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Abstract
Direct optical force measurement is a versatile method used in optical tweezers experiments, providing accurate measurements of forces for a wide range of particles and trapping beams. It is based on the detection of the change of the momentum of light scattered by a trapped object. A digital micromirror device can be used to selectively reflect light in different directions using an appropriately defined mask. We have developed position-sensitive masked detection (PSMD) for measuring transverse (radial) and axial forces. The method is comparable in performance to the fastest split detectors, while maintaining the linearity and customizability similar to duo-lateral position-sensitive detectors (PSD) and cameras. We show an order of magnitude increase in the bandwidth compared to a conventional PSD for radial forces. We measure axial force and verify the measurement using the Stokes drag for the particle. Combining both detectors (PSMD and PSD), we can perform full 3-D optical force measurements in real time.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since its invention [1], optical tweezers became a useful tool for applying and measuring forces on nano- and micro-objects. The optical force measurements can be done indirectly — by tracking the position of the trapped particle. However, to convert the position of the trapped object into a force, a calibration is required. There are a number of excellent methods to perform this calibration: using the equipartition theorem and the distribution of the position of the object in the trap [2], by analysing the power spectral density [3,4], using drag force [5,6], etc. While these methods for force calibration are extensively used, they are not only limited mostly to spherical particles and small displacements, but also require a recalibration with every new particle being used.
Another method is to measure the change in the momentum of the light scattered by the particle [7–9]. The important requirement for accurate measurement is the collection of all of the scattered light, but in practice, it is not feasible to collect the light at all possible angles. However, it was shown that for the most particles used in optical tweezers, light is mostly scattered in the forward direction [10] and a condenser with a high numerical aperture will be able to collect most of this light. Also, the accuracy can be improved by detecting the back-scattered light which is especially important for the determination of the axial force [11]. Alternatively, a dual-beam optical trapping system [7] can use low-NA objectives which reduce the scattering at large angles. As this method is based on the detection of the momentum of the light, it is independent of the physical properties of particles or mediums used and thus, calibration-free. If the condenser satisfies the Abbe sine condition [12] (which is the case for the objectives with correction for spherical aberrations), the force, F, can be determined from the light distribution in the back focal plane of the condenser [11]:
where I(x, y) is the light intensity distribution in the detector’s plane, CA = r/sin θ is a constant of the condenser, the radius at which light that has been scattered at 90 degrees to the optical axis is detected in the back focal plane, H is the constant which includes magnification of the optical system and all absorptions/reflections in the optical elements, and F0 corresponds to the momentum flux of an empty trap [7,8,10,11].The radial components of the force (x and y) are proportional to the unweighted centroid (which is proportional to the power) of the pattern (see Eq. (1)). By definition, the weighted centroid is:
The weighted centroid is used as a measure of the position of a laser beam in a given plane. As can be seen from the Eq. (2), the information about the position is in the numerator and the denominator represents the total power which is position independent. It is often important to know the position of a beam, and instruments — position detectors — have been developed to measure it. Since the task of measuring the radial components of the momentum can be expressed in terms of a measurement of the beam position (i.e., the centroid of the intensity distribution) and the beam power, and typical position detectors measure both of these quantities, position detectors can be, and have been used to measure the radial components of optical forces in traps. Often, it is not explicitly stated that the beam momentum is being measured, but since the position of the beam (for radial forces) or the spread of the beam (for axial forces) is what is measured, the measurements are indeed optical momentum measurements.
The most commonly used position detectors are suitable only for radial force measurements (that is, a position detector by itself — with the addition of a spatially varying mask it can be used to measure axial forces [9]). It is possible to get some information about the axial position by using an aperture [13]; however, this is a rather approximate estimation. In many cases, the detector and/or the particular experimental implementation does not allow one-off calibration of the detector for momentum measurement, and a new calibration must be performed for each type and size of particle.
1.1. Beam position detectors for direct optical force measurements
There are three types of position detectors which may be potentially used for such direct force measurements: position-sensitive detectors (PSDs), split detectors (SD) (although they are far from suitable, as we discuss below) and cameras.
