Copyright 2006 Society of Photo-Optical Instrumentation Engineers. This paper was published in Proceedings of SPIE 6326(2006)and is made avail- able as an electronic reprint with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited Preprint of: Robert N. C. Pfeifer, Timo A Nieminen, Norman R. Heckenberg, and Halina Rubinsztein -Dunlo Two controversies in classical electromagnetism Pro.SPlE6326,63260H(2006)
Copyright 2006 Society of Photo-Optical Instrumentation Engineers. This paper was published in Proceedings of SPIE 6326 (2006) and is made available as an electronic reprint with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. Preprint of: Robert N. C. Pfeifer, Timo A. Nieminen, Norman R. Heckenberg, and Halina Rubinsztein-Dunlop “Two controversies in classical electromagnetism” Proc. SPIE 6326, 63260H (2006)
Two controversies in classical electromagnetism Robert N. C. Pfeifer, Timo A Nieminen, Norman R. Heckenberg and Halina Rubinsztein-Dunlop The University of Queensland, Brisbane, Queensland 4072, Australia ABSTRACT This paper examines two controversies arising within classical electromagnetism which are relevant to the optical trapping and micromanipulation community. First is the Abraham-Minkowski controversy, a debate relatin to the form of the electromagnetic energy-momentum tensor in dielectric materials, with implications for the momentum of a photon in dielectric media. A wide range of alternatives exist, and experiments are frequently proposed to attempt to discriminate between them. We explain the resolution of this controversy and show that regardless of the electromagnetic energy-momentum tensor chosen, when material disturbances are also taken into account the predicted behaviour will always be the same. The second controversy, known as the plane wave angular momentum paradox, relates to the distribution of angular momentum within an electromagnetic wave. The two competing formulations are reviewed, and an experiment is discussed which is capable of distinguishing Keywords: Abraham-Minkowski controversy, Khrapko's paradox, photon momentum, electromagnetic momen- tum. angular momentum 1 INTRODUCTION The ongoing drive towards miniaturisation which began in the latter half of the twentieth century has spurred the development of many new technologies capable of manipulating matter over a vast range of scales. One uch technology is that of optical micromanipulation, which may lie at the heart of the micromachines or microassembly plants of tomorrow. However, if this technology is to be widely and reliably used then the fundamental theory underlying its operation must be clearly understood. Optical tweezers offer tremendous new opportunities to probe the interactions between physical objects and electromagnetic radiation Two important questions of fundamental physics are raised in this paper, both of which have bearing upon the accurate modelling of behaviour of objects undergoing optical micromanipulation. Both have engendered some degree of controversy in the published literature, and both are amenable to resolution, one through theory and the other through experiment e shall first discuss the Abraham-Minkowski controversy, in which the form of the electromagnetic energy- momentum tensor in a dielectric medium has been debated for almost a century, and show that this controversy has been successfully resolved and poses no threat to our understanding of the behaviour of optical tweezers Secondly, we shall discuss Khrapko's paradox, which asks about the distribution of angular momentum within an electromagnetic wave and addresses the contentious question, Does a plane wave carry spin? " We examine a modification of Khrapko's original experiment which is capable of producing qualitative results Further author information:( Send correspondence to R.N.C. P) R.N.C. P. E-mail: pfeifer @physics uq. edu. au, Telephone: +61(0)7 3365 3463
Two controversies in classical electromagnetism Robert N. C. Pfeifer, Timo A. Nieminen, Norman R. Heckenberg and Halina Rubinsztein-Dunlop The University of Queensland, Brisbane, Queensland 4072, Australia ABSTRACT This paper examines two controversies arising within classical electromagnetism which are relevant to the optical trapping and micromanipulation community. First is the Abraham–Minkowski controversy, a debate relating to the form of the electromagnetic energy–momentum tensor in dielectric materials, with implications for the momentum of a photon in dielectric media. A wide range of alternatives exist, and experiments are frequently proposed to attempt to discriminate between them. We explain the resolution of this controversy and show that regardless of the electromagnetic energy–momentum tensor chosen, when material disturbances are also taken into account the predicted behaviour will always be the same. The second controversy, known as the plane wave angular momentum paradox, relates to the distribution of angular