Radiation pressure and momentum transfer in dielectrics: The photon drag effect

Transcription

1 Radiation pressure and momentum transfer in dieletris: The photon drag effet Rodney Loudon Eletroni Systems Engineering, University of Essex, Colhester C04 3SQ, United Kingdom Stephen M. Barnett and C. Baxter Department of Physis, University of Strathlyde, Glasgow G4 0NG, United Kingdom Reeived 28 Deember 2004; published 10 June 2005 The momentum transfer from light to a dieletri material in the photon drag effet is alulated by evaluation of the relevant Lorentz fore. In aordane with measurements on Si and Ge, the material is taken as a two-omponent optial system, with harge arriers desribed by an extintion oeffiient in a host semiondutor desribed by real refrative indies p phase and g group. The alulated momentum transfer to the harge arriers alone has the value p / per photon, the so-alled Minkowski value, found experimentally. The time-dependent Lorentz fore is alulated for light in the form of a narrow-band singlephoton pulse. When the pulse is muh shorter than the attenuation length, whih is muh shorter than the sample thikness, there is a lear separation in time between surfae and bulk ontributions to the fores. The total bulk momentum transfer harges plus host in this ase is found to be / g, the so-alled Abraham value. DOI: /PhysRevA PACS numbers: Nn I. INTRODUCTION Theories of radiation pressure and the momentum of light in dieletris have frequently involved arguments about the orret form of the stress tensor of the eletromagneti field in a material medium see 1 for a review. Several formulations of the stress tensor have been developed, but partiular attention has been given to the two ontenders in the so-alled Abraham-Minkowski ontroversy. The quantity at issue is the momentum per photon available for transfer from the light to the medium. Most theoretial work is onerned with nondispersive media, but the more general alulations presented here allow for material dispersion, with the phase refrative index p different from the group refrative index g. The aepted onvention assoiates the term Abraham momentum with the quantity / p or / g and the term Minkowski momentum with the quantity p /. Itis onvenient to follow this nomenlature, with Abraham and Minkowski used as labels for these forms of momentum transfer. The few measurements of radiation pressure in dieletri media appear to support the Minkowski value for the momentum transfer, and several alulations strive to produe this result from theories that obstinately seem to support the Abraham expression. Observations of radiation pressure in media rely on measurements of the fores exerted by light on some distinguishable omponent of the medium itself or on some objet immersed in the medium. We take the view that the Lorentz fore provides the fundamental desription of radiation pressure effets and use it as the basis for the present alulations. This same approah has been used previously 2,3 in treatments of the radiation pressure on a general semi-infinite dieletri. It has the advantage that no prior assumptions are made about the magnitude of the optial momentum in the medium. However, the results of the alulation do provide information on the transfer of momentum per photon to the observed objet. Moreover, the alulated fore represents the measured quantity, and not some subsidiary quantity that may not itself be diretly measurable. As for the ontroversy between different formulations of eletromagneti theory, we believe that all formulations are equally valid and that they produe the same preditions when properly applied to speifi problems. Any ontroversy arises from the improper isolation of momentum densities from other ontributing terms in the relevant ontinuity equations. The photon drag effet, observed in ,5, is one of the simplest manifestations of radiation pressure. The main effet is the generation of urrents or eletri fields in semiondutors, notably germanium and silion, by the transfer of momentum from an inident light beam to the harge arriers. The requirements of energy and momentum onservation generally forbid the absorption of photons by free arriers, and the proess an only take plae by interband transitions or with the assistane of phonon absorption or emission. The magnitude and sign of the transfer depend on the detailed band struture of the material in ompliated ways 6. However, for optial angular frequenies and harge arrier relaxation times suffiiently small that 1, the details of the band struture beome irrelevant 7. The momentum transfer to the harge arriers was observed to approah a value given by p / per photon for suffiiently long optial wavelengths, although the onditions of the experiments were suh that the phase and group refrative indies ould not be distinguished. This same form of limiting momentum transfer was found experimentally for both n- and p-type germanium and silion 7. The observed effet of the material is thus to multiply the free-spae photon momentum / by an additional fator p, and this was taken as evidene in support of the Minkowski momentum transfer. The alulations presented here show that the transfer of momentum to the harge arriers is aompanied by an additional transfer of momentum to the host semiondutor. The outstanding advantage of the photon drag measurements /2005/716/ /$ The Amerian Physial Soiety