1.1.1. Position-Sensitive Detectors
The position-sensitive detector (PSD) consists of a photodetector with resistive layers on the front (tetra-lateral) or both front and back (duo-lateral) sides. The position measurements along one axis requires a set of two electrodes placed near the opposite edges of a rectangular photodiode [14]. In these devices, the measured signal at each electrode is proportional to the photoresistance current as a function of position. Another set of two electrodes near the orthogonal edges allows the measurements in two axes simultaneously. The output from the detector is:
where 2L is the length of the detector, and are signals from the electrodes on the opposite sides of the detector, ℜ and G are the responsivity and transimpedance gain of the detector, respectively.These detectors are relatively slow due to large surface areas, but provide excellent linearity and the output is independent of the spot shape or size. This makes these detectors an excellent choice for direct force measurement in the directions transverse to the beam propagation.
1.1.2. Split detectors
Another class of detectors, often called split detectors, use spatial sampling of light to obtain signals. In split detectors, light is separated by a sharp edge into two beams and the power of each is measured. Quadrant photodetectors (QPD) are 2-D examples of the split detector. They consist of four photodetectors — each one makes up a quadrant of the detector. Each segment can only measure the light impinged on itself. Thus only a part of the intensity distribution can be measured with the difference between two sets of signals (for each axis) becoming the ‘position’ of the spot. However, this is true only for some types of distribution. In a general case the signal is:
where are the signals from the quadrants. This signal is not equal to the centroid. It will be proportional to the small displacement of the beam for some light distributions, including the common and useful case of a Gaussian distribution.Also, the output will depend on the size of the spot. The size of the spot will depend on the particle that is trapped, and to use a split detector for direct optical force measurement therefore requires a new calibration for each different type or size of particle that is trapped. This makes split detectors unsuitable for direct optical force measurement on unknown particles where such per-particle calibration is difficult or impossible. In addition, the light distribution in the back focal plane of the condenser is irregular [10] and the size of the spot is continuously changing due to the axial part of Brownian or other motion of the particle, which means that split detectors are further unsuitable.
Despite these problems, split detectors have been commonly used for optical force measurement. Noting that typical per-particle calibration methods yield the spring constant of the optical trap (in the approximation of the trap as a linear spring), the force measurements become equivalent to measurements of the particle position. Thus, split detectors can be used for particle tracking in optical traps in two different modes: the particle itself can be imaged onto the detector, or the detector can be placed in the Fourier plane providing a measurement of the beam momentum. Measurements made with the latter method are often described as measurements of the particle position, but are fundamentally measurements of the optical force. While such detectors are often used for particle tracking purposes, including high-speed measurements [15], they are not the best choice for the direct force measurements, since they need per-particle calibration and measurements are affected by axial motion.
1.1.3. Camera
The third type of position detectors are CCD and CMOS cameras which are an array of photodetectors [16]. The image of the light distribution is recorded and the required parameters of the beam can be calculated in a post-processing using a variety of numerical methods [11]. This type of position detectors offers good linearity and can be used with any spatial distributions of the recorded signal.
As noted above, many position detectors are suitable only for radial force measurements. A camera allows the calculation of axial forces as well as radial forces, providing a full 3-D force measurement, as an arbitrary function can be applied to the image. However, a camera has comparatively low bandwidth and the frame rate for a full image typically will not exceed few kilohertz. Also, there are millions of pixels in a single camera chip, and cameras therefore have much higher noise than a single-element photodetector.