momentum within an electromagnetic wave. The two competing formulations are reviewed, and an experiment is discussed which is capable of distinguishing between the two. Keywords: Abraham–Minkowski controversy, Khrapko’s paradox, photon momentum, electromagnetic momentum, angular momentum 1. INTRODUCTION The ongoing drive towards miniaturisation which began in the latter half of the twentieth century has spurred the development of many new technologies capable of manipulating matter over a vast range of scales. One such technology is that of optical micromanipulation,1 which may lie at the heart of the micromachines or microassembly plants of tomorrow. However, if this technology is to be widely and reliably used then the fundamental theory underlying its operation must be clearly understood. Optical tweezers offer tremendous new opportunities to probe the interactions between physical objects and electromagnetic radiation. Two important questions of fundamental physics are raised in this paper, both of which have bearing upon the accurate modelling of behaviour of objects undergoing optical micromanipulation. Both have engendered some degree of controversy in the published literature, and both are amenable to resolution, one through theory and the other through experiment. We shall first discuss the Abraham–Minkowski controversy, in which the form of the electromagnetic energy– momentum tensor in a dielectric medium has been debated for almost a century, and show that this controversy has been successfully resolved and poses no threat to our understanding of the behaviour of optical tweezers. Secondly, we shall discuss Khrapko’s paradox,2 which asks about the distribution of angular momentum within an electromagnetic wave and addresses the contentious question, “Does a plane wave carry spin?” We examine a modification of Khrapko’s original experiment which is capable of producing qualitative results. Further author information: (Send correspondence to R.N.C.P.) R.N.C.P.: E-mail: pfeifer@physics.uq.edu.au, Telephone: +61 (0)7 3365 3463
2. THE ABRAHAM-MINKOWSKI CONTROVERSY 2.1. Background "What happens to the momentum of a photon when it enters a dielectric medium? This seemingly innocuous question lies at the centre of a century-long debate. The answer is tied to the closely-related question, " What is the energy-momentum tensor of an electromagnetic wave traversing a dielectric material? The energy-momentum tensor describes the propagation of energy and momentum in a four-dimensional space-time. To derive the appropriate expression for an electromagnetic wave propagating in free space is relatively straightforward and the result is known as Maxwells tensor. Its derivation may be found in more advanced texts on electromagnetism or special relativity ,5 It is only within material media that uncertainty arises The first expression for the electromagnetic energy-momentum tensor in a dielectric medium was proposed by Minkowski in 1908., According to the Minkowski electromagnetic energy-momentum tensor, the momentum Aux density of an electromagnetic wave increases from p in free space to np in a material medium, where n is the refractive index of the medium, and the electromagnetic momentum density is given by d B. However, the tensor which Minkowski proposed came under heavy criticism on account of its lack of diagonal symmetry, a fact which was held incompatible with conservation of angular momentum. In response to this, first Einstein and Laub, 9 and later Abraham10, II developed symmetric energy-momentum tensors. However, under these tensors the momentum of an electromagnetic wave decreases on entering a dielectric medium from p to p/n, and the electromagnetic momentum density is given by(1/c)Ex H. The tensor of Einstein and Laub did not purport to be valid outside of the rest frame of the material medium, whereas that of abraham was rigorously developed in accordance with the principles of special relativity and hence rapidly gained favour. Numerous arguments and thought experiments were proposed attempting to discriminate between the two tensors(for example, Refs. 5, 12-15) though neither gained a convincing upper hand. In 1954, Jones and Richards performed an experiment 6 to measure the momentum transferred by a beam of light to a reflecting surface suspended within a dielectric medium, and demonstrated that the magnitude of the momentum transfer ras np, but Jones comments that this is consistent with both the Minkowski and the Abraham tensors, if one recognises that under the Abraham tensor, the electromagnetic wave is also accompanied by a disturbance within the dielectric medium carrying momentum In 1973, another significant experiment was performed by Ashkin and Dziedzicl? in response to a theoretical paper by Burt and Peierls, 8 in which they considered the behaviour of a liquid interface traversed by a laser beam. As the electromagnetic wave traversed the air-liquid