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Abstract
Structured-light displacement detection method is an innovative approach with extremely high sensitivity for measuring the displacement of a levitated particle. This scheme includes two key components, a split-waveplate (SWP) and a single-mode fiber. In this work, we further investigated the influence of SWP installation on this method regarding the sensitivity of displacement detection. The results indicate that the sensitivity increases with the expanding of SWP offset in the effective range. In addition, we found this method has a significant tolerance rate, with an extensive SWP offset effective range of 5%-25%. However, an excessive offset can render this method ineffective. More interestingly, we demonstrated the feasibility of rotating the SWP to detect displacement in different directions. Our research contributes to guiding the structured-light detection methods in practical applications and expanding their applications in fundamental physics.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical tweezers (OT) are prominent tools for levitating and controlling micro- and nanoscale objects using tightly focused beams [1–4]. Initially, OT found widespread use in the biological field, such as trapping tobacco mosaic viruses, characterizing the mechanical properties of biomolecules, and observing single-molecule dynamics [5–7]. In the last decade, OT in vacuum has rapidly evolved, becoming a workhorse in precision measurement field. This advancement has led to remarkable achievements, including nano-g acceleration detection [8], zeptonewton force sensing [9], femtogram mass measurements [10], and so on [11,12]. These advances owe much to advanced displacement measurement technology.
Two primary methods exist for determining the centroid of the trapped particle. Video-based position detection, typically used for measuring low-frequency particle displacement, offers the advantage of direct observation of particle states [13,14]. Conversely, laser-based displacement measurements are commonly employed in applications demanding high resolution and rapid response rates. This method can be further divided into the quadrant photodetector (QPD) method [15,16] and the balanced photodetector (BPD) method [17,18] based on detector types. Both ways utilize the scattered light of the trapped particle to extract displacement information. The QPD method can obtain three-dimensional position information with only one detector, and the sensitivity can reach pm/Hz1/2. The BPD method mainly includes a D-shaped mirror and balance detector. Li et al. first applied this method to the OT system, achieving a detection sensitivity of 39 fm/Hz1/2 in a wide frequency range [17]. This method is widely employed in OT with high vacuum owing to its broad bandwidth and extraordinary signal-to-noise ratio [12,19]. Despite its advantages, the BPD method requires the addition of a Dove prism for displacement measurements in different directions using one photodetector (PD) [20]. In addition, it involves the re-adjustment of PD coupling. Moreover, the intensity of the trapping light often exceeds the PD saturation threshold in almost all OT systems. A common strategy to prevent saturation involves placing attenuators between the OT and the PD in the beam path. However, this approach also attenuates the information-containing field, subsequently compromising the ability of the system to detect particles effectively [21]. Alternatively, it is necessary to customize detectors to meet noise and high-power saturation thresholds. A promising solution was proposed by Lars S. Madsen et al. in 2021; they introduced a structured-light displacement detection method using a split half-waveplate (SWP) and a single-mode fiber (SMF) with a detection sensitivity of 1.5 fm/Hz1/2 [22]. This sophisticated method can detect the information-containing field while filtering out the un-scattered trapping field. However, the influence of SWP installation on structured-light detection remains unexplored.
In this paper, we conducted structured-light detection experiments in a counter-propagating dual-beam OT system. We compared the displacement sensitivity of the structured-light detection and BPD methods under identical conditions. At the same time, we examined the influence of SWP offset on the photodetector’s receiving power and particle displacement sensitivity. Finally, we demonstrated the potential of rotating SWP to detect particle displacement in different directions. Beyond studying the Brownian motion of the particle, our work serves as a reference for the practical application of the structured-light detection method and improves the performance of OT in precision measurement.
2. Principle
2.1 Scheme principle
Figure 1 presents the schematic of the structured-light detection method. The critical components of this method are SWP and SMF. We have defined the z-axis as the propagation direction of the laser beam, with the x-axis parallel to the optical table and orthogonal to the y-axis. The P-polarized beam, scattered by the trapped microsphere, serves as a probe beam that enters the detection optical path. This probe beam passes through a SWP and a SMF in turn, finally illuminates the PD.