1.2. Amplitude filter mask
We propose a new detection method for performing a fast measurement of the optical forces which at least matches the accuracy of the PSD, attains the high bandwidth and low noise of the split detectors and gains the camera’s flexibility in the choice of the weighting functions. It is based upon a reflective filter with a specific transmittance function which is placed in front of the photodetector to modulate the intensity of the light transmitted through the optical system (Fig. 1). A similar approach (but using an attenuator with a spatially varying attenuation) was developed by Smith and Bustamante [9] for measuring axial forces. In this method only a single detector is needed which implies that the method is simpler to implement experimentally. It is not possible to generally implement Eq. (1) with a transmissive mask and a single detector because a change in the centroid position is not distinguishable from a change in power, which affects the x and y components in Eq. (1). In particular, the signal at zero force will be non-zero and, therefore, a change in power will appear as a force since this signal will change in proportion to the power. Therefore, a second detector is used to measure the power reflected from the mask. This gives in effect zero signal at zero force. In our method, the reflective mask allows the detection of both transmitted and reflected beams as complementary information is contained in the two paths. This doubles the signal and, unlike the detection with the attenuator, preserves a uniform distribution of the detection noise over the mask. Measuring both the reflected and transmitted light provides the radial optical force even when the beam power changes, without requiring additional measurement of the power.
Fig. 1 Schematic representation of position-sensitive masked detection. M is a reflective filter with a specific spatially varying transmittance function; PD1 and PD2 are photodetectors, respectively detecting signals, ST and SR.
For the axial component of Eq. (1), the left-hand term can be implemented with a transmissive mask and a single detector. However, the right-hand term, F0, can still cause difficulty, which we discuss below.
For an appropriately defined mask function at the filter plane, M, the transmitted and reflected light patterns from an initial intensity, I, are:
Assuming the diffracted light all falls onto a single element detector, gives the detected signals of: with the complete measurement being: In particular, to estimate the force acting on a particle, the mask is best placed in the far-field zone of light scattered by the particle. This can be achieved by placing the mask at the back focal plane (or its image) of the collection optics.1.2.1. Radial force
According to Eqs. (1) and (2), the radial force is proportional to the unweighted centroid of the light distribution in the back focal plane of the condenser. Here we assume the following expression for a masking transmission function:
where krad ranges from 0 to 0.5 is a transmission gradient of the filter normalised by the size of the pattern, and maximum sensitivity is achieved when the size of the filter is matched with the size of the light pattern.The resultant transmitted and reflected intensities IT and IR are integrated by corresponding detectors and subtracted either numerically after the measurement (two separate photodetectors) or electronically in the circuit using balanced photodetectors (BP). The resultant signal in volts is then given by:
where ℜ and G are the responsivity and transimpedance gain of the detector, respectively, krad is assumed to be the same for both x and y components.The components of the radial force then becomes:
As ℜ, G and H are constant for a given optical system, we will combine them into a new constant, H̃. Thus, the final equation for the radial components of the force is:There are two approaches to calculate the constants in the Eq. (11). We can precisely measure all the parameters and simply calculate them [17]. However, this is a challenging task as every element needs to be precisely characterized. Alternatively, we can calibrate our system using drag force or the statistics of the Brownian motion. The result will depend on the accuracy of the measurement of these forces but offers an easy implementation and can be quickly repeated if needed. We will use the equipartition theorem and the position of the trapped particle from the camera to obtain the constants:
where kB is Boltzmann’s constant, T is the absolute temperature, σcam is the standard deviation of the position of the particle measured in meters, is the standard deviation of the signal from the detector measured in volts. The calibration constant Crad = H̃/krad will directly convert the output voltage from the detector into an optical force acting on the trapped particle, circumventing the need of finding H. Crad can be found through observation of distributions of position of the particle and force that the particle experiences at room temperature (low trapping power regime).For the best performance of the detectors, it is important to balance the transmitted and reflected signals. This can be easily achieved by a spatial shift of the pattern (which is equivalent to adding a constant value to the Eq. (8)) without the loss of validity of Eq. (11).
1.2.2. Axial force
For axial force measurements we need to apply a spherical function centered on the optical axis . The mask for axial force measurements, Max, is defined as:
where kax is a balancing coefficient.Similarly to the radial force measurements, the signal from the detector for axial force measurement is:
The axial force is:
The second term in Eq. (14) represents the total power of the laser beam. Unlike the radial force measurements, the axial component of the momentum of the beam, i.e., the F0 term in Eq. (1), is not zero for an empty trap. This term can be removed by subtracting the momentum of the empty trap as done in Eq. (15), as long as the beam power does not change after the empty trap measurement is made. Alternatively, one can measure the total power of the beam.