2 LOUDON, BARNETT, AND BAXTER et al. is their ability to separate the ontributions from the momentum transfer to the harge arriers, assoiated essentially with the extintion oeffiient, from the unobserved momentum transfer to the host material, assoiated with the real refrative index p. The alulated momentum transfer to the harge arriers has the Minkowski value of p / per photon, in agreement with experiment. Calulations equivalent to those reported here ould also be performed in the framework of lassial eletromagneti theory, with similar onlusions. However, the quantum theory is easily applied and it has the advantage of providing results diretly expressed in terms of momentum transfers per inident photon. The general formulations of the eletromagneti energymomentum stress tensor by Abraham 8, Minkowski 9, and others are disussed in Se. II, where the equivalene of the superfiially different theories for the photon drag effet is demonstrated. The relevant optial properties of semiondutors are summarized in Se. III, and the Lorentz fores on the harge arriers and host semiondutor are alulated in Se. IV. The time dependenes of the fores on the two omponents produed by a narrow-band single-photon pulse are alulated in Se. V, and the results are approximated for some limiting values of pulse length, attenuation length, and sample thikness. In partiular, when the lengths as listed are in asending orders of magnitude, it is shown that the bulk material aquires an Abraham total momentum transfer of / g per photon. Our onlusions are summarized in Se. VI. II. ELECTROMAGNETIC STRESS TENSORS A. History Two varieties of stress tensor for the eletromagneti field in a material medium were formulated early in the last entury by Abraham 8 and Minkowski 9. This setion is devoted to a brief review of these formulations and others, in order to provide a bakground for the results on the photon drag effet derived in the following setions. These results are based on evaluations of Lorentz fores and are independent of any speifi formulation. However, it is emphasized here that the Abraham and Minkowski theories are equally valid when they are properly applied to speifi problems. The refrative index and the eletri suseptibility = 2 1 are taken to be real and independent of frequeny for the alulations of this setion. The stress tensors appear in equations expressing linear momentum onservation, whih are often written as g A j 3 ij TA j + i=1 x i = f A and g M j 3 ij TM + i=1 x i =0, 2.1 where the subsripts A and M indiate Abraham and Minkowski. The tensorial omponents of the momentum flux density of the light beam are 1 T A ij = 1 2 ij E D E i D j E j D i ij H B H i B j H j B i T M ij = 1 2 ij E D E i D j ij H B H i B j, 2.2 where the A form is symmetri in the spatial oordinates i, j=1, 2, 3 but the M form is not. The dependene of the field vetors on spae and time is not shown expliitly. The momentum densities are g A = E H/ 2 and g M = D B 2.3 and the so-alled Abraham fore density is f A = 0 E B. 2.4 The assoiation of eah tensor with a different momentum density has been regarded historially as signifiant 1, as has the presene of the fore density in the Abraham formulation. To add to the ontroversy, from time to time other eletromagneti tensors, suh as those of Einstein and Laub 10 and Peierls 11,12, have been onstruted. All these tensors exist against the bakground of standard eletromagneti theory, ontaining the well-known symmetri, Maxwell and anonial tensors 13. The Einstein-Laub and Peierls tensors are not formulated from the anonial theory, but diretly from the physis of speifi material systems, and we leave onsideration of these tensors until Se. II B. Many of the arguments of the Abraham-Minkowski ontroversy have entered on features that are either irrelevant for partiular experimental onditions, or for optial media in general, or are inonsistent with the fundamental harateristis of the anonial formulation of a tensor from a Lagrangian density. So it is the issues involving the presene of physial boundaries, inhomogeneity, and anisotropy, although important in radiation pressure alulations, ought not, in priniple, to arise from a rigorous tensor treatment. For example, the variation of the Lagrangian density used to formulate the tensor assumes that the mathematial boundary is idential to the boundary of the material medium. Indeed, sine any eletromagneti tensor is a omponent in a differential onservation equation, it is not surprising that tensors are indiative of bulk material only, where physial boundaries are removed from the oordinates of interest. Furthermore, the anonial determinations of Abraham 8 and Minkowski 9 stress tensors assume that the material s relative