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Abstract
The use of the radiation pressure of a laser field, as an effect of the momentum transfer of the absorbed and re-emitted photons, suggests rather a complementary than an alternative possibility for metrology to generate calibration forces or to calibrate the optical power directly traceable to the International System of Units (SI). This paper reports a method and experimentally measured evidence on options to extend the effective use of radiation pressure for generating optical forces in the sub-microNewton (μN) range. Among other features and results presented, we emphasize the variability in controlling the accuracy of these forces through the proper utilization of the power of a multi-pass laser beam (semi- or completely) locked within confined systems. The direct measurements of these forces, augmented due to the partial or total momentum transfer of the photons of a multi-pass laser beam extended from several hundreds of picoNewton (pN) up to sub-μN range for the same power of laser source, are done by a precision force measurement system and compared with basic theoretical computations. We also discuss the opportunities to probe the fundamental physical limits associated with this method and to the considerable extent other competing contributing effects that might be regarded as sources of errors in metrological tasks.
© 2017 Optical Society of America
1. Introduction
In recent years, the force sensing in optomechanics and in a wide variety of quantum motion experiments have garnered significant attention [1–3]. The interest on these subjects has intensified after the detection of gravitational waves by the efforts of the multinational Scientific collaborative projects, such as LISA, LIGO, and Virgo [4, 5]. These initiatives have led to a new leap in the developments of precision optomechanical systems [6–10], thereby impacting on technological progress and advancement in other practical aspects of regular life. In these regards, the technical capabilities of accurate and precise measurements undeniably play a crucial role in many scientific discoveries.
From a metrologist viewpoint, these advancements can be grouped into the following interrelated research areas:
- In addition, a special attention requires the broad field of the quantum mechanics where breakthrough studies are continually being reported. For instance, the measurement of a 1.7 yg mass (corresponding to the mass of single proton) using a nanotube resonator [18]. The cooling of the molecules to the ultra-cold temperature states (in the sub-mK to 1 K range for the needs of biological and medical applications) by using tailored laser fields [19]. These all are included in the growing field of “Cavity optomechanics” [3, 20–26].
Without undermining advances achieved in the aforementioned fields, still there are several actively ongoing disputes within the National Metrological Institutes and International Committees for the measurement of the forces (or masses) below μN level. Historically it is developed such that the concept of force is defined as rate of change of the momentum, whereas in order to obtain a quantitative knowledge about it, the measure of the force should contain at least its magnitude, uncertainty, and units of this measure taken within some scale system. To a certain degree of the simplification in the SI system the original definition of the force unit reads as: “one Newton is the force that is needed to accelerate one kilogram of mass at the rate of one meter per second squared in direction of the applied force” [27], considering also that the mass of the object on which the measure is taken remains unchanged.
Hence, to perform accurate and precise measurement of the forces, irrespective of whether it is in the order of meganewton or in the order of several pN or less, the measurements of the ground-based quantities, first, must be made with necessary accuracy and precision traceable to the SI system. In the case of meter (base unit of length) and second (base unit of time), the measurements have been already connected to the fundamental constants within the framework of quantum mechanics, providing a reliable means for SI-traceability. For the base unit of mass, the kilogram is still connected to a real physical artifact known as International Prototype of the Kilogram (IPK). The IPK by itself does not have uncertainty because the unit of mass and this artifact are fundamentally considered to be the same, and only the smaller weights (submultiples of this artifact) do possess uncertainty with respect to the IPK due to the real technical limits of their manufacturing and successive weighing executions. These technical limits impose uncertainty in determination of the mass of the IPK itself, according to measurements realized at PTB it is determined with 2.8·10−8 relative uncertainty (see Fig. 1). Hereafter, the uncertainties of ever smaller series of weights are growing and the relative uncertainty in terms of the IPK artifact amounts to be on the order of unity by the extent when the scale of these small artifacts reach the size of a μg [28–32] (see Fig. 1). Therefore, the accuracy and the precision of the force measurements are directly subjected to these uncertainties. Moreover, in the weighing and force measurement processes, the proper evaluation of the obtained results depends also from the value of the local gravity at the site of measurements. In other words, the force to be measured is the dead weight of the artifact resulting from the vertically acting gravity force. Therefore, all these measurements are directly connected to the gravitational field at and around the site of the measurements [6–10]. The gravitational field in itself is connected to the physical constant Big G - the gravitational constant, whose numerical value still requires better determination. In spite of the efforts made by research metrologists around the world, some of the best obtained values of the gravitational constant are determined with relative standard uncertainty below 2·10−5, whereas, the relative difference between the most of the reported values exceeds the 4.7·10−4 according to CODATA (Committee on Data for Science and Technology). In contrast, the relative standard uncertainties of the values obtained for other fundamental natural constants are notably better, for example the mass of the proton, the mass of the electron, the Planck and the Avogadro constants are all measured to have a relative standard uncertainties of 1.2·10−8. In the framework of weighing science and technology, however, this problem is partially solved by adaptation of the balancing/compensation measurement principle [12, 13] and by the use of apparatuses such as gravimeters [10, 33] and Watt balances. Besides of these considerations, such measurements are recognized as a static measurements for the invariant masses. Therefore, this approach hides many dynamically occurring effects, or in some cases introduces to the scientific community a measurement tasks, which are sometimes challenging to reproduce or lacking a convincing agreements after the measurement data are post-processed [34–36].
