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Determining the Increase or Decrease of Photon Momentum: Dielectric or Plasmonic Mie Object at Air-Liquid Interface

Though the interfacial tractor beam experiment supports the increase of photon momentum (i.e., Minkowski momentum), it is still a matter of investigation whether, inside matter, the photon momentum always increases. Considering the inhomogeneous or heterogeneous background, we have demonstrated that if the background and the half-immersed object are both non-absorbing, the transferred photon momentum to the object can be considered as the one of Minkowski exactly at the interface. In contrast, the presence of loss inside matter, either in the half-immersed object (i.e., plasmonic or lossy dielectric) or in the background, changes the whole situation. For such cases, our several demonstrations and proposed thought experiments have strongly supported the decrease of photon momentum instead of the usual perception of its increase. Although almost all the major radiation pressure experiments have so far supported the linear increase of photon momentum, our proposed simple experimental set-ups may introduce a novel way to observe and verify the exactly opposite proposal: the Abraham momentum of photon. Finally, as an interesting sidewalk, based on several parameters, a machine learning based system has been developed to predict the transferred momentum of photon within a very short time avoiding time-consuming full simulation.


INTRODUCTION
In 1908, physicist Hermann Minkowski, proposed that the total momentum of an electromagnetic field inside any matter is equal to ∫ D×B dv [1]. This suggests that the photon momentum should actually increase and takes the value p = nħk 0 according to quantum description. Here n is the refractive index of the medium, k 0 is the wave number of the electromagnetic wave in air and ħ is the reduced Planck constant. One year later, the German physicist Max Abraham proposed a different argument that the photon inside a medium would have a lower/ decreased momentum that is equal to ∫ [(E×H)/c 2 ] dv, or according to quantum description the value of photon momentum, p = ħk 0 /n [2].
Nearly 100 years later, there is still no clear answer as to which of these formulae is correct [3][4][5].
According to [5]: "both the Abraham and Minkowski forms of the momentum density are correct, with the former being the kinetic momentum and the later the canonical momentum." Notably, there is no simple experiment that supports the decrease of photon momentum (i.e., Abraham momentum of photon) inside a matter except for a few recent reports [6][7][8]. As a result, it would be certainly a great achievement if some very simple but realistic experiments are designed to measure the decrease of photon momentum inside a matter.
One of the recently popularized ideas in the area of photon momentum transfer or optical force is known as Tractor beam [9][10][11]. A tractor beam is a customized light beam that exerts a counter intuitive negative force on a scatterer [9][10][11][12][13][14][15][16][17][18], pulling it opposite to the propagation direction of light, in contrast to the conventional pushing force. Tractor beam experiments, which involve the material background [12][13][14], can also be investigated in details to understand the persistently debated roles of different photon momenta such as those of the Abraham-Minkowski controversy [3][4][5]. In this article, we make an attempt to investigate the Abraham-Minkowski controversy based on the tractor beam like effects.
We have investigated the light momentum transfer and related optical force on a scatterer (both absorbing and non-absorbing) floating on an interface of two or three different media (both absorbing and non-absorbing) as shown in Fig. 1(a), (b) and Fig. 2(a)-(d). Prior to this current work, for the interfacial tractor beam experiment, detailed calculations by ray tracing method and stress tensor equations showed that the pulling force is natural in air-water scheme due to the linear increase of photon momentum in the infinitely long water medium [12,14]. Interestingly, non-Minkowski formulations [14] show a pushing force contradicting the experimental observation in [12]. However, the suggested interpretation in favor of Minkowski photon momentum for interfacial tractor beam experiment has been questioned in [19]. Two different recent experiments have supported the observation of Abraham photon momentum for air-water interface [6,7]. In addition, some theoretical works have suggested in favor of the possible existence of Abraham photon momentum [4,5,[20][21][22][23] for different other situations.