interface it would either increase or decrease in momentum, and conservation of momentum required that an equal and opposite quantity of momentum be imparted to the fluid interface, which would then either bulge inwards or outwards accordingly. Burt and Peierls argued that the material disturbance in the medium in the Abraham case described above would propagate far lower than the electromagnetic radiation, and hence the initial response of the interface would depend upon the electromagnetic momentum alone. Ashkin and Dziedzic performed this experiment, and discovered that the interface bulged outward, into the medium of lower refractive index. However, they also recognised the then-unpublished work of Gordon19who performed a more detailed analysis and showed that the role of the material disturbance was not negligible and hat the predictions of the two tensors would be identical after all the gordon recognised that the disturbance in the dielectric medium arises as a result of interactions between medium and the electromagnetic wave. It is established by the leading edge of an electromagnetic pulse as it traverses the medium, and restored to normal by the trailing edge. The material disturbance therefore propagates at the speed of the electromagnetic wave, and not at the speed of sound in the medium, as previously supposed. Pressure effects arising from the edges of the beam were also found to play a significant role Gordon's work provided a very practical demonstration of the equivalence of the Abraham and Minkowski tensors but was confined to fuids with a dielectric constant e 1. It was later extended to elastic solids by
2. THE ABRAHAM–MINKOWSKI CONTROVERSY 2.1. Background “What happens to the momentum of a photon when it enters a dielectric medium?” This seemingly innocuous question lies at the centre of a century-long debate. The answer is tied to the closely-related question, “What is the energy–momentum tensor of an electromagnetic wave traversing a dielectric material?” The energy–momentum tensor describes the propagation of energy and momentum in a four-dimensional space–time. To derive the appropriate expression for an electromagnetic wave propagating in free space is relatively straightforward and the result is known as Maxwell’s tensor. Its derivation may be found in more advanced texts on electromagnetism3 or special relativity.4, 5 It is only within material media that uncertainty arises. The first expression for the electromagnetic energy–momentum tensor in a dielectric medium was proposed by Minkowski in 1908.6, 7 According to the Minkowski electromagnetic energy–momentum tensor, the momentum flux density of an electromagnetic wave increases from p in free space to np in a material medium, where n is the refractive index of the medium, and the electromagnetic momentum density is given by D × B. However, the tensor which Minkowski proposed came under heavy criticism on account of its lack of diagonal symmetry, a fact which was held incompatible with conservation of angular momentum. In response to this, first Einstein and Laub8, 9 and later Abraham10, 11 developed symmetric energy–momentum tensors. However, under these tensors the momentum of an electromagnetic wave decreases on entering a dielectric medium from p to p/n, and the electromagnetic momentum density is given by (1/c) E × H. The tensor of Einstein and Laub did not purport to be valid outside of the rest frame of the material medium, whereas that of Abraham was rigorously developed in accordance with the principles of special relativity and hence rapidly gained favour. Numerous arguments and thought experiments were proposed, attempting to discriminate between the two tensors (for example, Refs. 5, 12–15) though neither gained a convincing upper hand. In 1954, Jones and Richards performed an experiment16 to measure the momentum transferred by a beam of light to a reflecting surface suspended within a dielectric medium, and demonstrated that the magnitude of the momentum transfer was np, but Jones comments that this is consistent with both the Minkowski and the Abraham tensors, if one recognises that under the Abraham tensor, the electromagnetic wave is also accompanied by a disturbance within the dielectric medium carrying momentum n − 1 n p. (1) In 1973, another significant experiment was performed by Ashkin and Dziedzic17 in response to a theoretical paper by Burt and Peierls,18 in which they considered the behaviour of a liquid interface traversed by a laser beam. As the electromagnetic wave traversed the air–liquid interface it would either increase or decrease in momentum, and conservation of momentum required that an equal and opposite quantity of momentum be imparted to the fluid interface, which would then either bulge inwards or outwards accordingly. Burt and Peierls argued that the material disturbance in the medium in the Abraham case described above would propagate far slower than the