WO2022249127A1

WO2022249127A1 - Photonic system for detecting transverse displacements

Info

Publication number: WO2022249127A1
Application number: PCT/IB2022/054969
Authority: WO
Inventors: Vincenzo D'ambrosio; Raouf BARBOZA; Filippo CARDANO; Lorenzo Marrucci; Corrado DE LISIO
Original assignee: Universita' Degli Studi Di Napoli Federico Ii
Priority date: 2021-05-28
Filing date: 2022-05-26
Publication date: 2022-12-01
Also published as: IT202100013949A1

Abstract

A photonic system for detecting a relative displacement between two objects (P, TS), comprising a source assembly (S, HWP1, Pol1) configured to generate a radiation beam (R) having a linear polarization status, a first and a second Pancharatnam-Berry phase grid (GP1, GP2) associable with a first and a second object (P, TS), respectively, movable with respect to each other along a transverse direction (x), in which first and second Pancharatnam-Berry phase grids (GP1, GP2) are configured to rotate the polarization status of the radiation beam (R) by a given rotation angle Δθ, depending on a relative displacement Δx between the first and second objects from a reference position, and a detector assembly (HWP2, Pol2, L, I, PM) is configured to detect the radiation beam (R) and provide a signal indicative of the angle of rotation Δθ.

Description

Photonic system for detecting transverse displacements

The present invention relates generally to photonic techniques for detecting displacements, even of very small magnitude.

The measurement and tracking of the position of a system are of fundamental importance in different fields such as microscopy, mechanical engineering, quantum physics, materials science, semiconductor or general relativity industries. For this purpose, light has proven to be a valuable tool, as it allows rapid, non-invasive and accurate measurements. In photonic systems, detectable displacements may be parallel or transverse to the main propagation direction of the optical beam. While triangulation measurements or interferometric archi tectures may be used in the former case, the measurement of transverse displacements (TD) typically relies on the detection of differential current signals emitted by photodiodes. This provides a practical but limited solution in terms of sensitivity and resolution. Other tech niques allow for better performance, exploiting, for example, grid interferometry, diffrac tion-based overlay or imaging with fluorophores. While such methods allow sub-nanometer resolutions to be achieved, they are often based on complex equipment and therefore are limited in terms of versatility.

Techniques based on structured light, that is, an optical field that has a spatially variable distribution of amplitude, phase and/or polarization, have recently emerged as a resource in this area. Taking advantage of structured illumination, TDs may be measured by, for exam ple, directional scattering from a nanoantenna or detection of the position of the centroid of the distribution of the scattering field. Moreover, by properly structuring the phase profile of a light beam via a metasurface, an optical ruler exploiting super-oscillations achieves a resolution far below the diffraction limit. Although they allow sub-nanometric resolutions to be achieved, such methods rely on high-magnification imaging systems, with coupling of the wavelength of light to specific resonances of a nanoantenna, or post-production of images by reconstruction algorithms. These factors impose limits in terms of versatility, cost, and speed, which are all desirable features in an ideal sensor, in addition to sensitivity.

An object of the present invention is to provide a photonic technique that enables the meas- urement of transverse displacements with sub-nanometric resolutions while using relatively simple instrumental architectures.

To achieve this object, the subject matter of the invention is a photonic system for detecting a relative displacement between two objects, comprising: a source assembly configured to generate a radiation beam having an initial linear polarization status and being able to propagate along a propagation axis, a first and a second Pancharatnam-Berry phase grid arranged one after the other along the optical axis, wherein: each of said first and second Pancharatnam-Berry phase grids comprises a birefrin- gent medium acting on the polarization of the light as a delay plate of which the fast and slow optical axes are oriented in a periodic way along a direction lying in the plane orthog onal to the propagation axis of the light, said first and second Pancharatnam-Berry phase grids are respectively as sociable to a first object and a second object movable relative to each other along said transverse direc tion, said first and second Pancharatnam-Berry phase grids are configured to rotate the polarization status of the radiation beam by a given rotation angle AQ, as a function of a relative displacement Ax between said first and second object from a reference position, and a detecting assembly arranged along the propagation axis, said detecting assembly comprising a polarizer configured to project the rotated polarization status of the radiation beam along said initial polarization status of the radiation beam, and an optical power meter configured to provide an optical power signal of the radiation beam, the value of which is indicative of said rotation angle AQ.