4
Considering the inhomogeneous or heterogeneous background, we have theoretically demonstrated that if the background and the half-immersed object are both non-absorbing, the transferred photon momentum to the object can be considered as the one of Minkowski exactly at the interface in time averaged scenario. Remarkably, we have shown that even if the background's width is only a few nanometer (extremely small in comparison with the object), the half-immersed object experiences an optical pulling force. Several intuitive thought experiments have also been put forwarded to establish this fact.
In contrast, the presence of loss inside the matter, either in the half-immersed object (i.e., plasmonic or lossy dielectric) or in the background or in the extremely short background, changes the whole scenario. For such cases, our several demonstrations have strongly supported the decrease of linear momentum of photon instead of the usual perception of increase thereof. Though we have demonstrated several full wave simulation based results (along with analytical and qualitative explanations) in favor of linear decrease of photon momentum for distinct situations (i.e. absorption in the object or in the background or in the short background), we have concluded that: the easiest possible experiment to verify the decrease of photon momentum can be done by replacing the silicon object with a gold or a silver object in the well-known interfacial tractor beam experiment [12] and shining the usual Gaussian beam wave on it [12].
Finally, based on several parameters, a machine learning algorithm has been applied to predict the transferred momentum of photon at a shortest possible simulation time (instead of time consuming full wave simulations). Apart from our motivation of getting a quick prediction of the force type without a time-consuming full wave simulation, this artificial intelligence based investigation may open up a novel research avenue by providing us with yet another useful tool/approach to investigate other problems related to optical force and photon momentum inside a matter.

METHODS
Throughout this paper, we refer to 'exterior' ('interior') or 'outside' ('inside') forces as those evaluated outside (inside) the volume of the macroscopic objects. We have done all the full wave electromagnetic simulations using COMSOL MULTI PHYSICS software [24].
The proposed set-ups are illustrated in Figs. 1 (a)-(c). The source is a simple z -polarized (cf. Fig. 1 Here 'out' represents the exterior total field of the scatterer; E , D , H and B are the electric field, electric displacement field, magnetic field and magnetic induction field respectively, represents the time average and I is the unity tensor. In contrast, the 'internal optical force' of Minkowski [27,32] is calculated by the integration of time averaged Minkowski stress tensor at r=aemploying the interior fields of the scatterer of radius a as follows: Here 'in' represents the interior field of the scatterer. Notably, the 'internal optical force' of Minkowski represents the total conducting force (due to the interaction of the photon with the free charges inside the absorbing objects) inside an object. For lossless objects, Eq (2) always yields zero force inside the objects [27,32].

RESULTS AND DISCUSSIONS
Though the interfacial tractor beam experiment supports the linear increase of photon momentum (i.e. Minkowski momentum), the following question is still a matter of investigation: inside the matter, does the photon momentum always increases; and even if it increases, does it increase throughout the whole medium or just at the interface? In [33], it has been argued that the increase of linear momentum usually occurs at the interface of two different media due to the reduced impedance mismatch. But no conclusive proof is still presented regarding this very interesting argument given in [33]. Based on a few proposed 'thought experiments', we shall first check this proposal for interfacial tractor beam set-up at first. width, where object diameter is 4000 nm], still an optical pulling force occurs. So, the main point is that even for an extremely short background, the proposition of the linear momentum increase remains valid. Now, the obvious question arises whether there is any alternative way to verify the robustness of such a proposal that claims an increase of photon momentum just at the interface of two different media. This is addressed next along with few other distinct issues.
In Fig. 2  if the lower long background is air but the short background (width around 10 nm) is just water, the force is found to be a pulling force in Fig. 1(b) in this article [also cf. Fig. 2(b) for another case]. So, the short background is indeed an imperative criterion which plays a significant role in such set-ups.
Although for the non-absorbing object and non-absorbing background, the transferred momentum of photon usually increases, this scenario changes when applied in the presence of absorption; either in the sub-merged object or in the embedding lower background. At first, we can check the case of half immersed absorbing object. If we replace a half immersed non-absorbing dielectric by a lossy dielectric or a plasmonic object, it is clearly observed that the pulling force fully vanishes as shown in A simple fact is that, if the upper medium is a high refractive indexed lossless medium but the lower one is air, then the lossless object experiences an optical pushing force as shown in Fig. 3 (d), can be conducted very easily, which also support that the linear momentum of photon decreases in the presence of absorption. The easiest possible experiment to verify the decrease of photon momentum can be done by replacing the silicon object with a gold or a silver object in the well-known interfacial tractor beam experiment [12] and shining the usual Gaussian beam wave on it [12]. The findings and demonstrations in this article suggest that optical pushing force should be observed instead of optical pulling force in such a set-up.  Fig 10 (a), (b) and Fig 12 (a) given in ref. [38])] and the linear increase of the photon momentum occurs exactly at the interface of two different media [Helmholtz/ Minkowski force also suggests so]. But when absorption is introduced inside the half-immersed object, the force inside the absorbing object has a non-zero value equal to Eq (2) [cf. ref [27] for more detail on Eq (2)].