permittivity and permeability are real parameters. Notwithstanding, the literature ontains apparent generalizations of these tensors, supposedly obtained by retrospetive introdution of material inhomogeneity or anisotropy. Alternatively, one sometimes reads the laim that a partiular tensor is unsound beause it failed to inlude these generalizations in the first plae 11. The anonial determination of a stress tensor of a field involves the Hamilton derivative, and it takes plae at a level independent of the equations of motion 14,15. The proedure is to find the onditions for an extremum of ation by varying the Lagrangian density with respet to the spaetime metri, hene the name stress tensor. The Hamilton derivative method is appliable to urved and Eulidean spae-times, whereas the standard determination of the a

3 RADIATION PRESSURE AND MOMENTUM TRANSFER IN nonial tensor is restrited to flat spae-time. In general, the Hamilton derivative produes a tensor that obeys a onservation law, in that its ovariant derivative vanishes. A speialization to Eulidean spae-time redues the ovariant derivative to the ommon form involving the four-vetor gradient operator. A Lagrangian for the eletrodynamis of a material medium an be onstruted in two distint forms, depending on whether the phenomenologial relative permittivity and permeability of the material are introdued by means of matterindued urrents or by means of an effetive modifiation of the spae-time metri. The indued-urrents formalism gives rise to a minimal-oupling term in the Lagrangian, and it is perhaps better known than the formalism that involves an effetive metri. It also has diret experimental relevane to a material medium at rest on the optial benh. However, the effetive-metri formulation is fully ovariant and enables a natural introdution of the four-vetor veloity of the medium. The effetive metri is introdued formally; it is a funtion of the material four-vetor veloity as well as its phenomenologial parameters, and it performs the usual metri role of shifting between ovariant and ontravariant vetors by raising and lowering indies. After speialization to Eulidean spae, the Hamilton derivative of the indued-urrents Lagrangian yields where g S j 3 ij TS + i=1 x i = f j L, 2.5 T S ij = ij E 2 E i E j ij B 2 B i B j 2.6 is the symmetri stress tensor whih, with a hange of sign, is idential to the Maxwell stress tensor. The momentum density is g S = 0 E B 2.7 and f L is the Lorentz fore density, whih for our purposes may be written in the onventional forms f L = P E + P B = E2 + E B, 2.8 as given by 16. Note that the momentum density g S is the same as g A in 2.3 for a nonmagneti material with B = 0 H. The indued-urrents Lagrangian ontains an inherent matter-field separation; it is therefore unsurprising that it yields the physially signifiant Lorentz fore. This ontrasts with the effetive-metri Lagrangian, whih treats the eletromagneti field and the material medium as a single entity. It is only the effetive-metri Lagrangian that anonially produes the Abraham and Minkowski tensors. Therefore, at the enter of the Abraham-Minkowski ontroversy lies a Lagrangian whose division into field and matter omponents is ambiguous, and perhaps impossible. It is in any ase pointless to follow this route as the minimal-oupling induedurrents Lagrangian supplies all that is required for the kind of problem of interest here. To ounter the laim sometimes made that only the Abraham tensor is orret, we devote the remainder of this setion to an outline of the historial reasons for the laim and to the justifiation for safely ignoring it. The mathematial detail an be found elsewhere 14,17 19 and is omitted here. The Hamilton derivative proedure for the effetivemetri Lagrangian reveals what has sometimes been seen as a problem, whose apparent resolution is the origin of the laim for the unique orretness of the Abraham form. The variation of the Lagrangian with respet to the effetive metri, in fat, produes the Minkowski tensor, with a momentum onservation equation involving the vanishing of its four-divergene, as in the M part of 2.1. The laimed problem with this proedure is that the real spae-time metri is not the effetive metri but one that, loally at least, is Minkowskian. A orretion is supposedly made to ensure that the effetive-metri Lagrangian is varied with respet to the Minkowskian metri. This orretion results in the introdution of the Abraham tensor, whose divergene vanishes only in ombination with the Abraham fore, as in the A part of 2.1. The formalisms involving the indued-urrents and effetive-metri Lagrangians would be expeted to lead to similar results when the spae-projetion omponent of the medium s four-veloity vanishes, and this is indeed the ase. The Abraham and Minkowski momentum onservation equations desribe the same situation and lead to idential results in a spatially stationary frame. This an be appreiated immediately by noting that the effetive metri in this ase differs from the Minkowski metri only by the presene of the refrative index in its