Fig. 1. (a) The schematic of structured-light detection method. (b) The offset δ and rotation angle α between SWP and the probe beam.
A quartz half-wave plate was cut and reassembled to create one SWP with split lines. This design allows the SWP to maintain the light on one side of the split line unchanged, while introducing a π-phase shift on the other side. The probe beam can be decomposed into two distinct fields: the trapping field and the information-containing field [22]. The former, symmetrical mode 1, is dominant in the probe beam, while the information-containing fields constitute anti-symmetrical mode 2 (Fig. 1(a)). After passing through the SWP, the probe beam’s symmetry gets reversed. Mode 1 becomes an anti-symmetric flipped mode 1, and mode 2 changes into a symmetric flipped mode 2. Since the guide mode of the SMF is Gaussian and symmetrical, the anti-symmetric mode gets filtered out [22]. In essence, the SWP and SMF together form a spatial filter optimized to suppress trapping field, while maximizing the transmission of information-containing field.
The offset and rotation angle of the SWP are displayed in Fig. 1(b). The red dotted line signifies the central axis of the probe beam along the y direction. r denotes the radius of the probe beam, and d is the radial deviation of SWP relative to the center of the probe beam in the x direction. We define the offset δ as the ratio of d to r, i.e. δ = d/r. α represents the rotation angle of SWP around the clockwise rotation in the x-y plane.
2.2 Principles and simulation of displacement measurement
The structured-light detection method is sensitive to radial direction in OT, with x and y being orthogonal to each other. Therefore, our focus is solely on displacement in the x direction. The electric field at the back focal plane position can be expanded into the trapping field and the particle information-containing field. Similarly, the photocurrent ${i_T}$ detected by PD can be expanded to the following:
where $\langle{i_T}\rangle$ is the average total photocurrent generated by the beam striking the target surface of the PD, ix is the photocurrent response to particle displacement measurement. i0 is almost attenuated to 0 when the SWP is perfectly aligned, while ix is amplified as the offset δ increases [22]. The incident optical field can be expanded using the incident and exit vector spherical wave functions (VSWFs) [23,24]. The VSWFs expansion coefficients and transformation matrices can be calculated using the T-matrix method [25]. Thus, the probe beam can be expressed as follows [26]:In Eq. (2), A and B represent the transformation matrix, while a, b, p, and q are the function coefficients. θ refers to the numerical aperture angle of the condense objective, while φ signifies the gird angle of the measurement plane. The Optical Tweezers Toolbox can solve photocurrents in radial directions as follows [18,27]:
Here, E denotes the probe optical field, ${\varepsilon _0}$ is the vacuum permittivity, and $\hbar $ is Planck’s constant. Since the number of photons in the information-containing field is significantly smaller than that in the trapping field, iT ≈ i0[22]. The minimum resolvable displacement can be obtained by defining a gain Gx with the coefficients of an N-th order polynomial and the shot noise limit [28].
In the numerical model, we set the laser wavelength to 1064 nm and probe beam power to 20 mW. Both objectives have a numerical aperture of 0.4. The silica particle has a radius of 1.5 µm and its refractive index is n1 = 1.46, and the refractive index of the medium is n0 = 1.
Figure 2(a) illustrates how forward offset affects photocurrent as a function of the particle’s displacement along the x direction when α = 0°. The photocurrent in the x direction gradually increases with δ increases. SF(x) is the slope of the detector response curve at each point, and SF(0) is the detection sensitivity x = 0 near the axis center [29]. Since stimulated results, we speculated that the sensitivity also raises step by step with δ increases. The photocurrent as a function of the particle displacement at different α are also demonstrated in Fig. 2(b) (δ = 5%). The curves of α = 0° and α = 90° correspond to the photocurrent in the x and y directions of the particle, respectively. These two cases have the same slope at 0 µm along the radial direction owing to the circular symmetry of the probe beam. It means that rotation only changes the detection direction of particle displacement without changing sensitivity. However, they will diverge when the beam becomes asymmetrical.
