizón Five, Méndez Grand, Sánchez-de-La-Llave D. Authentic encoding of arbitrary circuitous fields with aamplitude-but liquid crystal spatial light modulators. Opt. Express. 2005;13:7913–7927. doi: x.1364/OPEX.13.007913. [PubMed] [CrossRef] [Google Scholar]

Manufactures from Light, Scientific discipline & Applications are provided here courtesy of Nature Publishing Group

guerraabroves.blogspot.com

Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6107032/

Guerra Abroves

A H Dorrah M Zamboni-rached and M Mojahedi Physical Review a 93 063864 (2016)

Experimental demonstration of tunable refractometer based on orbital angular momentum of longitudinally structured calorie-free

Ahmed H. Dorrah

Michel Zamboni-Rached

Mo Mojahedi

Abstract

Introduction

Background

Concept

Results

Theoretical framework

Experimental measurements

Improving the sensor's performance by increasing the interaction length

Extending the dynamic range of sensing

Word

Future considerations and outlook

Materials and methods

Electronic supplementary material

Acknowledgements

Notes

Conflict of interest

Footnotes

Electronic supplementary material

References

Postar um comentário for "A H Dorrah M Zamboni-rached and M Mojahedi Physical Review a 93 063864 (2016)"