In most cases, the power of the trapping beam is constant during the measurement and we can determine the axial force without any additional operations by finding the momentum when the trapped object is at its equilibrium axial position.
Since it is easiest to calibrate the detector using radial forces, it is useful to use the same calibration for axial forces, given by:
It should be noted that, unlike the radial force, it is important to know the CA constant to produce a pattern of the proper size. It is related to the maximum size, rmax, of the light distribution in the back focal plane of the condenser. CA is the radius at which light that has been scattered at 90 degrees to the optical axis is detected in the back focal plane. For a condenser with sufficiently high NA (i.e., greater than or equal to the refractive index of the trapping medium) this is equal to rmax:
where θmax is a maximum angle accepted by the condenser, nm is the refractive index of the trapping medium and NA is a numerical aperture of the condenser. If the NA is lower than the refractive index of the medium, rmax < CA and the right-hand term should be used to calculate CA.Thus, the force detection task is reduced to the measurement of the intensities of the two beams after the separation. This can be done with separate photodetectors. The best performance will be achieved with a balanced photodetector (BP) as it directly outputs the difference between the two signals and has common-mode noise reduction capabilities [18,19].
1.2.3. Bandwidth
Since the noise in an optical tweezers system is closely related to the bandwidth of the system, some further discussion of noise and bandwidth is warranted. We define the bandwidth of a system for the measurement of optical forces in an optical trap as the frequency range where the optical force acting on a particle can be measured over the noise floor. The low-frequency performance is limited by long-term drift, but typical force measurement systems are sufficiently stable over typical measurement times, and we will not consider low-frequency performance further. The high-frequency limit typically results from one of two factors: first, the maximum rate that the detector (e.g., PSD, split detector, or camera) is capable of. For example, if a camera is used as a beam position detector, the maximum frame rate of the camera can limit the highest frequency at which force measurements can be made. Second, the power spectrum density of motion of a trapped particle due to Brownian motion falls off as 1 / f2, where f is the frequency. Since, for small displacements from the equilibrium position, the optical force is linearly proportional to the position, the same 1 / f2 behaviour occurs for the optical force acting on a trapped particle. This decrease in the optical force with increasing frequency means that at some frequency, the force will no longer be measurable above the noise floor. That is, the high-frequency limit of the bandwidth can result from the noise floor. Generally, the lower of these two frequencies — the maximum frequency of the detector and the frequency at which the force meets the noise floor — will determine the upper limit of the bandwidth.
For both of these reasons, the choice of detector affects the bandwidth of the force measurement system. The distinction between the use of a single detector and balanced photodetector can have practical implications for measurements. All else being the same, the common-mode noise reduction obtained using a balanced photodetector will result in lower noise, potentially increasing the bandwidth. Noting that the reduction in noise can be over 50dB [19], the improvement can be significant (depending, of course, on other sources of noises). The principal and typical reason for choosing a balanced photodetector is that photocurrent cancellation is achieved, allowing an efficient digitization of signals such that bias subtraction is unnecessary [18]. The effective photocurrent for a two detector system is higher than for a single detector measurement yielding a better sensitivity of the detector. The measurement of the force at higher laser powers will be also improved as the resulting signal will be unbiased. Also, once the beam has passed the filter mask, we can use lenses to focus the beams onto the detectors, which means that smaller detectors can be used. This enables a measurement of large-size beams without a significant reduction in the bandwidth of the detector. Moreover, the BP contains well matched detectors and shares the amplification circuit which insures the balance of the measurements in both arms of the optical system. While two individual detectors can also be used, the measurements will contain a much bigger error due to mismatch in the responsivity and amplification of each individual detector.