Fig. 1 Expanded uncertainties (k = 2) of the PTB’s secondary standards for the realization of the mass scale. Absolute values U are shown by the blue triangles (in mg); Relative values U/m are denoted by the red circles. Reproduced from [29].
In this respect, the questions to be tackled are as follows: a) how does the measured force in the μN, nN, pN ranges and below correspond to the commonly known forces that are being measured with a high level of confidence and traceability with respect to the IPK? [5, 11–13, 18, 29–31, 33]. More importantly, b) in the absences of the force standards at these small scale levels, how to relate to these infinitesimal forces and make them of more accurate and precise use?
As it has been reported by Kim, Pratt et al. [37, 38] in “SI traceability: Current status and future trends for forces below 10 microNewtons” and “Report on the first international comparison of small force facilities: a pilot study at the micronewton level”, there are several methods and facilities realizing force measurements below the μN level. The most recognized facilities are:
- The use of the influence of the earth’s gravitational field on the mass artifacts (widely accepted by all NMIs)
- Optical radiation force, which is the force produced by the momentum of the photon.
In our view, from the all listed above the optical radiation force based method is attractive and relatively straightforward to employ for precision force sensing. Moreover, in the light of ever growing trend, both in academy and in industry, towards the integration of the laser-based applications with the electronics units this method has a potential to be further studied in detail. In addition, such a direct measurement of the momentum of the photons when they exert a force on the real physical object can be considered as an integral means for improving the link between the classical and quantum mechanics, meanwhile fulfilling the important traceability requirement. As noted earlier, rather a complementary than an alternative possibility provided by this method is constituted additionally by the fact that the produced forces can be generated also dynamically.
While in the field of theoretical quantum mechanics there are various debates about the behaviour of the momentum of the light at the edge of the light-matter interaction [39–41], the other fields of science and technology consistently provide supportive evidences on application and utilization principles of some of its effects, such as the transfer of the weighted momentum of the laser fields to the object with reflective surface [42–53]. At the footsteps of this new paradigm for “Measuring laser power as a force” (as have been coined by Williams, Lehmann et al. [45, 46, 53] in line with similar earlier works reported by Wilkinson, Nesterov, and others [42, 43, 47, 49, 54, 55]) we took the next steps forward to realize measurements differently and extend the effective use of the radiation pressure primarily on the behalf of optical and mass metrology fields.
In this paper we demonstrate a method of optical force measurements, based on a relatively simple configuration of the experimental setup including the optical components for achieving multiple reflections and the force measurement setup. The comparison of the initial experiments with the theoretically calculated values illustrate the wide range of possibilities suggested by this method, that also are compatible with the results obtained from other methods which use the same effect of optically generated forces. In the earlier reports (see Refs. [43–47, 53, 54, 56, 57]), the primary investigations concerning the metrological use of photon momentum for force generation, force and laser power measurements have employed single reflections. In our case, the initially demonstrated experimental setup makes use of multiple reflections of the laser beam within two quasi-parallel flat surface mirrors suspended from the precision force measurement system. By this the generated forces and their further detection are enabled in the sub-μN range which can already be compared with the lowest limit of SI traceable mass artifacts.
The paper aims also at reassessing the results of earlier works which can be further exploited to systematize the research efforts on the metrological aspects of using the optical radiation force produced from single or multiple reflected laser field systems, and to outline some future possible research directions. In rest, the paper is organized as follows. In Section 2, we provide the derivation of the basic theoretical considerations for the radiation pressure and the underlying importance of the geometrical configuration. In Section 3, a discussion on the conventional geometrical models of the laser beam reflections is outlined. Section 4 briefly reviews the main approaches and the experimental methodology used for precise measurement of the small forces produced by radiation pressure. Section 5 is devoted especially to present the actual implementation of our method and the results achieved from the experiments.