The invention stems from the realization that structured light may be a resource for ultra sensitive detections even without high magnifications by resorting to so-called “photonic gears,” in which a mapping between the polarization status and a suitably adapted vector mode of a paraxial light beam allows an increase in sensitivity in measuring an angle of rotation [1]

By combining such a principle with a Moire-type detection scheme [2-4], the inventors have achieved a new optical transduction method that enables increased sensitivity in TD meas urements with a compact, fast, stable, and inexpensive architecture. The concept is based on mapping the displacement in an optical polarization rotation by a pair of Pancharatnam- Berry phase grids. The transverse displacement may thus be immediately measured simply by monitoring the optical power after a linear polarizer. The sensitivity of the method is controlled by the polarization rotation rate per unit of length in the direction of the TD. The inventors have experimentally tested this principle with a minimal setup, under ambient conditions and temperature, and found TD measurements with sub-nanometric resolution. It is expected that under more controlled, but still realistic, conditions, the resolution may be reduced to a scale of tenths of a picometer.

In the following description, reference will be made for simplicity to an example in which the birefringent medium is a liquid-crystalline material having a periodically defined mo lecular director arrangement along a transverse direction (e.g., the direction x). The inven tion is not limited to such materials, however, and comprises all birefringent media designed to act as a delay plate having the slow and fast optical axes oriented periodically along a direction lying in the plane orthogonal to the axis of propagation of the light. For example, the invention comprises, as a birefringent medium, also metamaterials configured to act on the polarization of the light as a delay plate having the slow and fast optical axes oriented periodically along the transverse direction.

The features and advantages of the system according to the invention will become clearer from the following detailed description, made in reference to the accompanying drawings, provided purely for illustrative and non-limiting purposes, wherein:

- Fig. 1 is a diagram representing the concept behind the invention. The dashed circles rep resent the polarization status of a laser beam at three different positions along the laser beam, corresponding to | H), |A), and |0), respectively. Box A shows the geometry of the polari zation grids; the arrows represent the orientation of the fast (or slow) axis of the birefringent medium. The spatial period is determined by the parameter L. Box B shows the microscopic image of a polarization grid placed between two linear polarizers. The graphic scale corre sponds to 50 pm. Box C shows the super-resolving Malus’ law for the pair of Pancharatnam- Berry phase grids; - Fig. 2 reports experimental results: a) For each value of A, the normalized measured power of the horizontal polarization component is reported as a function of the displacement be tween the two polarization grids. The zero displacement point for each curve has been ad justed to allow visual comparison between different pairs of grids b) The same measure ment is repeated in the linear part of the calibration curves, with a displacement step of 20 nm. For each configuration, the input optical power has been set at a reference value P₀ = 1 mW. In both boxes, the points represent experimental data, the solid lines are best- fit curves according to the equations given in the description, and the experimental data correspond to the average power over 20 independent measurements (the error bars are smaller than the size of the data point). Different colors are associated with respective spatial periods L (expressed in pm in the legend);

- Fig. 3 shows measurements for L = 50 pm. Each graph reports the measured optical power for 1 second (250 points) before and after a controlled step Dc of the translation stage. The solid lines represent the average power calculated over time intervals of 0.1 s, while the dotted lines indicate the total average power calculated before and after each step. Also shown on the right side of each graph are the power difference DR and step amplitude;

- Fig. 4 shows a possible experimental apparatus;

- Fig. 5 is a diagram showing a possible development of the invention.

Referring to Fig. 1, a conceptual diagram of a photonic system according to the invention, configured to detect a relative displacement between two objects, is shown. Specifically, this relative displacement is a transverse displacement with respect to an axis z, defined by the propagation direction of a laser beam R. In the figure, the two objects between which the relative displacement is to be measured are a first object denoted P and a second object denoted TS. In the example shown, the transverse displacement to be measured is a hori zontal displacement along an axis x, and therefore denoted by Dc.

A first and a second polarization grid arranged one after the other along the optical axis z and denoted GP1 and GP2 in the figures are respectively made integral to the two objects P and TS.

The laser beam R generated by a source assembly (not shown in Fig. 1) is a collimated beam having a polarization status | H), that is, polarized uniformly along the horizontal direction. This specific orientation of linear polarization was chosen for simplicity of discussion, and is not necessary for the purposes of the invention.

The laser beam R is passed through the first polarization grid GP1, which is integral with the first object P. The first polarization grid GP1, or Pancharatnam-Berry phase grid [5], is a patterned liquid-crystal plate, in which the orientation of the molecular director of the liquid-crystal material is defined periodically along a direction in a plane orthogonal to the reference axis z, in the example, the direction x. Specifically, the arrangement of the mo lecular director of the material of the polarization grid is defined by the relation a(x,y) = ^x + a_Q, (1) where cr(x), defined as the modulus 7G, is the angle formed by the molecular director of the material of the polarization grid with the transverse direction (x), a₀ is the residual angle at x = 0 and A is the spatial periodicity of the angle (see Fig. 1A). In the more general case of a birefringent medium, the relation (1) defines the orientation of the fast (or slow) axis of the birefringent medium of the polarization grid.