However, the value of high bandwidth should not be overstated. It is important to keep in mind the distinction between the measurement of optical forces, and the measurement of other forces in optical tweezers. If the goal is to measure some force Fmeas using an optical trapped particle as a transducer, the force measurement task is often simplified to
where Foptical is the optical force acting on the transducer particle. However, a more complete description is where we have included the thermal forces driving Brownian motion, and viscous drag. If a single measurement of the optical force is made, the thermal and viscous forces are additional sources of noise as far as measurement of Fmeas is concerned. Of course, the effect of thermal and viscous forces on the measurement can be reduced by average multiple measurements of the optical force over time (which requires sufficient bandwidth), but enough time must be allowed for the particle to move about the position at which Fmeas and the optical force are in equilibrium.It is when one is directly interested in the optical force, or seeking to measure the viscous drag — when these forces, which are sources of noise above, become the forces of interest — that the bandwidth of the optical force measurement becomes especially valuable. This is why low noise and high bandwidth systems are often used for Brownian motion and fluid measurements [20,21].
1.2.4. Digital micromirror device as a dynamic filter mask
A digital micromirror device or DMD is a micro-electrical-mechanical system which consists of millions of micron-sized mirrors which are able to switch between two stable positions [22–24]. Each mirror can be set to the “on” or “off” state individually. We can use this property of the DMD to reflect the light into two different directions to perform the optical force measurements. As a DMD is a binary modulator, we will use dithering [25] to represent different transmission levels. We use two types of patterns: a linear gradient, Eq. (8) for radial forces and a spherical gradient, Eq. (13), for axial force measurements. The linear gradient was created with each line of the array (row or column) containing randomly distributed “on” pixels with the number of pixels corresponding to the required transmission level. The spherical gradient was created with the same algorithm but using circles instead of lines.
2. Methods
2.1. Experimental setup
A fiber laser (YLR-10-1064-LP, 10W, 1064nm, IPG Photonics) is focused by a high-NA objective (Olympus UPlanSApo 60×, water immersion, 1.2 NA) to create an optical trap (see Fig. 2). Light, scattered by a trapped object, is collected by a condenser (Olympus UPlanSApo 100×, silicone oil immersion, 1.35 NA). The back focal plane of the condenser is imaged on the force detectors using a telescope (lenses L1 and L2) and a beamsplitter BS. A CMOS camera is used to track the position of the trapped particles. The detectors are: PSD (duo-lateral PSM2-10 with OT-301DL amplifier, 15kHz, On-Trak Photonics); PSMD which includes DMD (infrared chip from DLP4500EVM, Texas Instruments) and balanced photodetector (PDB210A/M, 1MHz, Thorlabs). The signal from the detectors was recorded using a data acquisition card (NI PCIe-6351, National Instruments) and a custom made program in LabView (National Instruments).
Fig. 2 Setup for optical trapping. DM1 and DM2 are dichroic mirrors used to separate the laser beam from the illumination. An objective (NA 1.2, water immersion) creates an optical trap and a condenser (NA 1.35, silicon oil immersion) collects the scattered light. Lenses L1 and L2 image the back focal plane of the condenser on the PSD and DMD.
The sample is made from two coverslips with double-sided adhesive tape as a spacer. This provides an enclosed ≈ 100μm high sample chamber. The sample is held and moved by a 3-D piezo-stage (PI P-563.3CD, Physik Instrumente).
2.2. Alignment and calibration of the detectors
The CMOS camera was calibrated by moving and tracking a fixed spherical particle with a nanometer precision piezo-stage. The calibration constant for all measurements is Ccam = 0.170 ± 0.002μm/px.