2. Basic theoretical consideration
The use of the multiple reflected laser beam in the calibration and standardization procedures requires that both the laser power and the reflectivity coefficient of the mirrors are determined precisely in a traceable manner as it was also noted in [38]. However, the hypothesis in the same report that “a high-power laser beam is needed to produce a force at the micro- or nanoNewton level, high enough to be compared with another force produced from other methods, such as electrostatic or gravitational force for verification of optical radiation force standards” has set an underestimated statement expressing a comparatively segmental viewpoint in relation to the actual experimental possibilities and theoretical limits when considering multiple reflections. Typically, the difficulties are accounted to several other effects, which may occur and thereby produce spurious forces (one of them is reported in [56]). However, during the measurement tasks, these competing actuation mechanisms can be suppressed by carefully controlling the system (as it was reported in [42]).
Remark: There is a well developed practice widely used in the cavity optomechanics for various proposes employing a spatially arranged impedance-matched Fabry-Perot cavities [48, 50] or generally using a stable resonators where the locked portion of the laser fields are circulating and being compared with the other portion of transmitted or injected field. In this method a biconcave type of mirrors are typically employed, provided also that they have a transmitting back plane or an option for the injection or ejection of the laser field inside or outside the cavity. However, it can be also used a plane-parallel, hemispherical, astigmatic, and other types or mirror assemblies. Indeed, these options with stable resonators are also a candidates for further investigations from the metrological standpoint. The measurements described in the literature have certainly a potential for closer reexamination in terms of tractability to the SI system. Most of these apparatuses and the optical techniques are taking the analysis of the measurements based on obtained mechanical transfer function of the resulted motion of the reflectors or of the trapped object (atom, molecule, particle, etc.) as a function of frequency. Thus, the detectors are essentially probing the shape of this transfer function on the frequency domain mainly in comparison with the parameters of the applied laser source. It is important to note, that these special cases also we attempt to include in our future works, however, based on metrological aspects which are systematically discussed in this paper. Similar to works reported in the literature [42–47, 53, 58], which are given as an approach for direct and traceable force and laser power measurements, using only the case of single reflection of the laser beam from the surface of different type of objects.
We found that the discussions against the use of the laser pressure and the use of the enhanced method of optically generated forces due to multiply occurring specular type of reflections are rather based on several technologically imposed barriers from other class of apparatuses, than a kind of a fundamental physical limit. However, we admit that the generalizations following the experimentally achieved evidences from our side could be further refined. In the light of ever improving technical capabilities, it would be important to probe and characterize each effect or method experimentally for identification of all possible competing contributions beforehand and only then apply for broad applications and for use of it to fulfill ever growing practical needs (for instance, in the standardization tasks, AFM spectroscopy, etc.).
Below we revisit the basics of the theoretical formulation for the optical radiation force. One of the conceptual advantages of this method that is also adopted in this work (see section 5) is based on comparison of experimentally obtained results with this initial theoretical model that consist minimal mathematical formalism. The main idea is to compare the measured and derived force values, which are produced by the momentum of the photon once it is being absorbed and re-emitted by an object. Here, we focus our interest on the case when the object/material has a surface with near to ideal reflectivity. On the account of this ideal case, the energy/momentum losses of the active laser field can be considered negligible after the photon’s are absorbed and re-emitted. The mathematical formulation is given as follows:
here E is the total energy of the particle, p is the momentum of the particle, c is the speed of the light. Under the assumptions that the mass of the photon is m = 0 and the speed of the photon is v = c the Eq. (3) for the photon momentum in the vector form is: where n is a unit vector indicating the photon’s motion direction. The force produced by the photon can be derived from the time-derivative of its momentum; therefore, for a given power of the light (laser beam) it yields:Upon hitting the object, the laser beam can be reflected, which yield |n| = 2, absorbed |n| = 1, or transmitted |n| = 0. By denoting the corresponding fraction of the reflected light as RL, for the absorbed light AL and for the transmitted light by TL
and when considering for an object as a mirror (no transmission TL = 0), Eq. (5) yields while having Eq. (6) already in the form of RL + AL = 1 and assuming that the laser beam is impinging normal to the surface (otherwise the incidence angle should be considered additionally in a cosine term), then the Eq. (7) can be rewritten in the following form:As an example, for a perfect reflective mirror (assuming RL>99.9 % of reflectance) the force exerted by the laser source with 15 mW power can be approximated theoretically as 100 pN.