Fig. 2. Simulation of photocurrent (a) The photocurrent as a function of the particle displacement at different δ values when α = 0°. (b) The photocurrent as a function of the particle displacement at different α values when δ = 5%.
3. Experiment and discussion
Our counter-propagating dual-beam optical tweezers system utilizes two objective lenses (OJ1 and OJ2) to form an optical trap to bind particles, as shown in Fig. 3. A LED light source and a CCD form an illumination imaging optical path for observing particles. The P-polarized beam at the back-focal-plane is divided into two parts by the beam splitter (BS) as the probe beam. One part enters the structured-light detection optical path (blue dotted frame), and another part comes into the BPD optical path based on D-shape mirror for comparison (orange dotted frame) [17,18]. A quartz half-wave plate (HWP20-1064BM, LBTEK) is cut and reassembled into a SWP in structured-light detection optical path. One PD (DH-GDT-D002N, Daheng optics) connects SMF to receive optical signals.
In our experiment, we trapped a silica microsphere with a radius of 1.5 µm. The radius of the probe beam r = 1.8 mm, and the sampling rate is 500 kHz. The optical power of two optical trapped beams is 115 mW. However, owing to transmission losses of the objective, there’s only light with a power of about 20 mW entered the structured-light detection path. It is worth mentioning that the SWP at back-focal-plane was installed on a rotating adjustment connected by a three-axis translation stage, so that we can investigate the influence of the SWP offset and rotation angle on the structured-light detection method.
The relationship between the value and its error of the detector output voltage at different δ values, as shown in Fig. 4(a). Notably, the voltage falls below the detector’s electronic noise floor (0.12 V) when SWP has no offset [22,30]. As the absolute value of δ increases from 0% to 50%, the voltage rises continuously since more trapping fields are allowed into the PD. Additionally, the voltage fitting curve conforms to a Gaussian distribution. The yellow dashed line (1.84 V) approximates the 1/e2 line in the Gaussian beam, where the offset ± 32.5% appears as a unique point. In the offset range of 0%-32.5%, the intensity of the optical signal varies more significantly. These results provide a preliminary reference for selecting ranges and step sizes when investigating the impact of offset. The voltages at different SWP rotation angles are shown in Fig. 4(b). Nearly all voltage values fall below the electronic noise floor when the SWP is perfectly aligned because the trap field is completely suppressed. The voltage can exceed the detector’s electronic noise floor by moving the translation stage to introduce a mini offset (δ = 5%).
Fig. 4. The voltage measured by the structured-light detection method. (a) Voltages of different δ values. (b) Voltages of different α values.
We chose δ = 5% and 20% to compare with the BPD method since the voltage signal does not change significantly after exceeding the offset of 32.5%. Figure 5(a) shows the voltage measured by the structured-light detection and the BPD methods. In structured-light detection method, the average voltage is about 0.18 V and 1.10 V respectively, when the offset δ = 5% and 20%. The average voltage measured by the BPD method is about 0 V. Figure 5(b) demonstrates the corresponding voltage signal spectrum. It can be seen that the spectrums of the two methods are quite different, the low-frequency parts do not overlap each other. However, the high-frequency part of the BPD method overlaps with it when δ = 20%. In addition, the spectrums with an offset of δ = 5% and δ = 20% also have almost no overlap in structured-light detection method. The results in the figure draw the conclusion that two methods have the same noise floor, but the structured-light detection method has a larger signal amplitude (δ = 20%). It indicates this method has a superior signal-to-noise ratio because it receives a larger optical signal.
Fig. 5. Comparison of structured-light detection and the BPD method. (a) The measured voltage signal; (b) Voltage signal sensitivity spectrum.