The PSD was calibrated with a spherical silica particle with the diameter of ø1.70μm. The position of the particle (in meters) and the optical force (in volts) was recorded simultaneously at 5000fps (camera) and 15kHz (PSD) respectively. Then, the calibration constant of the PSD was calculated using Eq. (12) and has a value of CPSD = 1.93 ± 0.02pN/V
PSMD alignment and calibration are done in a few steps. Firstly we align the pattern on the DMD with the optical axis using horizontally/vertically divided patterns (analogue of the split detector). Then, the size of the light distribution is estimated by a circular aperture pattern with one of the inputs of the BP blocked. When the size of the aperture pattern is bigger then the size of the light spot no signal is observed. By changing the radius of the aperture on the DMD we record the size of the pattern when the signal is detected. We use a stuck particle to increase the scattering at large angles. The calibration constant for the radial forces (linear gradient pattern) is determined using the same procedure as for the PSD and becomes Crad = 0.70 ± 0.01pN/V.
Secondly, we use the radial force calibration and the size of the light distribution on the DMD to set the size of the pattern for axial force measurements (Rax = 290px ≈ 3.1mm). The axial calibration constant is then calculated using Eq. (16).
2.3. 3-D optical force measurement
Spherical silica particles (water solution, ø1.7μm) were allowed to settle to the bottom in the sample chamber and some of the particles adhered to the slide. The trap is placed at the center of the particle and the piezo-stage is used for a point-by-point scan of the particle. The 3-D force is measured with both the PSD (radial forces) and PSMD (axial force). The particle is scanned in a small displacement steps. To compare the simulations with the measurements we find the plane with a maximum radial force in both.
2.4. Tracking the particle
For radial displacements we determine the position of the particle by analysing its image with an algorithm for tracking the center of radially symmetric objects [26].
3. Results
We perform three experiments using different configurations of our apparatus involving either a PSD, PSMD or both devices. Firstly, we will show the high-bandwidth performance of the PSMD method and compare it with the duo-lateral PSD. Secondly, we investigate the accuracy of the axial force measurements and compare them with the Stokes drag force. Finally, by combining axial forces from the PSMD with radial force measurements using the PSD, we perform full 3-D force measurements on a stuck spherical particle.
3.1. Bandwidth
The real bandwidth of the force detection system is limited by various noise sources in the optical setup, e.g., mechanical noises of the optical components, thermal instability of the environment, etc. While these sources of noise can be suppressed, the primary limitation — the bandwidth of the detector — is much harder to deal with. To investigate the temporal performance of the PSMD we measure the optical force acting on a trapped silica microparticle (1.70μm) with the proposed detection system and compare results with a duo-lateral PSD. The power spectrum density is shown in Fig. 3. Our data shows the frequency response of the PSD begins to decrease faster than predicted by theories of Brownian motion. This is due to decreasing gain of the transimpedance amplifier that converts photocurrent to voltage. In our particular case, this is due to high-pass filtering in the feedback of the amplifier. This serves to lower high frequency noise contamination of lower frequency measurements to prevent aliasing, reducing the effective noise floor of the detector. Excluding this filtering we predict that the theoretical maximum frequency would be in the vicinity of 20kHz assuming no increase to the noise floor. However, removal of the filter from the circuit will likely increase the noise floor as well.
Fig. 3 Power spectrum densities of the radial optical force acting on a trapped silica microparticle ø1.70μm. The bandwidth of measurement for each experiment was chosen such that the noise floor meets the optical trap signals. The effect of surface capacitance on the PSD measurement can be seen with a rapid fall to the noise floor at about 7.5kHz.
For PSD measurements we reach the noise floor at 7kHz while with PSMD we get a higher cut-off frequency, at 100kHz. This gives us an order of magnitude improvement in temporal resolution: from 143μs to 10μs. For our system, the noise becomes dominant at frequencies greater than > 100kHz. Unlike the measurements using a PSD, which fail before the signal reaches the noise floor due to the surface capacitance of the PSD limiting high-frequency performance, with PSMD we are not limited by the detector’s bandwidth and can explore the full bandwidth of the optical trap, i.e., the frequency range where we can measure the optical force acting on trapped particle over the noise floor.
3.2. Axial force
The aperture calibration constant, CA, is required to create a pattern for the axial force measurements. Variations of the choice of CA leads to a spread of axial trapping position distributions. The effect of the different CA values on the axial force measurement of a trapped silica particle (ø = 1.70μm) is presented in Fig. 4. The particle is strongly trapped and will only make small Brownian excursions from the equilibrium. Thus, distributions of axial forces will have high correspondence with the distribution of an overdamped harmonic oscillator driven by Brownian motion.
