Infinitesimal optical singularity ruler for three-dimensional picometric metrology

Abstract

Optical metrology with picometer-scale precision in three-dimensional space is of considerable importance in modern physics and state of the art technology, optical interference is an effective method, but techniques with rapid spatial variation have the potential to enhance measurement precision, which will be required as measurement dimensions decrease. Here, the concept of the vanishingly small optical phase singularity ruler is introduced. Inspired by the well-known plumb-line technique used to locate the centroid, an analogous singularity line technique is proposed to locate the optical singularity with a precision of ~4.5 pm (~λ/140000) in the transverse direction and ~24.2 pm (~λ/26000) in the longitudinal direction. This precisely positioned singularity can serve as a ruler to detect displacement signals with an accuracy approaching ~60 pm.

A picometer-scale optical ruler is proposed based on precise location of a vanishingly small transverse optical phase singularity. This ruler is capable of measuring three-dimensional picometric displacements within optical interferometric systems.

Introduction

Optical metrology has been critical in the development of science and technology, an important example being the application as a high-precision ruler¹. As early as 1887, optical Michelson interferometer was used in the well-known Michelson-Morley experiment for the measurement of light velocity. Until today, optical interferometers still play a very important role in a lot of research fields, such as gravitational wave detection², star observation³, topological insulator detection⁴, microscopic particle vibration⁵, semiconductor inspection⁶, gravitational constant measurement⁷, etc. With the advances of scientific researches from macroscopic to microscopic dimensions, especially in molecular and atomic scales, the accuracy of optical metrology needs to be improved urgently^8,9. For instance, the capillary condensation phenomenon of the liquid molecular layer¹⁰, and the overlay control for high-precision semiconductor manufacturing^11,12 both need to be measured with the accuracy of 10 picometers.

The precision with which one can measure a small displacement is determined by two factors namely the sharpness of the variation of the optical field and the signal to noise ratio. Consider an optical interferometer where the minimum detectable displacement, $\mathrm{\Delta }x$, for, say, a Michelson interferometer, can be represented as:

$$\mathrm{\Delta }x=\frac{\lambda }{4\pi }\sqrt{SNR}=\frac{1}{d\varphi /dx}\sqrt{SNR}$$ 1

where $\lambda$ is the wavelength, the $4\pi$ arises from the standing wave in the Michelson type interferometer and SNR is the signal to noise ratio of the measurement. The first term is basically the reciprocal of the phase gradient as shown in the last equality. If the phase gradient can be increased the minimum detectable displacement will decrease provided the SNR is not degraded too much. In regions where the phase gradient is high the intensity inevitably deceases, and the SNR will suffer in the case of shot noise limited detection. In many cases, however, the limiting noise factor is not shot noise so provided so the achievable SNR is good even in regions of low intensity. In these circumstances increasing the phase gradient is highly desirable to achieve good sensitivity in situations of non-optimal noise conditions. To obtain a larger phase gradient than can be achieved with a conventional interferometer such as shown in Fig. 1a we seek the optical field with sharper phase gradient.

Fig. 1. Schematics illustrating the OAM singularity method.

a A conventional interference fringe of light (bottom) and the corresponding gradient distribution of the intensity of the fringe (top). b Phase distribution of a vortex field around the centered phase singularity (bottom) and the sharp phase gradient distribution (top) close to the singularity. The peak value of the phase gradient is theoretically infinite. c The diagram depicts the generation of the transverse OAM field. The polarization marked as the white arrows is perpendicular to the plane of the light source, i.e. along the y-direction, and the focal field generates a transverse OAM field around the phase singularity. The singularity line technique of OAM enables precise positioning of singularities at the picometer level. d Schematic of the experimental setup. BE: Beam expander; P1, P2: Polarizers; λ/2 Half-wave plate, SLM Spatial light modulator, L1, L2 Lenses, A Aperture, MO1 MO2 Microscope objectives, ET Extension tube, CCD Charge-coupled device.