The voltage signal V(t) = βx(t), where β is the voltage-displacement calibration factor, and x(t) is the displacement of the microsphere. β can be calibrated by the energy equipartition theorem [31]. In the structured-light detection method, the voltage-displacement calibration factors of δ = 5% and 20% are 1.37 V/µm and 4.21 V/µm, respectively. The voltage-displacement calibration factor based on the BPD method is 3.02 V/µm. Using β to transform the signals of Fig. 5, we can get the characteristics of the particle’s displacement with a unit of nanometer, seeing Fig. 6.
Fig. 6. Comparison of the structured-light detection (δ = 20%) and BPD methods. (a) and (b) The histogram of the particle displacement measured by the structured-light detection and BPD methods. (c) and (d) The displacement spectrums measured by the structured-light detection method and the BPD method.
Figure 6(a) and (b) illustrate histograms of the particle’s displacement measured by the structured-light detection (δ = 20%) and BPD methods. The solid green and black lines are their corresponding fitting curves. Both histograms conform to the Maxwell–Boltzmann distribution [32]. The full width at half maximum (FWHM) of fitting curve 1 is 7.47 nm, closely matching that of fitting curve 2 (8.15 nm). In addition, the displacement distribution range of the particle is ±20 nm, consistent with energy equipartition theorem calculations [31]. These results indicate that this method effectively measures the Brownian motion of the trapped particle. The displacement spectrums of particle’s signals measured by the structured-light detection and BPD methods are depicted in Fig. 6(c) and (d), respectively, where the pink and blue solid lines are their fitting curves. The corner frequency of the trapped particle obtained by the structured-light detection and BPD methods are 4.16 kHz and 4.54 kHz, respectively. On the basis of the power spectrum analysis method, which has a standard deviation of about 6% [33], some differences between the two resonant frequencies are acceptable. According to the power spectrum analysis method [31], the stiffness of the optical trap in the x direction obtained by the structured-light detection method is 21.25 pN/µm, while the stiffness measured by the BPD method is 26.80 pN/µm. The insets are their magnification of sensitivity in high frequency. It can be clearly seen that the displacement spectrum of the two methods begins to flatten from approximately 200 kHz. It indicates that we achieved a detection sensitivity of 244.8 fm/Hz1/2, greater than the BPD method of 789.4 fm/Hz1/2 at 200 kHz. Moreover, the optical power of the mW level reaches the saturation threshold of the commercial balanced detector in the BPD method. However, the structured-light detection method allows for the use of standard commercial detectors in any strong optical traps without attenuating the signal field [22].
To thoroughly investigate the impact of offset on the structured-light detection method, we varied the offset from 0% to 35% in steps of 5%. Figure 7 demonstrates the histograms of laser noise and particle displacement under different offsets. The histogram of the laser noise signal exhibits a random distribution, as illustrated in Fig. 7(a). The histogram of the SWP no offset is quite different from the perfect Maxwell-Boltzmann distribution, and the FWHM of fitting curve 2 is 4.01 nm (Fig. 7(b)). On the contrary, the histogram of particle’s displacement conforms to the Maxwell-Boltzmann distribution when the offset δ ranges between 5% and 25% (10% and 20% are not shown in Fig. 7). The FWHM of fitting curve 3 (δ = 5%) is 6.53 nm, fitting curve 4 (δ = 15%) is 8.09 nm, and fitting curve 5 (δ = 25%) is 6.20 nm). However, the FWHM of the fitting curve of displacement distribution decreases to 3.16 nm when the offset increases to 30%, significantly lower than the 8.16 nm FWHM measured by the BPD method. The result implies that some motion information on the trapped particles is lost, and the structured-light detection method fails when δ = 30%, which is very close to the 32.5% corresponding to the1/e2 line. Besides, the voltage-displacement calibration factors of 5%, 10%, 15%, 20%, and 25% are 1.37 V/µm, 2.19 V/µm, 3.48 V/µm, 4.21 V/µm and 5.01 V/µm within the effective range. It is similar to the trend of the 0-point slope SF(0) of the detector response curve to increase as δ rises in Fig. 2(a).
Fig. 7. Displacement distribution of trapped particle with different offset δ. (a) laser noise. (b) δ = 0%. (c) δ = 5%. (d) δ = 15%. (e) δ = 25%. (f) δ = 30%.