This idea, as discussed, is well known and has been already demonstrated experimentally in the field of precision metrology by several research groups and authors [42, 43, 57]. Equally obvious, however, less accountable in all previous studies is the fact that the remaining portion of the reflected laser power (in the above example case the remaining 99.9 %) has still a momentum that may be measured as a force. Therefore, considering that there is a strictly defined geometrically confined system of reflectors such that the path of the laser beam could be folded/trapped within by big number (N → ∞) of reflections then the remaining power of the laser beam after each reflection can be harvested completely in controllable manner. This is reminiscent to a certain class of optical resonators, also known as; Fabry-Perot cavity [48, 50], Herott cells, White cells and other common optical configurations alike. In the scenario described above, if the borders/surfaces of the confined system have the same coefficient of reflectivity (this is simplified configuration, otherwise, in the case of different coefficients the key point would be to define the coefficient of the reflectivity of the every reflector) then at every next reflection the power of the laser beam will fall in accordance to some geometric series until it vanishes completely. Thus, the total sum of forces produced by the N number of reflections of the laser beam would approaches asymptotically to some finite value.
In Fig. 2(a), the residual power of the laser beam as a function of the number of reflections for the two different values of reflectivity is presented. At the lower limit it can be theoretically extrapolated down to single quanta (power/energy of a single photon Eend−1) of light, then after the last reflection (Eend) to the absence of it at all. In Fig. 2(b), the total force produced by the 100 mW laser source as a function of number of reflections calculated by Eq. (9) is presented. Figure 2(c) displays the typical dependence of the generated total sum of forces for the different number of reflections as a function of coefficient of reflectivity.
Fig. 2 a) Residual power of the laser beam as a function of number of reflections, for reflectivity of 97.5 % (blue circles) and 99.5 % (red triangles). b) Total force produced by the 100 mW laser source as a function of the number of reflections, for reflectivity of 97.5 % (blue circles) and 99.5 % (red triangles). c) Total force produced by the 100 mW laser source as a function of the reflectivity, for different number of reflections N = 3,...,21,...10000.
In accordance to this theoretical calculations, the saturation force produced by the 100 mW laser source approaches the 133 nN asymptote after of about 1500 reflections for the surfaces with 99.5 % reflectivity, and to the 26.35 nN after of about 250 reflections for the surfaces with 97.5 % reflectivity, whereas during the first reflection it produces only 663 pN, and 650 pN forces, respectively. From here, some of the prospective questions that need to be answered are: whether is it possible to completely utilize the power of the laser beam to the necessary accuracy and precision and reach saturation force limit? Otherwise, to what extent the power of the laser can be used in metrology for the force sensing needs in relation to different parameters of the system? For instance, what is the range of interest in using the high power lasers (1 mW up to 1000 mW or higher) or possibilities to increase the number of reflections and thereby improving such factors as; reproducibility, range, precision and the stability of the force and laser power measurements?
3. Geometry of the specular reflections and the shape of the mirrors
In general, there are various types of mirrors and their geometrical configurations, which may provide multiple (multipass) reflections. The typical 2-D configurations used in many different optical applications are displayed in Figs. 3 and 4 [11, 16, 55, 59–83]. Here, we address initially the geometry of the flat-flat, concave-concave and concave-convex surfaced mirrors facing by their reflective coated sides to each other. These configurations are similar to the different types of optical resonators [84].
Fig. 3 Typical 2-D arrangement of the flat surface mirrors. a) single mirror, incidence angle of the laser beam is parallel to the normal of the surface, b) single mirror, incidence angle is θ, c) two quasi-parallel mirrors with relative α angle, incidence angle is θ, d), e) and f) are the same as in c) with α>θ, α=θ and α<θ respectively.
Fig. 4 Typical 2-D arrangement of the concave and convex surface mirrors. a) single reflection of the laser beam inclining normal to the surface of the concave mirror at the point of reflection, b) inclining at an angle θ, c) multiple reflection of the laser beam in the concave-convex mirror assembly, d) multiple reflection of the laser beam in the concave-concave mirror assembly, e) typical arrangement of the concave-concave surface mirrors, conventionally known as the Herriott Cell. This cell is primarily used in multipass or long path absorption cells (a similar configuration of this kind are the White Cells).
As illustrated in Fig. 3, for commonly known 2-D configurations of the flat surface mirrors the most important parameters of interest are the incidence angle of the laser beam θ and the intersection angle between the planes of the mirror surfaces α (Dihedral angle). In Fig. 3(f), can be identified collective representation of all the simplified cases from a) to e). In the case of flat surfaced 2-D configuration of the quasi-parallel mirrors, if the first and the last incidence angles of the laser beam and the dihedral angles are defined then the number of reflections can be calculated with minimal complications:
where i = 1,2,...,n is a natural number for designation of the number of the reflections, and γi is assigned to be the incidence angle of the corresponding reflection. The non-integer values obtained for i from Eq. (10) should be truncated towards the nearest integer.Above adopted approach for calculating the angles or the number of reflections may describe an ideal 2-D arrangement and it may be invalid in most cases of the 3-D mirror configurations, where additional trigonometric considerations are required. In the case of the laser beam directed within the plane perpendicular to the intersection line of the mirror planes, the calculations still can be made by Eq. (10) and as sketched in Fig. 3. This kind of consideration in the experimental work cannot be applied directly; therefore, it would be required to make distinct angular and position adjustments with an adequate level of accuracy and precision. The 3-D sketch of the planes of two flat mirrors, which are arranged relative to each other with α and β angles, developing along the x and y axis, respectively, is illustrated in Fig. 5. The path of the reflections is dependent from the incidence angle (or its projection in the x − z and y − z planes) and from the dihedral angle. The computations are usually performed with the help of numerical models, see [78].