electromagnetic radiation, and hence the initial response of the interface would depend upon the electromagnetic momentum alone. Ashkin and Dziedzic performed this experiment, and discovered that the interface bulged outward, into the medium of lower refractive index. However, they also recognised the then-unpublished work of Gordon19 who performed a more detailed analysis and showed that the role of the material disturbance was not negligible and that the predictions of the two tensors would be identical after all. Gordon recognised that the disturbance in the dielectric medium arises as a result of interactions between the medium and the electromagnetic wave. It is established by the leading edge of an electromagnetic pulse as it traverses the medium, and restored to normal by the trailing edge. The material disturbance therefore propagates at the speed of the electromagnetic wave, and not at the speed of sound in the medium, as previously supposed. Pressure effects arising from the edges of the beam were also found to play a significant role. Gordon’s work provided a very practical demonstration of the equivalence of the Abraham and Minkowski tensors but was confined to fluids with a dielectric constant ε 1. It was later extended to elastic solids by
Peierls, 20,21 who demonstrated a number of interesting effects including the generation of phonons in an elastic solid by a beam of finite width However, in 1968 and 1975 two experiments were performed which were taken to strongly support the Abraham electromagnetic energy-momentum tensor over the Minkowski case. When the force density arising as result of the Abraham and Minkowski energy-momentum tensors is calculated, the two tensors yield expressions which are identical except for a single term yielded by only the Abraham tensor, and hence known as the abraham f E c2 at Both James2,23 and Walker et al. 24,25 successfully detected the existence of the Abraham force. It was not ntil 1978 that Israel pointed out that these results were nevertheless compatible with the Minkowski energy momentum tensor prouided the Minkowski tensor was also accompanied by an appropriate disturbance in the neat. Israel's conclusion placed the Minkowski and Abraham tensors on an equal footing once more, and consid- eration of the material counterparts to the two electromagnetic energy-momentum tensors have subsequently yielded a number of explicit proofs of equivalence not only for the Abraham and Minkowski electromagnetic energy-momentum tensors, 26-30 but also for other tensors proposed by Grot and de Groot and Suttorp32(see Ref These were not the first proofs of equivalence but they were the first to be published in a succinct and comparatively accessible format. They were, however, preceded by Penfield and Haus, 4 who also considered he tensors arising from the boffi, 35 Amperian, 36 and Chu,37 formulations of electromagnetism, and also by de Groot and Suttorp, upon whose work we shall now draw to demonstrate the theoretical foundations underlying these proofs 2.2. Theoretical Analysis The argument pursued by de groot and Suttorp, and also by many of the later authors cited above, arises from considerations of global conservation of energy and momentum. From the law of conservation of linear momentum we obtain a constraint on the total energy-momentum tensor Ttot, a TaR=0 (3 Similarly, from conservation of angular momentum we may obtain (4) However, these constraints only apply to the total linear and angular momentum of a syst When an tromagnetic wave enters a dielectric medium, momentum may be carried both within the electromagneti itself, and within excitations of the electromagnetic dipoles of the material. We may therefore write EM+ T where TEM and Tmat are the electromagnetic and material energy-momentum tensors respectively Upon examination of the full expression of the total energy-momentum tensor (which may be found all the above proofs of equivalence, with Refs. 26, 32, 34 being perhaps the clearest) it becomes apparent that the distinction between electromagnetic and material terms is not absolute. There exist terms which describe an excitation of the dipoles of the material medium, but which are dependent upon the magnitude of the electromagnetic wave. If these terms are included in the electromagnetic energy-momentum tensor, one obtains a tensor resembling the Minkowski tensor, whereas if they are assigned to the material tensor, one obtains a tensor resembling the abraham tensor