The two-dimensional profile of the probe beam after passing through the SWP for different offset distances δ are shown in Fig. 8(a) and (b). While the beam profile almost maintains symmetry at δ = 5%, it gradually deteriorates as δ rises to 25%. Figure 8(c) depicts the contrast of displacement spectra between δ = 5% and 25%, with the insert showing a magnified view of their displacement spectrums. It is evident that δ = 25% offers superior sensitivity at 200 kHz compared to δ = 5%. From Fig. 8(d) it can be seen that the displacement sensitivity improves from 382.6 to 210.3 fm/Hz1/2 as δ increases from 5% to 25%. The minimum resolvable displacement curve obtained by solving Eq. (4) is shown in the solid blue line. It is observed from the numerical results the sensitivity rises as the offset δ increases. While the sensitivity measured experimentally does exhibit a certain numerical deviation from simulated results, the overall trend of change remains consistent. In short, this method cannot function without an offset in the SWP, and it begins to operate effectively when δ exceeds 5%. Within the range of 5% to 25%, sensitivity improves as δ increases. However, if δ continues to increase, this method fails.
Fig. 8. Beam profile and sensitivity at different δ. (a) Beam profile after SWP at offset δ = 5%. (b) Beam profile after SWP at offset δ = 25%. (c) The contrast of sensitivity spectra of δ = 5% and 25%. (d) Experimental results and simulation of different δ.
We further explored the effect of SWP rotation on the structured-light detection method. For structured-light detection to work, we introduced the same offset (δ = 5%) using a three-axis translation stage. The particle displacement measured at α = 0° and 90°, as illustrated in Fig. 9(a). It is evident from the figure that the root mean squared of α = 0° is greater than that of α = 90°. Figure 9(b) demonstrates the displacement spectrums for α = 0° and 90°. Its solid pink and green lines represent the corresponding fitting curves, respectively. This method measures displacement in the x direction within the OT when SWP remains at an angle of α = 0°. However, this method measures displacement in the y direction when the SWP rotates 90°. It may be noticed from the figure the primary difference between these two cases lies mainly in the low-frequency band. Moreover, the detection sensitivity in the high-frequency band aligns substantially in both scenarios, primarily because the rotation does not change the symmetry of the probe beam. When the SWP is not rotated, the corner frequency of the particles is 4.02 kHz, resulting in trap stiffness kx = 19.79 pN/µm. Conversely, when the SWP rotation angle is α = 90°, the corner frequency increases to 4.68 kHz, and the corresponding trap stiffness ky rises to 26.87 pN/µm. These results underscore the potential role of achieving displacement detection in different directions simply by rotating SWP. It eliminates the need for additional photodetectors or detector coupling re-adjustments when detecting particle displacement in different directions.
Fig. 9. Experimental results of different rotation angles. (a) Particle displacement at α = 0° and α = 90°. (b) The contrast of sensitivity spectra of α = 0° and α = 90°.
4. Conclusion
In summary, we investigated the influence of SWP installation on the structured-light detection method. This work yielded several key findings drawn from both simulations and experimental results.
The optical power received by the detector increases with the offset δ in the range of 0-50%. Furthermore, this method has a large working range, and it can operate effectively within an offset range of 5%–25%. Surprisingly, the detection sensitivity improves with increasing offset within this range. However, if δ expand beyond this range, this method ceases to function optimally. Finally, we demonstrated that displacement measurements in different directions could be accompanied by simply rotating the SWP and slightly adjusting the SWP-mounted translation stage. These findings offer valuable insights for enhancing sensitivity with the structured-light detection method and new approaches for measuring different directions within OT systems. Our research addresses a gap in displacement measurements using the structured-light detection method and will benefit the further fields of precision measurements and physics.
Funding
Natural Science Foundation of Hunan Province (2021JJ40679); National Natural Science Foundation of China (61975237).
Acknowledgment
The authors wish to thank the anonymous reviewers for their valuable suggestions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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