In addition, as shown in Fig. 5, the path (pattern) of such reflections is formed as a curvature that cannot be expected to achieve in 2-D model. Note that a similar character of the curved line appears when the mirrors are of convex or concave type. Such a feature can be used for several purposes; for instance, as an additional useful means for the investigations or for the visualization of the points of the reflections. It is noteworthy to mention that a critical consideration of the parameters of the laser beam and respectively their influence on the final result of the measurement are yet to be considered in further experiments in details. In the listing below, we describe the possible sources of errors and parameters that may influence the measurements:
- Defining the incidence angle dependent losses (including the scattering of the light beam due to the flatness of the surface, typically given in terms of the proportional ratio from the wavelength λ/4, λ/10, λ/20,...λ/200)
- The development of a method for achieving maximally possible dense pattern of the reflection points therefore improving the light power harvesting strategies,
- Application limits in terms of reflection phase shifts at planar interfaces, convergent or divergent laser beams,
- The focus distance of the laser beam and means to control it,
- Laser profiling, cross-sectional area of the laser beam,
- Polarization effects (s– or p–type, dependent from the type of the laser), angle of incidence, the material of the mirror coating, the surface roughness of the mirror, etc.,
- The uniformity, angular accuracy of the faces and the surface smoothness of the coating material,
- The operational wavelength, intensity, and the power ranges of interest for laser sources,
- The size and weight limitations of the mirrors,
- The tolerance requirements for the mount and the electro-mechanical guidance systems,
- The general environmental conditions in which the experiments are being performed. Parameters including the vacuum conditions, distribution of the thermal and optical radiation inside and around the mirrors.
In addition, there might be other unknown factors of influences that can induce uncertainty or be the hidden parties in overall uncertainty budget of the measurements, therefore we note that they are not only limited to the cases listed above.
4. Direct measurements of the photon’s momentum as a force signal
In order to perform the force measurements at an adequate level of accuracy and precision, a special setup is necessary to fulfill several basic (but fundamentally unavoidable) requirements. Theoretically, it can be assumed that the laser beam/spot is produced from a point source and is being reflected from the surface of the mirror/object at a point with negligible size (see Section 3). Therefore, in terms of classical mechanics, the mirror/object can be considered as a point mass and consecutively conduct the force measurement of the photon’s momentum as a respective change in the position of the point mass, which may lead further simplification of theoretical computations. Apparently, due to objective reasons in the real experimental conditions neither the spot of the laser beam nor the object on which the laser is being reflected can be assumed such. In fact, there are known setups (in PTB and NIST and elsewhere) by which the measurement of the photon’s momentum has been performed. These setups adopt deflection-based force measurement method, for example with a reflective coated cantilever [43, 54, 58]. A short summary about their basic parameters are provided in table 1 and Fig. 6.
Fig. 6 Simplified mechanical schematics of the active probes of both experimental setups. Depicting the principle of the optical force–displacement measurements. Reproduced from [42] for cantilever and [43] for disc-pendulum.
There is also known a precision optomechanical apparatus, reported in [44], which uses an injected low power laser beam in an optical cavity thereby generating a low forces in the order of several hundreds of fN. The method uses a deliberately tuned coherent reflections within the cavity, similar to a principle employed in coherent etalons. This study differs from the discussion we present, because in our case we encounter to non-overlapping specular reflections.
An important aspect of the laser beam reflection in several experimental investigations have been yet addressed indirectly, namely the change of the cosine term due to the change of the incidence angle of the laser beam (see sketch in Fig. 6 and table 1) during the continual deflection of the probes. In accordance to the information gathered from both studies (table 1) and calculation of the developed angle by d-displacement and l-hypotenuse, it can be only assumed that for the single reflection case, the change and the influence of the cosine angle term is insignificant. However, in case of the multiple reflections, it may introduce additional considerations. Contrariwise, in other research fields (demonstrated in [16, 59, 79–81]), this approach of multiple reflections and change of the incidence angle are deliberately manipulated in order to achieve the required sub-pN resolutions in the length measurements or for the measurements of the rotation angles at fractions of nrad.