Peierls,20, 21 who demonstrated a number of interesting effects including the generation of phonons in an elastic solid by a beam of finite width. However, in 1968 and 1975 two experiments were performed which were taken to strongly support the Abraham electromagnetic energy–momentum tensor over the Minkowski case. When the force density arising as a result of the Abraham and Minkowski energy–momentum tensors is calculated, the two tensors yield expressions which are identical except for a single term yielded by only the Abraham tensor, and hence known as the Abraham force: fAbr = εµ − 1 c 2 ∂S ∂t . (2) Both James22, 23 and Walker et al.24, 25 successfully detected the existence of the Abraham force. It was not until 1978 that Israel pointed out that these results were nevertheless compatible with the Minkowski energy– momentum tensor provided the Minkowski tensor was also accompanied by an appropriate disturbance in the medium. Israel’s conclusion placed the Minkowski and Abraham tensors on an equal footing once more, and consideration of the material counterparts to the two electromagnetic energy–momentum tensors have subsequently yielded a number of explicit proofs of equivalence not only for the Abraham and Minkowski electromagnetic energy–momentum tensors,26–30 but also for other tensors proposed by Grot31 and de Groot and Suttorp32 (see Ref. 33). These were not the first proofs of equivalence but they were the first to be published in a succinct and comparatively accessible format. They were, however, preceded by Penfield and Haus,34 who also considered the tensors arising from the Boffi,35 Amperian,36 and Chu34, 37 formulations of electromagnetism, and also by de Groot and Suttorp,32 upon whose work we shall now draw to demonstrate the theoretical foundations underlying these proofs. 2.2. Theoretical Analysis The argument pursued by de Groot and Suttorp, and also by many of the later authors cited above, arises from considerations of global conservation of energy and momentum. From the law of conservation of linear momentum we obtain a constraint on the total energy–momentum tensor T αβ tot , ∂αT αβ tot = 0. (3) Similarly, from conservation of angular momentum we may obtain T αβ tot = T βα tot . (4) However, these constraints only apply to the total linear and angular momentum of a system. When an electromagnetic wave enters a dielectric medium, momentum may be carried both within the electromagnetic wave itself, and within excitations of the electromagnetic dipoles of the material. We may therefore write Ttot = TEM + Tmat (5) where TEM and Tmat are the electromagnetic and material energy–momentum tensors respectively. Upon examination of the full expression of the total energy–momentum tensor (which may be found in all the above proofs of equivalence, with Refs. 26, 32, 34 being perhaps the clearest) it becomes apparent that the distinction between electromagnetic and material terms is not absolute. There exist terms which describe an excitation of the dipoles of the material medium, but which are dependent upon the magnitude of the electromagnetic wave. If these terms are included in the electromagnetic energy–momentum tensor, one obtains a tensor resembling the Minkowski tensor, whereas if they are assigned to the material tensor, one obtains a tensor resembling the Abraham tensor
We now note that Eqs. (3)and(4) apply only to the total energy-momentum tensor, and not to the electromagnetic or material components in isolation, i.e EM TaB ao ( Teg It therefore follows that the lack of diagonal symmetry in Minkowski's electromagnetic energy-momentum tensor does not violate conservation of angular momentum as a complementary asymmetry in the material energy- momentum tensor ensures that the total energy-momentum tensor is symmetric. Any choice of electromagnetic energy-momentum tensor is equally valid provided the corresponding material counterpart is also taken into consideration, as it is only the total energy-momentum tensor which is uniquely defined consequences The consequences for optical tweezers are threefold. First, there is the recognition that the form of the elec- tromagnetic energy-momentum tensor in a dielectric medium is not of crucial importance. This also serves to validate the usage of the relative refractive index for a body suspended within a dielectric fluid: This change has no effect on the total energy-momentum tensor, and so although the medium is eliminated by converting to an equivalent situation in vacuum, no physically relevant information is lost Second, there is the important recognition that the Minkowski electromagnetic energy-momentum tensor has a material counterpart. Historically this was overlooked for a long time because the counterpart carries no linear momentum and hence is unnecessary in experiments such as Jones and Richard 16 mentioned above. However the material counterpart does carry angular momentum and this is illustrated extremely well by the thought experiment described by Padgett et al.38 where a laser beam carrying orbital angular momentum is passed through a glass disc. The Abraham tensor pair demonstrates a transfer of angular momentum to the glass disc whereas if the Minkowski electromagnetic energy-momentum tensor is considered in isolation, this does not take plac Third, we have discovered that the material disturbance may have significant physical effects on experimental predictions, and travels at the same speed as the electromagnetic wave. 9 It is therefore inappropriate to treat material objects as rigid bodies when analysing their behaviour, as this corresponds to instantaneous, i.e. superluminal, transfer of momentum throughout the body, whereas in reality the traversal of the electromagnetic wave is inevitably accompanied by physical pressures and deformations within the medium, and at the boundaries of both the medium and the beam. It is similarly wrong to neglect the material counterpart entirely or to assume that its propagation is negligibly slow 3. KHRAPKOS PARADOX 3.1. Background In 2001, Khrapko asked in American Journal of Physics, "Does plane wave not carry spin? "[sic]. 