Fig. 4 a) The axial force distribution for different amplitude mask sizes. The thick black line corresponds to the correct pattern size. b) Change in the measured standard deviation for different sizes of the pattern.
Inaccurate determinations of aperture constant are a systematic uncertainty for the direct measurements of forces. The linear dependence of the position variation with aperture is shown in Fig. 4(b). The correct radius of the pattern was measured to be CA = 290px ≈ 3.1mm (see procedure outlined in Methods section 2.2). When the mask is smaller, the detector becomes more sensitive, which leads to a wider Gaussian distribution of the force. To balance the BP we need to increase kax. If we continue to shrink the radius, the mask will reach the limit where kax → ∞ and become a split detector–circular aperture. Created in this way, the mask has a benefit that both signals — from the inner circle and the outer ring — are measured, while in conventional aperture detection the later is discarded. For bigger masks, kax → 0.5 when R → ∞ and the mask will become a beamsplitter with no position sensitivity. In all cases, except with the correctly sized pattern, the detector will not be universal for different particles and a recalibration will be required. In the case of small deviations from the exact mask radius we get an error in the force of ≈ 0.6%/px. Given that the pixel size is 10.8μm, precise measurements are required to get the correct values of the force.
To evaluate the accuracy of the detection of the axial force with the correct size of the mask, we drag the particle with a constant velocity and measure the axial drag force. The results for the drag force measurements with different velocities ([10, 20, 30, 40, 50, 60, 70, 80, 90, 100] μm/s) are presented in Fig. 5. The error bars on the plot are the standard deviations in the force measurement due to the Brownian motion of the particle. A bigger deviation at the higher velocities is due to the shorter drag time (6.0s to 0.6s). The measurements show good linearity and correspond to the theoretical calculations of the drag forces within 5%. The underestimation in the force is expected as we are not measuring the back-scattered light which plays an important role in the axial force [11]. Although the silica particle has low reflectivity, back-scattered light contributes twice its original momentum to the axial force while forward scattered light only contributes a small change in momentum.
Fig. 5 Axial Stokes drag force measurements of the spherical silica particle (ø1.70μm) using PSMD. The corresponding velocities are [10, 20, 30, 40, 50, 60, 70, 80, 90, 100] μm/s. Error bars are the standard deviations of the Brownian motion of the particle.
3.3. 3-D force measurements
For the second experiment we use a silica particle which is stuck to a glass substrate by the materials’ intrinsic adhesive contact forces. We perform a 3D scan of the particle using a piezo-stage and measure optical forces using a PSD for the radial components (X and Y) and PSMD for the axial force measurement. Since the optical forces are measured from the deflection of the beam and the change in the divergence of the transmitted beam, the optical forces can be measured even if the particle is stuck to the slide. A particle moving with Brownian motion is used for the initial calibration, but once calibrated, the system can be used for other particles, and Brownian motion is no longer required. The Optical Tweezers Toolbox [27] is used for a simulation [28] of the optical force distribution (Fig. 6). The simulation parameters were chosen to match the experimental values and are within uncertainty of the parameters of our system. Here we use a silica (n = 1.45) spherical particle with diameter of 1.72μm in a linearly polarised Gaussian beam. The full experimental and theoretical 3D force distribution is shown in the Fig. 6. The measurements show excellent correspondence with the simulation.
Fig. 6 3-D optical force measurements of the silica microparticle (ø1.70μm) stuck to the slide. a). Radial component of the optical force Fy. To match the simulation and experiment the plane with a maximum radial force was chosen for both cases. b). Axial force in the same plane as a. c). The comparison of the measured and calculated 3-D distribution of the total optical force.