In reference to these early developments, in force sensing by the use of the optical force of a single reflection of the laser beam, we considered the following: by increasing the power of the laser beam and by folding the laser beam such that multiple reflections can occur at first without overlapping onto each other, it is theoretically not prohibited to expect that a forces with larger magnitude can be produced therefore making their measurements accessible to commercially available and commonly used ultra-precise force and weight measurement systems. This argumentation and the necessity to develop a new but different type of experimental setup are also based on the facts that neither the cantilevers nor the disc-pendulum type facilities satisfy the following required parameters:
- The geometrical dimensions of the reflective surfaces should be big enough in order to achieve accurate allocation of the reflection points, which can be seen also as a combination of parameters necessary for achieving stable resonators. However, one should consider to avoid or oppositely to achieve a distinct control over the optical interference effect due to overlapping laser field of modified character, because the magnitude of their influences are yet unknown. Also, this would certainly be dependent at least on the actual type of laser in use, its power, and the cross-sectional area of the laser spot. The cross-sectional area (considering for simplicity of a circular type) typically vary from several μm to several mm in diameter for most of conventionally used laser systems.
- The object with the reflective surface (mirror) should be enough robust and bulky in respect to both of the following competing aspects; the stability of the thermo-mechanical properties of the coating material and of surrounding environment against the power of the applied laser beam, and the actual physical weight of the object that is needed to be minimal in order to diminish gravimetric influences during the force measurements.
In our measurements (presented next in Section 5), the initial choice for the reflective surfaces were made in favor of the conventional optical mirrors due to economical and maintenance reasons. Depending on the material type, geometrical size and shape (flat, concave, convex), the mirror weight varies from several tens of grams (1/2″ mirror) to a kilogram (4″) range. One of the options to minimize gravitational influences caused by such a dead weights of reflective objects is to make the force measurements in horizontal direction while the mirrors are suspended from the Force Measurements System, or FMS. Therefore, a FMS with closer to nN resolution would be required for the measurements of horizontally directed force, which may carry meanwhile a dead weights of these unusual orders. Such a system has already been developed in the Technische Universität Ilmenau (TU Ilmenau), where two Electromagnetic Force Compensation, or EMFC, balances have been incorporated together (see sketch in Fig. 7). A detailed description of the FMS has been reported in [85, 86] with an earlier prototype model in [87], the actual application where the FMS was used is reported in [88]. This FMS is unique by its characteristics, among others; the most important factor of interest is the resolution of the relative force measurements. It is the ratio of the standard deviation of the measured signal of the horizontally directed forces σ(FH) < 20 nN over the dead weight FG = 13.6 N (σ(FH)/FG ≈1.47 ppb) of the object on which the horizontally directed forces are acting. In this FMS, the calibration of horizontally directed forces has been done in relation to the applied forces produced by the external voice coil actuator. This method was chosen because the conventional method involving the use of the influence of the earth’s gravitational field on the μg mass artifacts is a) impractical to apply for horizontal directed force measurements, and b) due to the limitations for the dynamical characterization of the acting forces. Series of extensive crosschecking reference measurements were performed to compare the force scales of the external voice coil actuator with the force scale of the internal voice coil actuator of the FMS. The input parameter for the voice coil actuator is the applied electric current (in steps of 100 nA) generated by a precision current source.
Fig. 7 Functional diagram and geometrical configuration of the FMS, mirrors and the laser source (see body text). Presented as two parallel shifting independent pendulums, dashed line in the right EMFC balance indicates initial state. (1) common bearing plate, (2) separate elbows for mechanical adjustment of EMFC balances in horizontal plane, (3) internal assembly of EMFC balance joining the three main internal components the internal voice coil actuator, proportional lever arm, and the positioning sensor, (4) EMFC balance, (5) load carrier, (6) External voice coil actuator located either at A or B for generating a horizontal force on the load carrier during reference force scale comparisons, (7) two quasiparallel 1″ square mirrors adjusted from loading carrier (the size is exaggerated for better visibility).
By this calibration method, at the levels of 20 nN (resolution/floor noise) up to hundreds of nN an inaccuracy of approximately 5 % to 7 % was introduced. Therefore, a more rigorous calibration method (force sensing) is required to improve the accuracy of the horizontally directed force measurements particularly for this FMS and for the forces below 10 nN in general. The accomplishment of this task of a practical importance was one of the original motivations behind of the idea of using photon’s momentum (utilizing power of the multipass laser beam) to generate forces of this order and/or manipulate the dynamical performance of the setup. The initial realization of this idea suggests an intriguing perspective, as is presented in this paper, to use the photon’s momentum in more broad-based metrological and scientific applications.