At the output of the first polarization grid GP1, the optical polarization assumes the follow ing expression:

I H) ® |A) = cos[2a(x)] \H ) + sin[2cr(x)] |F) (2)

The polarization status \V) represents a status of vertical polarization, along the axis y. The status |A) thus represents a structured beam of light where the polarization direction varies linearly along the axis x with a period that corresponds to half the period of the first polari zation grid GP1.

The beam then passes through the second GP2 polarization grid, identical to the first but displaced by the amount Dc relative to a reference position. Thus, the output field from the second GP2 polarization grid is:

I A) ® \q) = cos(A0)| H) + sin(A0)|y) (3) which is a linearly polarized beam with the polarization direction rotated by an angle AQ with respect to the input status at the first polarization grid GP1 and where:

2pAc

Dq A (4)

The equation (4) represents the map between the displacement Dc and the polarization ro tation DQ. It is important to note that this rotation may be amplified by reducing the value of the spatial period A of the polarization grids GP1 and GP2. Note that this is true for any linearly polarized status; as mentioned above, the initial status | H) was chosen only for reasons of simplicity. Furthermore, to omit the diffraction of the beam in the derivation of equation (3), the distance D between the polarization grids may be chosen to be sufficiently small, i.e., D « w₀

A, where w₀ is the width of the beam at the waist position and l is the optical wavelength. Alternatively, for large values of D, a lens system may be used to form an image of the first polarization grid on the second polarization grid.

To measure Dc, the optical power after a projection onto the initial polarization status may be read, leading to a super-resolving Malus’ law (Fig. 1C):

R(Dc) = R₀\(H\Q)\² = P₀ (cos (^ ))² (5) where P₀ is the laser beam input power to the first polarization grid GP1. In Fig. 1 this is represented by a detector assembly comprising a polarizer Pol2 and an optical power meter

PM.

The maximum sensitivity S may thus be obtained by working in the linear regions, where

dP

This corresponds to a sensitivity S dAx = P₀ — , which may be increased by reducing L.

As an added advantage, by rotating the input polarization direction (or, alternatively, the polarization projector Pol2 in the detector assembly, it is always possible to set the “zero displacement” point of the system (Dc = 0) in the center of the linear region. This is im portant because, as the sensitivity is increased, the working range of the system is simulta neously reduced, which is given approximately by L/4 (monotonicity interval for Malus’ law). For larger displacements there would, in principle, be ambiguity in estimating the cor rect TD. However, this limit may be surpassed in at least two ways. First, the operating point may always be kept in the linear range of the system by dynamically rotating the polariza tion analyzer. Degeneration may then be removed by keeping track of said rotation. Second, an additional pair of polarization grids with a period L' large enough to remove degeneration may be exploited, while the desired resolution is provided by the original pair of polarization grids. In this way, two or more pairs of polarization grids may be used in parallel, each providing a different range in TD measurement. Referring to Fig. 5, a diagram of a system comprising multiple pairs of polarization grids, denoted GP1-GP2, GP1’-GP2’ and GP1”- GP2” is precisely depicted. The beam generated by the source S is then subdivided, in a manner known per se, into several beams R, R’ , R” respectively directed at the aforesaid pairs. The polarization grids GP2, GP2’ and GP2” are carried by the same displaceable object so that they are movable integrally. Each pair of polarization grids is associated with a respective detector assembly, denoted Pol2-PM, Pol2’-PM’ and Pol2”-PM”. The value L of each pair of grids is sized, in the manner described above, to have different sensitivities and measuring ranges for each measuring branch of the system.

With the system according to the invention, it is also possible to measure displacements both along the axis x and along the axis y. In practice, two parallel-propagating light beams and two systems as described above may be used, one oriented along x and one along y.