salar omponent. Suh an alteration in the speed of the salar eletromagneti field is simply a gauge transformation and an lead to no new physis. In this situation, the Abraham, Minkowski, and symmetri formulations are equally valid, and 2.1 and 2.5 are equivalent expressions of momentum onservation. In a spatially stationary frame it is meaningless to laim that one or the other tensor is orret. The soure of the Abraham- Minkowski ontroversy lies in a failure to understand fully the anonial onstrution of the relevant tensors. The differene between the A and M momentum densities in 2.3 an be a soure of onfusion only if they are detahed from the other ontributions to the ontinuity equations in 2.1 and individually given physial signifiane. The induedurrents formalism, but not, of ourse, the effetive-metri formalism, plaes no restrition on the material effetive permittivity and permeability, as these parameters are impliit within the indued polarization urrents. Material inhomogeneity or anisotropy ould therefore be introdued in the symmetri or Maxwell stress formulation. For onsisteny, the equivalene between symmetri, Abraham, and Minkowski formalisms for vanishing spae-projetion omponent of the four-veloity is restrited to homogeneous and isotropi media. A possible signifiane of the Abraham tensor in general relativity, whih the Minkowski tensor laks, is not an issue here. We note, however, that a reent review 20 of 83 years of progress and problems in general relativity and osmology fails to mention the ontroversy

4 LOUDON, BARNETT, AND BAXTER et al. B. Stress tensors and photon drag Photon drag is a phenomenon of the bulk semiondutor. The light fields vary with only the one oordinate z for the experimental arrangement onsidered here, with E and B parallel to the x and y axes, respetively, and Maxwell s equations simplify to where g A j 3 ij TE + i=1 x i = f j L, T E ij = ij E 2 E i D j ij H 2 H i B j, E z + B H = 0 and z + D = With material dispersion and loss ignored, the fields are funtions of the argument z t/ and the solution B =E/ of 2.9 allows the momentum flux densities from 2.2 and 2.6 to be written T zz A = T zz M = 0 2 E 2 = U and T zz S = E 2, 2.10 where U is the energy density of the eletromagneti field in the bulk semiondutor. The momentum densities from 2.3 and 2.7 have the values g A z = U/ = g S z and g M z = U/ These are the onventional Abraham and Minkowski momenta and, for onsisteny with other authors, we follow this nomenlature here to denote the momentum transfers from single photons of energy to the material medium. The equivalene of the Abraham, Minkowski, and symmetri formulations is expliitly demonstrated by use of 2.9 to rewrite all three momentum onservation equations from 2.1 and 2.5 as the single form U + U z =0, 2.12 when the fore omponents f z A and f z L from 2.4 and 2.8 are ombined with appropriate terms on the left-hand sides of the relevant equations. The total momentum flux density of the light beam is idential to the energy density U in the semiondutor, and the momentum density is idential to the Minkowski value in It may therefore seem that we have simply onfirmed the assertion of Gibson et al. 7 that the photon drag experiment supports the Minkowski rather than the Abraham formulation, but this is not the ase. We have only demonstrated that the Minkowski formulation represents the simplest but not the only valid matter-field division of momentum within a bulk material medium. Provided that the momentum density from 2.3 is not detahed from 2.1 and treated in isolation, the Abraham formalism produes exatly the same momentum onservation equation 2.12 as the other approahes. The Einstein-Laub 10 E tensor formalism is also onsistent with the photon drag results. The tensor is not onstruted anonially by means of a Hamilton derivative but from a onsideration of the reation of a moleular dipole to eletromagneti fields. The momentum onservation equation is the momentum density is idential to the Abraham value in 2.3, and the fore density has the Lorentz form from 2.8. The momentum onservation equation an again be written in the form of The Peierls theory 11,12 uses loal, rather than marosopi, fields with a dieletri funtion or refrative index that satisfies the Clausius-Mossotti or Lorenz-Lorentz formula. The use of loal fields is inappropriate for semiondutors, where the mobile valene and ondution eletrons are not loalized on individual atoms, and we onsider the Peierls theory no further. The analysis given here provides a somewhat different perspetive on the various formulations of the eletromagneti stress tensors to that presented by Brevik 21, but our onlusions are substantially the same. The interpretation of the photon drag experiments is independent of the formalism, whether that of Abraham, Minkowski, or Einstein-Laub. All of these provide valid frameworks for analysis of the measurements, whih do not themselves favor one formalism or the other. III. OPTICAL PROPERTIES OF SEMICONDUCTORS The optial properties of semiondutors are desribed in the