5. Experimental setup and force measurements
While the geometry of a suitable mechanical arrangement has been carefully adapted to the existing FMS [85, 86] illustrated in the sketch and images in Figs. 7 and 8 omitting most of detailed particularities, it is to be mentioned beforehand that the rearrangements made on the existing FMS are yet temporary and being far from the desirable and complete state of experimental conditions. The results achieved and presented in this section are only reflecting the proof-of-concept measurements in terms of general confirmation. We plan to further improve and refine the experimental setup. Also, all measurements presented in this section were carried out in air and only a partial use of passive thermal insulation was considered for shielding the setup (in the images the setup is uncovered from thermal insulators for better visibility). Further improvements in the force measurements in the vacuum conditions are yet expected. The effects such as the air buoyancy, the non-uniform temperature distribution, or the thermal-radiation due to the scattered light and other effects arising from material-laser interaction were considered indirectly. Also, in the laboratory during the 30 min period there are non-linear temperature fluctuations typical having maximum amplitude of 10 mK (uncontrolled conditions). These fluctuations transfer through the thermal insulation of the FMS and as a result around the FMS the temperature changes linearly with absolute value of about 5 mK (see [85]. The cumulative effect of all these influences are included in the total uncertainty of the force measurements as a single value of the standard deviation, i.e. the “floor noise” of the measurements.
Fig. 8 a) Typical pattern of reflections occurring on the single mirror due to the relative angles between mirrors (α and β) and incidence angle of the laser beam θ, b) Simplified mechanical arrangement of the measurement setup. Both EMFC balances are working independently from each other, fastened to the common bearing plate and carrying mirrors whose reflective surfaces are facing to each other. c) Image of the typical configuration for the measurements where 21 (10 and 11) reflections are achieved. d) Image of the 9 (4 and 5) reflections. Due to safety considerations and insufficient visualization capacities during the photographing of c) the laser power was preset to lower values as 10 mW to 15 mW, in d) the laser power was 814 mW which shows the reflection spots visually more rotund.
Figure 8 displays a) the typical pattern of the reflection spots on a single mirror, b) the schematics of the simplified mechanical arrangement of the FMS with mirrors and the laser source. For visualization purposes, two different images of the experiments are provided for 21 (10 and 11 reflections on each mirror) and 9 (4 and 5) reflections, in c) and d), respectively.
As a laser source, we used a visible laser with a wavelength of λ ≈ 420 nm (±10 nm). The optical power was measured to be approximately Power = 814 mW. Due to the low output stability of the laser power (arising from the unstable frequency/amplitude modulation led power fluctuations) and the inaccuracies in the reference power measurement, the total uncertainty is estimated to be about 10 %. The mechanical alignment of the laser source was adjusted to achieve different incidence angles of the beam to the surface of the mirrors. We used dielectric 1″ square mirrors whose coating material provides a reflectivity of RL ≈ 99.9 % for the corresponding optical bandwidth of the laser (as provided by the manufacturer). Due to geometrical limitations and the incomplete state of the experimental setup, it was possible to achieve a maximum of 21 reflections on both mirrors. The following computation principle has been considered to analyze the measured force signal. Let us consider 21 reflections as an example. At each mirror, a group of 10 and 11 reflections occur. Each group produces a force on each mirror. The sum of all forces produced by impinging laser beam at each mirror is described as FP10 = ∑ F2,4,...,10 and FP11 = ∑ F1,3,...,11 (see Fig. 8, Eqs. (8) and (9)). Therefore, the measured signals from first and second EMFC balances contain F1 = −FP10 + Ferr, and F2 = FP11 + Ferr (see Figs. 9(a) and 10), where Ferr – is considered to be the common error in both EMFC balances originating mainly from the ground vibration noise. Thus, taking the difference of the signals, the total sum produced by the laser source would be equal to FP(total) = F2 − F1 = FP11 + FP10 which is required to measure experimentally (see Figs. 9(b) and 10) and compare with the theoretical calculations from Eqs. (8) and (9).
Fig. 9 Measurements of forces obtained for 21 reflections of the laser beam. a) Measured force signals from the EMFC balances, b) the difference of the signals, c) the change of the air temperature inside the housing (surrounding the mirrors).
For initial test measurements, the following relatively simple and straightforward procedure was considered. The laser power was switched on and off periodically, and the signal from the FMS was recorded. An example of the measurements is provided in Figs. 9(b) and 10, for 21 and 3 reflections, respectively. Three Pt100 temperature sensors were used to measure changes in the air temperature surrounding the mirrors, during the course of each experiment (see Fig. 9(c)). The sensors were arranged inside the housing at different heights along the center axis; in relation with the position of the mirrors they are located as approximately 25 cm above (black curve), 20 cm below (blue curve), and 30 cm below (red curve). Yet, we did not observe any distinguishable or significant evidence showing a corollary between the force signal due to the laser signal modulations and the change of the temperature at the measurement points. In future, sensors with better characteristics of response time and precision of the temperature measurements will be required in order to obtain a complete map of the temperature distribution in the close vicinity of the mirrors.