2 This question, which is not as simple as it first appears, arises from the existence of two separate expressions for the total angular momentum of an electromagnetic wave. These are (r×S/e2)dv A Reli 2u (9) where L represents total angular momentum, r is the position vector, s is the real instantaneous Poynting vector, c is the speed of light, w is the angular frequency of the electromagnetic wave, and e is the complex electric fiel We shall term Eq( 8)the macroscopic expression, and Eq.(9) the microscopic expression. The former arises from the obvious construction of r x p, where p represents linear momentum, and the latter may be obtained
We now note that Eqs. (3) and (4) apply only to the total energy–momentum tensor, and not to the electromagnetic or material components in isolation, i.e. ∂α T αβ EM + T αβ mat = 0 (6) T αβ EM + T αβ mat = T βα EM + T βα mat . (7) It therefore follows that the lack of diagonal symmetry in Minkowski’s electromagnetic energy–momentum tensor does not violate conservation of angular momentum as a complementary asymmetry in the material energy– momentum tensor ensures that the total energy–momentum tensor is symmetric. Any choice of electromagnetic energy–momentum tensor is equally valid provided the corresponding material counterpart is also taken into consideration, as it is only the total energy–momentum tensor which is uniquely defined. 2.3. Consequences The consequences for optical tweezers are threefold. First, there is the recognition that the form of the electromagnetic energy–momentum tensor in a dielectric medium is not of crucial importance. This also serves to validate the usage of the relative refractive index for a body suspended within a dielectric fluid: This change has no effect on the total energy–momentum tensor, and so although the medium is eliminated by converting to an equivalent situation in vacuum, no physically relevant information is lost. Second, there is the important recognition that the Minkowski electromagnetic energy–momentum tensor has a material counterpart. Historically this was overlooked for a long time because the counterpart carries no linear momentum and hence is unnecessary in experiments such as Jones and Richard’s16 mentioned above. However, the material counterpart does carry angular momentum and this is illustrated extremely well by the thought experiment described by Padgett et al.38 where a laser beam carrying orbital angular momentum is passed through a glass disc. The Abraham tensor pair demonstrates a transfer of angular momentum to the glass disc, whereas if the Minkowski electromagnetic energy–momentum tensor is considered in isolation, this does not take place. Third, we have discovered that the material disturbance may have significant physical effects on experimental predictions, and travels at the same speed as the electromagnetic wave.19 It is therefore inappropriate to treat material objects as rigid bodies when analysing their behaviour, as this corresponds to instantaneous, i.e. superluminal, transfer of momentum throughout the body, whereas in reality the traversal of the electromagnetic wave is inevitably accompanied by physical pressures and deformations within the medium, and at the boundaries of both the medium and the beam. It is similarly wrong to neglect the material counterpart entirely or to assume that its propagation is negligibly slow. 3. KHRAPKO’S PARADOX 3.1. Background In 2001, Khrapko asked in American Journal of Physics, “Does plane wave not carry spin?”[sic].2 This question, which is not as simple as it first appears, arises from the existence of two separate expressions for the total angular momentum of an electromagnetic wave. These are L = Z V (r × S/c2 ) dV (8) and L = Z V Re iε (E∗ × E) 2ω dV, (9) where L represents total angular momentum, r is the position vector, S is the real instantaneous Poynting vector, c is the speed of light, ω is the angular frequency of the electromagnetic wave, and E is the complex electric field. We shall term Eq. (8) the macroscopic expression, and Eq. (9) the microscopic expression. The former arises from the obvious construction of r × p, where p represents linear momentum, and the latter may be obtained