Recently, optical surface waves were proposed to reduce the size of the optical field and enhance the accuracy of displacement measurement, benefiting from the shorter wavelength^13,14. For example, surface-wave interferometry on metallic surfaces can achieve displacement measurement with a high resolution of 0.734 nm (~λ/1000)¹⁵. Another technique that enables the generation of deeply subwavelength optical fields is optical super-oscillation^16–20. A representative work for this technique used a metasurface to construct a complex super-oscillatory optical field. By utilizing the sharp local wavevector peaks formed by the high phase gradient carried by the singularities in the super-oscillatory field, it created a precise optical ruler for the first time, instead of traditional interference fringes. This significantly improved the resolution of displacement measurement to 1 nm (~λ/800)^21,22. Furthermore, assisted by a deep learning algorithm, the resolution of the super-oscillation technique recently further improved to 92 pm (~λ/5300)²³. However, the theoretical results of the super-oscillatory optical ruler based on the high phase gradient carried by the singularity show a resolution at the sub-nanometer scale. How to further compress the spatial size of the optical ruler to improve its precision and thus enhance the accuracy of optical metrology to the picometer scale remains a challenge.

Apart from featuring a sharp local wavevector peak, the infinitesimal spatial scale of optical singularities^24,25 is another crucial characteristic that significantly enhances their potential as optical rulers. The spatial scale aspect of singularities has recently been utilized to extract precise deep subwavelength localization in MINFLUX microscopy²⁶. In comparison to interference fringes, optical singularities have infinitesimal feature sizes and very sharp phase gradients, as illustrated in Fig. 1b, theoretically providing high accuracy for displacement measurement. However, the localization of singularities is limited by the discretization of measurements. Since singularities are positioned in the dark regions of the light field, improving the precision of their localization presents a significant challenge. Here, borrowing the plumb line technique for centroid measurement in classical physics, we propose a singularity line technique to realize picometric location of the singularity, using the optical orbital angular momentum (OAM) density distribution around the singularity. By employing a transverse optical singularity generated through four-wave interference, we demonstrate that the singularity can be precisely located in three dimensions at the picometer scale (~λ/140000 in the transverse direction; ~λ/26000 in the longitudinal direction). We further confirm that the precisely positioned optical singularity can function as an extremely precise ruler for detecting three-dimensional displacement signals.

Results

The optical singularity ruler is created using a four-wave focusing configuration (Fig. 1c) [see Supplementary Note s1]. It produces a compact transverse singular light field with deeply subwavelength-scale circulation of energy flow and OAM around the singularity. This field features a highly sensitive response to both transverse and longitudinal displacements of the optical field, allowing it to be used as a sharp pointer for three-dimensional displacement measurements.

To detect the phase distribution around the singularity, an interference field between the four-wave focusing field and an extra plane wave optical field is formed at nanoscale, and the interference fringe is recorded with a magnifying imaging system (×1250)²², as shown in Fig. 1d. A piezoelectric displacement platform is used to sweep the field at nanometric scale to ensure high-accuracy measurements [see Methods and Supplementary Note s2].

The Richards–Wolf vector diffraction theory is used to calculate the four-wave focusing field [see Supplementary Note s1], and its interference field with an extra plane wave is shown in the x-z plane (Fig. 2a). The corresponding experimental results are presented in Fig. 2e. To retrieve the phase distribution of the four-wave focusing field, the Fourier transform method²⁷ is employed [see Supplementary Note s3]. The phase clearly varies from −π to π around the singularity (the white circles in Fig. 2b, f). With the singularities oriented perpendicularly to the propagation direction, the OAM direction is also transverse (along y direction). To visualize the transverse OAM characteristics within the circle, the annular energy flow distribution is analysed (Fig. 2c, g). The energy flow distribution is enclosed within a vortex, the boundary of which is determined by the zero points of the phase gradient, known as the saddle points. Beyond the saddle points, no circulation can form [see Supplementary Note s3]. The maximum range of the vortex energy flow is shaped like a “cup” (the green lines in Fig. 2c, g). Within the cup region, the OAM density distribution surrounding the singularity (Fig. 2d, h) can be calculated by the cross-product of the energy flow and position coordinates²⁸. As depicted in Fig. 2d, the obtained OAM value (Fig. 2d, h) is always greater than 0 in the cup region when the origin of coordinates is at the singularity. This arises through the anticlockwise energy flow around the singularity.

Fig. 2. Distribution of the transverse OAM field, from theory (top row) and experiment (bottom row).

The transverse OAM field has only an E_y component because of the choice of TE-polarized incident light. Panels a and e display the intensity distribution of the interference field between the transverse OAM field (E) and the plane wave (E_p). Panels b and f show the corresponding phase distributions of the transverse OAM field. Panels c and g depict the energy flow distributions surrounding the singularity within the white circles of panels b and f. Green lines mark the maximum energy flow loop adjacent to the two saddle points (red dots). Panels d and h show the OAM density distribution obtained by taking the singularity (green dot) as the origin coordinate, and calculating the cross-product of the position coordinates with the energy flow.