EXPERIMENTAL PROOF OF PRINCIPLE

The experimental setup depicted in Fig. 4 was constructed. A collimated He-Ne laser beam (L = 633 nm) generated by a source S is initialized in the status \H) by a linear polarizer Poll while a half-wave plate HWP1, arranged before the linear polarizer Poll, is used to control the optical power P₀. The system is then implemented with two polarization grids GP1 and GP2. To control the transverse displacement Dc between the two devices, the po larization grid GP2 is mounted on a motorized translation stage. Depending on the maxi mum TD required, two different translation stages were used, one with a position accuracy of 100 nm and a long stroke, and the other, based on a piezoelectric positioner, with a posi tion accuracy of 2 nm. The power R(Dc) is recorded by a power meter PM arranged after a second polarizer Pol2 and a spatial filter (lens L + iris I in the focal plane). Said spatial filter is used to improve the visibility of Malus’ law, as it cuts unwanted components of light associated with high spatial frequencies (possibly due to inaccuracies in the tuning of po larization grids or their patterning). A second HWP2 half-wave plate is placed between Pol2 and GP2 to rotate the analyzed polarization direction to set the working point at the desired position.

Five different pairs of polarization grids were used in the experimental tests, respectively corresponding to L = 5000, 1000, 500, 100, 50 pm.

The measured normalized optical powers for different TDs are shown in Fig. 2a, together with the best-fit curves in accordance with the equation (5). The data obtained reproduce the expected oscillatory behavior well.

To evaluate and compare the sensitivity of different pairs, measurements were focused in the linear region. The optical power was then placed at a reference value P₀ = 1 mW and the calibration curve R(Dc) was measured in the linear region, for a total displacement of 2 pm with steps of 20 nm (see Fig. 2B). To improve the clarity of the graph, each dataset was measured in a region slightly displaced from the center of the fringe (but still abundantly in the linear region of the calibration curve). A comparison of the slopes of the curves clearly shows the advantage in terms of sensitivity that is gained by decreasing L. This may be easily quantified by performing a linear fit for each curve. The best sensitivity S = 124.0 ± 0.1 nW/nm was obtained, as expected, for polarization grids with L = 50 pm. The effective resolution of the system for this configuration was then evaluated. For this purpose, the optical power was repeatedly measured for a time interval of 1 s (250 points), before and after a controlled step of the translation stage. The results are shown in Fig. 3, where circles represent experimental data, while dashed lines indicate the average power calculated on each set of 250 points. Starting with a “large” displacement Dc = 100 nm, which gives an average power difference DR = 13.2 pW, the step width was gradually de creased. The smallest measured displacement Dc = 5 nm, corresponding approximately to is still clearly resolved, because DR = 1.0 pW is significantly greater than its error bar

( s = 0.3 pW), calculated as the sum of squares of the standard deviations of the power distribution before and after the step, respectively. Note that the standard deviation of power using the polarization grids with A = 50 pm was typically s_R = 0.2 pW, which is reduced to s_R = 0.1 pW when the grids were off.

An estimate of the resolution of the system is obtained in terms of the ratio R = s/S. In order to evaluate the effective resolution of the described setup, it is crucial to reduce fast signal oscillations mainly due to mechanical instabilities in the system. For this purpose, average power was considered over time intervals of 0.1 s (solid lines in Fig. 4), resulting in a typical standard deviation s_R = 0.05 pW. This corresponds to a sub-nanometric resolu tion R = 400

In an ideal, vibration-free case, this result could be further improved by minimizing optical power fluctuations, such as by using an ultrastable laser or a balanced detector. The value of L, on the other hand, may be decreased to a few microns or even less if polarization grids are replaced with dielectric metasurfaces [6]. That said, referring to current liquid crystal technology, one may consider polarization grids with L = 6 pm [7], which, for the power fluctuations described above (at P₀ = 1 mW), would give the remarkable resolution of

R = 50 pm, corresponding approximately to ~~^·

DETAILS OF THE OPERATING PRINCIPLE

The action of the device described above is determined by its birefringent optical delay <5, the value of which may be controlled by adjusting an external alternating voltage [8]. By positing d = 7G, the device may be described as a geometric phase grid, the action of which on the polarization status may be easily written in the circular polarization base as:

I L/R) ® and^±i2aM\R/L > (7) where | L) and | R) denote the left and right circular polarization statuses, respectively, and a(x) is a linear function of the transverse position given by the equation (1). Considering a Gaussian beam | H), polarized uniformly along the horizontal direction (axis x), and propa gating along the axis z, the device-induced transformation on the incoming beam provides:

Essentially, each circular polarization component is a Gaussian beam, propagating along a direction in the xz plane forming, with the axis z, an angle that is approximately given by = +L/L. As such, in the near field the beam retains its Gaussian envelope but accommo dates a spatially inhomogeneous polarization pattern (see Fig. 1). After passing through a second polarization grid, identical to the first but displaced laterally by an amount Ax, the optical field is described by a status:

2pAc where DQ = A

Bibliographical references

[1] V. D'Ambrosio et al. Photonic polarization gears for ultra-sensitive angular measure ments. Nature Communications, 4(1):2432, 2013.