Explanation of the observed results
According to [27,32], that same time averaged conduction force in c F in Eq (2) (3)].
The connection between Eq (1) and Eq (2) can be written as:  (5) According to Eq (4), on the boundary r=a of any object, the quantity:  (2)]) and the momentum of photon decreases.
In [42], the connection of optical pulling force due to the incidence of Bessel beam has been discussed

Application of Machine Learning Techniques for predicting the type of optical force
Finally, we take an interesting sidewalk from our primary focus and investigate whether machine learning based data analytics can provide us an accurate prediction on the transferred momentum of photon for the more generic cases based on the discussed complex set-ups in this article. The motivation behind this rather unusual investigation is as follows. A lot of parameters actually contribute in determining the optical force type (i.e., pulling vs. pushing) and a full wave simulation [24] takes long time. In contrast, a quick accurate result in this regard could be extremely useful for experimental physicists at least to make quick primary decisions in many situations. So, we have made an attempt to develop a machine learning based system, using WEKA workbench [43], which can accurately predict the outcome extremely faster. However, the accuracy of this prediction is naturally dependent on the dataset used for training the system. The dataset has been prepared based on some full wave simulations and is provided in Chart 1s in the supplement.
We have used the WEKA workbench [43] to develop the machine learning based classifiers and to conduct the computational experiments. The data given in the supplement (Chart 1s of the supplement) has been used to train a system that is able to predict the optical force type. In what follows, this data will be referred to as the training dataset. The training dataset in Chart 1s has been prepared based on several full wave simulations. The varying parameters used in the dataset are: particle radius/lambda, particle loss, refractive index of: the particle, background and short background; loss at: long background and short background; and short background width. Thus Chart 1s in the supplement article has the first 8 columns for the parameters with the last (i.e., 9 th ) column representing the time averaged force experienced by the particle; here '1' ('0') represents optical pulling ( For our classification task, we have used some distinct classifiers, namely, Naïve Bayes [44][45][46], Simple Logistics [44,46], J48 [44,48], Random Forest [44,49], Random Tree [44,49] and so on. Following the machine learning literature, we have employed 10-fold cross validations: the training dataset is randomly partitioned into 10 equal sized subsets; of the 10 subsets, a single subset is retained as the validation data for testing the model, and the remaining 9 subsets are used as training data. The cross-validation process is then repeated 10 times, with each of the 10 subsets used exactly once as the validation data. The 10 results can then be averaged to produce a single estimation. Table   1 reports the accuracy measures found after performing 10-fold cross validations for each of the classifiers. From the results reported in Table 1, J48, Random Forest and Random Tree classifiers have predicted the final results with the highest accuracy (more than 95%, which is quite satisfactory).    Fig. 1(a)) for different angles of incident light.