Similar to the plumb line technique hanging the same object twice with a plumb line, the lines intersect at the object’s center of gravity²⁹, a singularity line technique is proposed to precisely locate the singularity (Fig. 3a). In this technique, the singularity is treated as the centroid of OAM, and its position in the dark region is determined through the surrounding OAM density, just like the case of MINFLUX technology where the fluorescent molecule can be indirectly located through the surrounding targeted coordinate patterns²⁶. Details of the singularity line technique are shown in Fig. 3b. If the origin of coordinate system is shifted away from the singularity gradually (orange dot in Fig. 3b), the rotational sense of the energy flow reverses in some regions. In these regions, the OAM changes from positive to negative, resulting in a line of singularity with OAM = 0 at the boundary between positive and negative areas. As the distance between the singularity and the origin increases, the line of singularity changes from a closed loop to an arc, but it always passes through the singularity (green dot). Moreover, it passes through the singularity regardless of the origin chosen because of the null intensity at the singularity. Therefore, the intersection point of any two different lines of the singularity can be used to determine its location with sub pixel precision.

Fig. 3. Process and results of the singularity line technique.

a The comparison between the classical plumb line technique and the singularity line technique. For off-axis OAM, the OAM density distribution changes and creates a line of singularity with OAM = 0 between two regions of opposite sign OAM. Like the plumb line technique, the OAM centroid (or singularity point) is obtained by the intersection of all lines of singularity. In the first row of b, the OAM density distribution changes with shifts of the coordinate origin. Negative OAM values are depicted in blue. The dividing line between positive and negative values is referred to as the line of singularity of OAM, represented by orange lines. In the second row, locating the OAM centroid is performed by calculating the intersection points of the singularity lines. The green dot represents the theoretical location of the singularity. The orange and blue lines correspond to the lines of singularity when the origin of coordinates is defined on the right and lower sides, respectively. c shows the statistical distribution of the intersection points with different coordinates obtained from experimental results. d and e depict the statistical data points, which are counted to determine a precise location of the singularity, and fitted with a Gauss function. f displays the results of repeated experiments and the average precision of singularity location.

In theory, the retrieved singularity is an infinitesimal point. However, in practice, the recorded data are discrete and, as a result, lines of singularity are discontinuous. This leads to multiple intersection points between the lines of singularity, rather than a single point. Such measurement errors are inevitable. To reduce this error band, a multi-angle statistical method [see Supplementary Note s4] is used to improve the location precision. The final location of the intersection point is an extended spatial distribution (Fig. 3c). When used as an optical singularity ruler, the full width at half maximum (FWHM) of the diffused spatial distribution determines the position precision. Figure 3d, e plots the fitted Gaussian distributions along the transverse and longitudinal directions, respectively. The corresponding FWHMs of experimental results are 2.7 pm and 17.8 pm. For comparison, theoretical predictions are obtained using the same discrete precision as that in experiments (purple lines with FWHMs of 2.2 pm and 12 pm) [see Supplementary Note s5], which agree well with the experimental results. The FWHM in the longitudinal direction is larger than that in the transverse direction because of the inhomogeneous phase distribution around the singularity and the larger phase gradient along the transverse axis. The results of many repeated experiments are shown in Fig. 3f. The mean values of the FWHM obtained from these experiments are 4.52 ± 2.13 pm and 24.17 ± 4.64 pm in the transverse and longitudinal directions, respectively, ensuring the stability of the experimental measurements.

Benefitting from the picometric location precision for the singularity in both transverse and longitudinal directions, the optical transverse singularity can be used as a precise ruler in three-dimensional space. The pointer precision shown in Fig. 3f is about 4.5 pm (~λ/140000) in transverse direction and 24.2 pm (~λ/26000) in longitudinal direction, which can be sensitive to space variations at the atomic scale.

To verify the practical applications of the picometric singularity ruler, we further conduct displacement measurements using a piezoelectric displacement platform (PI p-612.2XY) to control the movement of objective lens. The singularity then moves along the transverse and longitudinal axes [Fig. 4a, b]. The OAM centroid method is used to retrieve the successive positions of the singularity. Experimental results and linear fitting curves for various step size movements along the transverse and longitudinal directions are shown in Fig. 4c~4f, respectively. The final repetitive accuracy in the transverse direction is found to be 68 pm and 60 pm with step sizes of 0.8 nm and 12 nm, respectively, as calculated using the standard deviation of the linear fits²². It should be noted this accuracy is an order of magnitude worse than the position precision shown in Fig. 3f, because of environmental disturbances and system errors during repeated measurements. The measurement process took approximately 10 min in our experimental system.