[2] K. Patorski and M. Kuhawinska. The Moire Fringe Technique. Elsevier, 1993.

[3] Isaac Amidror. The theory of the Moire Phenomenon Volume I: Periodic Layers, 2nd ed. Springer, 2009.

[4] K. Hane et al. Moire displacement measurement technique for a linear encoder. Optics and Laser Technology, 17(2):89-95, 1985.

[5] A. D'Errico et al. Two-dimensional topological quantum walks in the momentum space of structured light. Optica, 7(2): 108-114, Feb 2020.

[6] Dianmin Lin et al. Dielectric gradient metasurface optical elements. Science, 345(6194):298-302, 2014.

[7] G. Berkovic et al. Optical methods for distance and displacement measurements. Adv. Opt. Photon., 4(4):441-471, Dec 2012.

[8] B. Piccirillo et al. Photon spin-to-orbital angular momentum conversion via an electri cally tunable q-plate. Applied Physics Letters, 97(24):241104, 2010.

Claims

1. A photonic system for detecting a relative displacement between two objects (P, TS), comprising a source assembly (S, HWP1, Poll) configured to generate a radiation beam (R) having an initial linear polarization status and being able to propagate along a propagation axis (z), a first and a second Pancharatnam-Berry phase grid (GP1, GP2) arranged after each other along the propagation axis (z), wherein: each of said first and second Pancharatnam-Berry phase grids (GP1, GP2) comprises a birefringent medium acting as a delay plate of which the fast and slow optical axes have an orientation varying in a periodic way along a transverse direction (x) in a plane (x,y) orthogonal to the propagation axis (z), said first and second Pancharatnam-Berry phase grids (GP1, GP2) are respectively associable to a first object and a second object (P, TS) movable relative to each other along said transverse direction (x), said first and second Pancharatnam-Berry phase grids (GP1, GP2) are configured to rotate the polarization status of the radiation beam (R) by a given rotation angle AQ, as a function of a relative displacement Ax between said first and second object from a reference position, and a detecting assembly (HWP2, Pol2, L, I, PM) arranged along the propagation axis (z), said detecting assembly comprising a polarizer (Pol2) configured to project the rotated polarization status of the radiation beam (R) along said initial polarization status of the ra diation beam (R), and an optical power meter (PM) configured to provide an optical power signal of the radiation beam (R), the value of which is indicative of said rotation angle DQ.

2. The system of claim 1, wherein the orientation of the optical axis of the birefringent medium of the Pancharatnam-Berry phase grids (GP1, GP2) is defined by the relation cr(x) = ^x + a₀, where a(x), defined modulus p, is the angle formed by the fast and slow optical axis of the birefringent medium of the Pancharatnam-Berry phase grid with said transverse direction (x), a₀ is the residual angle at x=0 and L is the spatial periodicity of the angle.

3. The system of claim 2, wherein the rotation angle AQ varies linearly with the relative displacement Ax according to the relation

where Dq₀ is the rotation of the polarization associated to the displacement Dc = 0.

4. The system according to any of the preceding claims, wherein said detecting assem bly further comprises a half-wave plate (HWP2) rotatable to adjust the relation between the rotation angle AQ of the polarization status of the radiation beam (R) and the relative dis placement Ax, and in particular to set the value of the rotation angle A0 associated to the zero relative displacement, Dc=0.

5. The system according to any of the preceding claims, in which said polarizer (Pol2) is rotatable to adjust the relation between the rotation angle DQ of the polarization status of the radiation beam (R) and the relative displacement Dc, and in particular to set the value of the rotation angle Dq₀, associated with the zero relative displacement, Dc = 0.

6. The system of any of the preceding claims, comprising a plurality of pairs of Pan- charatnam-Berry phase grids (GP1-GP2, GP1’-GP2’, GP1”-GP2”) arranged in parallel, each pair of Pancharatnam-Berry phase grids comprising said first and second Pancha- ratnam-Berry phase grid, and a plurality of said detecting assemblies respectively associated to the pairs of Pancharatnam-Berry phase grids, wherein the pairs of Pancharatnam-Berry phase grids have different sensitivities and measurement range of the relative displacement Ax.

7. The system of any of the preceding claims, wherein said birefringent medium is a liquid-crystal material, the arrangement of the molecular director of the material being de fined in a periodic way along said transverse direction (x).

8. The system of any of claims from 1 to 6, wherein said birefringent medium is a metamaterial, configured in such a way to act on light polarization as a delay plate, the orientation of the optical axis of the metamaterial being defined in a periodic way along said transverse direction (x).