Several independent measurements were performed on different days and time periods. After every set of measurements, an angular adjustment of the mirrors and of the incident angle of the laser beam were made to achieve different number of reflections. In Figs. 9 and 10, we show only two examples from many consecutive measurements with the laser power modulation in the 10 s period. The mean values from all measurements with 10 s modulation periods, see the circles presented in Fig. 11(a), and the standard deviations (k = 2) associated with these values at each step response were collected after settling time of the FMS is reached. The results as a function of the number of reflections (as 3, 9, 15, and 21) are compared with theoretical predictions made by Eqs. (8) and (9). During these initial experiments, we observed reproducible values in the magnitude of the measured forces also for different modulation periods (0.5, 1, 2, 5, 10, 15, 20, and 30 s).
Fig. 11 a) Comparison of measured and theoretically calculated force values as a function of the number of reflections. In circles (dashed) are marked the measured mean values of each step response without offset correction and the error bars display the combined standard uncertainty of all measurements. The solid lines are obtained from calculations by Eq. (9). b) Standard deviations attached to the each measured mean value of filtered difference force signals in case of 21 reflections. Shaded region represent the estimated magnitude of the applied laser power.
The values of the standard deviations of these force measurements at each step response have been found to be about 10 nN (see Fig. 11(b)). After submitting reports [85] and [86] there have been several additional adjustments made on the FMS, which have led to improvements of the standard deviation of the force measurements from 20 nN to below 10 nN. However, to ensure the validity of these values the FMS must be cross-checked (re-calibrated) again against other force measurement methods that have direct SI-traceability. The manufacturer of the EMFC balances gives its floor value by the parameter of readability, which is 1 μg and roughly corresponds to the force values about 10 nN, see [86]. The method suggested in this paper can be potentially used to compare the measured force signals (including the filtered signal) with the force produced by the photon’s momentum).
6. Conclusions
In this paper, we present a method for generation or detection of forces at the sub-μN level using the total momentum transfer of the photons from multipass laser beam. Based on the comparison of the experimentally measured results with the theoretical computations it is confirmed that the multiple reflected laser beam can be used to extend the effective use of the optical forces from pN order to the several nN up to sub-μN level for the same laser source. Thus, enabling in perspective the comparison possibilities between the lowest limit of SI traceable mass artifacts with the optically generated forces. The minimum necessary components consisting this method are a) a relatively high power (from 100 mW up to 1–2 W) and stable laser source, b) a well-defined geometry of a confined system with surfaces near to ideal reflectivity, wherein c) the beam of the laser source can be folded continuously with minimal power losses (effective utilization with maximally achievable number of reflections) after every reflection occur, and d) a SI traceable method and a calibrated FMS in order to measure the augmented sum of forces produced due to multiple reflections. In the limiting theoretical case when an infinite number of reflections occur, the total force approaches to an asymptote of some finite force value that would be important to probe experimentally in the future studies. Also, given a calibrated laser source with sufficient accuracy and precision, the measurements of this generated forces may provide additional experimental evidence-based distinctions between coherent and incoherent reflection processes at the interaction edge of the light with matter.
On more practical and industrial oriented application bases the answers and the results of the current work, achieved and to be refined in the future, aims to fulfill and bridge several technological/technical aspects which are necessary for the improvement of various legal and procedural tasks in force and laser power calibration measurements. Here, we would like to outline the following reversible and complementary measurement approaches: either the calibration of the laser power linking it to precise/accurate force measurements, or calibrating the forces while having stable/accurate/precise laser sources. The verification of the method is provided by the comparison of the basic theoretical computations Eq. (9) with the experimental results obtained from the force measurements using high-precision FMS (Fig. 11). A summary of the initial test measurements based on the initial geometrical configuration (discussed in Section 3) is provided in Section 5 in accordance with the necessary technological requirement for force measurements with FMS (Section 4).
Funding
Bundesministerium für Bildung und Forschung (BMBF).
Acknowledgments
We thank the Deutsche Forschungsgemeinschaft (DFG - within framework of the RTG/1567) for financial support in developing the force measurements system at the TU Ilmenau, Germany. Also, the Bundesministerium für Bildung und Forschung (BMBF) for the financial support during the initiation of the experiments presented in this work. The EMFC balances provided by Sartorius Lab Instruments GmbH & Co. The authors are grateful to the colleagues from Institute of Process Measurement and Sensor Technology for all constructive discussions.
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