Fig. 4. Repetitive accuracy of a picometric ruler for displacement measurement.

Panels a and b show phase distributions before and after a shift in the singularity along the transverse and longitudinal directions, respectively. Panels c~f present measurement results in the transverse and longitudinal directions, respectively. The dashed lines represent linear fits of the data. e is the standard deviation.

For a 3D displacement measurement, a displacement in the longitudinal direction is also performed (Fig. 4e, f). The repetitive accuracy is found to be 958 pm and 989 pm for step sizes of 80 nm and 10 nm, respectively, both of which are not as good as that for the transverse direction. This is because the position precision of the singularity in the longitudinal direction is much lower than that in the transverse direction (Fig. 3f). Furthermore, the longitudinal result requires an additional piezoelectric platform for repeatable measurements and thus produces a larger error. Note that for transverse OAM, the y-direction measurement is equivalent to that for the x-direction by rotating the light field by 90°, and thus is not discussed here.

In conclusion, we demonstrate a picometric optical singularity ruler that possesses characteristics of far-field, three-dimensional, and high-accuracy in measurement. Because of the theoretically infinitesimal property of the singularity, the accuracy of the optical singularity ruler could be further improved by optimizing the experimental system and conditions, for example, placing it in a vacuum environment [see Supplementary Note s6]. Moreover, optical singularity exists extensively in various optical fields, hence the singularity ruler is not only limited to the experimental system in this work, but also could be combined with other optical interferometry systems by detecting optical singularities instead of interference fringes. It is thus potential for various applications from macro to micro scale, such as gravitational wave detection, star observation, observation of topological insulator, and vibration detection of microscopic particle. The optical singularity ruler realizes the exploration of picometer-scale displacement and provides a new tool for the picophotonics.

Methods

For the experimental system (Fig. 1d), a spatial light modulator (SLM, PLUTO-2-NIR-015) and a 4-f system were used to regulate the 632.8 nm He-Ne laser (Melles Griot 05-LHP-991) generating the incident light and to produce the singularity optical field at the focus of the micro-objective (MO1, UPLFLN 60×, NA = 0.9). The incident light was flexibly adjusted by regulating the phase mask encoded in the SLM, thereby enabling control over the singularities in the focus field.

The singularity optical field was imaged using a 1250× magnification imaging system, consisting of a second micro-objective (MO2, UPLFLN 60×, NA = 0.9), extension tube with lenses, and a CCD camera (GS3-U3-23S6C-C, Pixel Size 5.86μm). Since the camera has a maximum signal-to-noise ratio of 45.14 dB, we average the camera image every 20 × 20 pixels to suppress the noise signal. Importantly, the centroid of transverse OAM is a virtual pointer, the precision of which is higher than the pixel resolution. Moreover, to guarantee normal imaging of the light field, NA of the imaging objective MO2 (NA = 0.9) had to be greater than MO1. Therefore, the actual NA of MO1 used was approximately equal to 0.65.

Scanning of the focus field along the propagation direction was achieved by employing a piezo actuator (accuracy of platform movement 5 nm). Each time the platform moves, the camera takes an image of the optical field distribution to obtain 3D optical field data, from which the xz plane optical field distribution is extracted. To control the polarization accurately, a half-wave plate (λ/2) was utilized to rotate the polarization direction perpendicular to the plane where the four sources are located.

Author contributions

H. Ma conceived the original idea. Y. Zhang, C. Min, and X. Yuan refined the idea as primary mentors, where C. Min proposed the concept of centroid of OAM. F. Feng and M. Somekh provided knowledge and theoretical suggestions, and participated in the experimental design. H. Ma provided experiments and simulations. J. Zhou participated in some experiments and undertook data processing. H. Ma wrote the manuscript. Y. Zhang, C. Min, F. Feng, M. Somekh, and X. Yuan revised the manuscript. Y. Zhang, C. Min, and X. Yuan supervised the project. All the authors discussed the results and commented on the manuscript.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The data supporting the findings of this study are available in the Supplementary Information.

Code availability

All numerical codes are available upon request from the corresponding authors.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-55210-0.