We can see good agreement between the experiment and the simulations. It is important to note that we do not collect the backscattered light, which was shown in [11] to play an important role in the axial force determination. Some discrepancy can also be explained by the drift in the stage in the axial direction and proximity to the slide where multiple reflections at the particle–slide interface can result in errors in the force measurement compared to a floating particle.
3.4. Comparison of the PSMD with a split detector
Split detectors are widely used in optical tweezers to measure the position of a particle in both radial (QPD) and axial (aperture) directions. This measurement is often performed by measuring the momentum of the beam, i.e., the actual measurement is an optical force measurement, and converting this to the particle position using the spring constant of the trap. Therefore, it is important to compare the performance of the split detector with a PSMD. The main advantage of the split detector over the PSMD is its sensitivity. If the light beam has a symmetrical, monotonically decreasing profile, the split detector will provide the highest possible signal. The drawback is that this signal depends on the size of the pattern and its structure. Also, the response of the split detector to non-monotonic distributions will not be correct. This makes the usage of the split detectors for direct force measurements without re-calibration for each new particle or beam impossible as the universality of the method will be lost.
4. Discussion and conclusion
We have shown that the PSMD has a much higher bandwidth comparing to a typical PSD — resulting in one order of magnitude increase in the bandwidth in our system (although, as we pointed out earlier, higher bandwidth PSDs are available). We are limited by the noise but one can expect a much higher bandwidth (of the order of 10–100MHz) if other sources of noise are suppressed and a faster, low noise, photodetector is used. One of the factors that decreases the bandwidth of the PSD is the size of the chip which determines the capacitance of the sensor, which in turn limits the response time of the detector. In PSMD the beam is separated by the amplitude mask and this problem can be addressed by focusing the split beam onto a photodetector, so the smallest detectors can be used without loss in the resolution and linearity. Also, both PSD and BP in our system are silicon based detectors and have a low sensitivity at 1064nm. This may introduce an additional low-pass filtering of the recorded signal [29].
A linear gradient pattern is used for the radial force measurement and provides a signal equivalent to that of the PSD, as shown in Fig. 3, where the power spectrum densities for both the PSD and BP are the same for frequencies below 100 kHz. The linearity of the filter means that the detection is proportional to the force and once calibrated should in principle always measure the radial (transverse) optical force for a wide range of particles, both spherical and non-spherical.
The accuracy of the PSMD for the axial optical forces is estimated using the measurement of the Stokes drag force on a sphere. The results show excellent linearity for the different velocities and very little spread in the measurements demonstrating repeatability. However, the size of the pattern plays a crucial role and the deviation of the radius of the pattern of just 10% will cause a 19% error in the force measurement. Accurate calibration of the pattern is thus essential. However, this limitation is imposed by the optical system and any method that measures the axial optical force will encounter this same problem.
The theory behind the split detectors shows that they are the special cases of the masks for the radial (QPD) and axial (aperture) force measurements with infinite slope of the gradient. This leads to an increase in the sensitivity to the displacement at the cost of losing linearity. Therefore, split detectors cannot be used for the direct optical force measurement, but remain valuable for particle tracking purposes.
While the position-sensitive masked detector can be implemented with a stationary filter, the use of the DMD is preferable and has many advantages in the usability of the system. It allows the implementation and testing of different patterns under the same conditions; characterisation of the light distribution in the detector’s plane; convenient electronic alignment of the detector.
3-D forces can be measured using a combination of the detectors (Fig. 6) or by time-sharing the patterns on a single DMD. However, in the later case, the bandwidth of the measurements will be limited by the framerate of the DMD, so the actual 3-D force measurement bandwidth will not exceed a few kilohertz which is comparable with the bandwidth achievable with a camera.
The proposed method of measuring optical forces using amplitude filter masks (PSMD) expands the universal direct force measurement method to a high bandwidth region which is inaccessible with existing detectors. Moreover, it can improve the accuracy of force detection by accounting for the variations in the transmission of the condenser for different angles and aberrations in the optical system.
Funding
Australian Research Council’s Discovery Projects (project number DP180101002).
Acknowledgments
A.V.K. was supported by UQ International Scholarship.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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