usual way by a dieletri funtion and a omplex refrative index n, related by = n = + i. 3.1 The photon drag measurements were made at optial wavelengths out to 1.2 mm in Ge and Si. At this wavelength, 0.6 in both n- and p-type Ge, and therefore the ondition 1 for free-arrier absorption independent of the semiondutor band struture was not stritly met. However, the trend toward a limiting value of p /, rather than / p, for the momentum transfer was learly apparent. On the other hand, 0.2 for n-si and 0.1 for p-si at 1.2 mm, and the experimental results for this semiondutor were in very lose agreement with p / as the limiting value for the momentum transfer at long wavelengths 7,22. However, it is only for Ge that the optial properties in the submillimeter range have been arefully measured and interpreted 23, and the numerial values quoted below refer to this material. It an be assumed to good approximations that the harge arriers make a purely imaginary ontribution to, whereas the host material makes a purely real ontribution 23 in the regime where 1. The phase refrative index has the value p 4.0 for Ge and varies very slowly with frequeny for the onditions of the photon drag experiments. The frequeny dependene is, however, retained here for greater generality of the alulations that follow. The value of the extintion oeffiient is of order 0.1 or less, and we an write

5 RADIATION PRESSURE AND MOMENTUM TRANSFER IN =1+ H + C 2 +2i, with, 3.2 where the host and harge-arrier suseptibilities are given by H = 2 1 and C =2i. 3.3 Referene 23 provides more detailed justifiation for the approximations made here. The thikness D of the semiondutor sample in the photon drag measurements was 30 mm, and the limiting lowfrequeny power attenuation length was l = /22.4 mm. 3.4 Thus essentially all of the photon momentum was transferred to the semiondutor. The inident light was in the form of pulses with free-spae lengths L of the order of 60 m, with a pulse length L/ p in Ge of the order of 15 m, muh larger than D and l. The experiments were, in fat, done with the sample in open-iruit onditions 5, so that the transfer of momentum to the harges resulted not in a urrent flow, but in an opposing eletri field within the semiondutor. The measured open-iruit voltage aross the two ends of the sample was proessed to provide values for the momentum transfer per photon. IV. LORENTZ FORCES ON CHARGE CARRIERS AND HOST We onsider the propagation of a polarized light beam parallel to the z axis with its eletri and magneti fields parallel to the x and y axes, respetively. The Lorentz fores on bulk dieletris and their surfaes have been alulated quantum mehanially 2,3 on the basis of the suseptibility of the material. We perform a similar alulation of the momentum transfer to the harge arriers based on C from 3.3. It is also shown that, beause of the attenuation provided by the harge arriers, there is a further momentum transfer to the host, based on the suseptibility H from 3.3, whih was not observable in the photon drag experiments. The harge and host ontributions are niely separated for the two-omponent optial system used in the photon drag experiments. The essential features of the momentum transfer are reprodued with the simplifying assumption of an inident light beam of uniform intensity over a ross-setional area A. The fields within the beam then vary only with z, and the positive-frequeny parts of the quantized field operators are 2,3 Ê + 1/2 x z,t = d â i0 4 0 A and exp i nz 4.1 Bˆ + yz,t = i0 d 4 0 A 3 1/2 nâ exp i nz. 4.2 Here â is the photon destrution operator at frequeny, with a orresponding reation operator â. The operator nˆ =0 dâ â 4.3 represents the number of photons in the light beam. The forms of the square-root normalization fators in 4.1 and 4.2 ensure that the time integrated energy flow at z=0 is given by 4.3 but with an additional weighting fator in the integrand 2,3. Note that, by omparison with a more general field quantization 24, only the real part of the omplex refrative index is retained in the normalization fators in view of the inequality in 3.2. However, the full refrative index n must be kept in the exponentials, as it ensures the proper attenuation of the light beam as momentum is transferred to the harge arriers. The samples used in the measurements 7 were suffiiently long that almost all of the optial momentum was transferred from the light beam, and a subsequent integration over z an aordingly be taken to extend from 0 to. The rates of hange of the momenta of the harge arriers and the host semiondutor, aused by their interation with the inident light, are determined by integrations of the Lorentz fore-density operator over time and over the illuminated spatial region. Only the seond term of the first form of the fore in 2.8 ontributes for the experimental geometry onsidered here, to give Pˆ z,t :fˆz,t:=: Bˆ z,t:. 4.4 The first fator on the right is the polarization urrent, where the operator Pˆ + xz,t is obtained from the eletri field operator in 4.1 by insertion of 0 in the integrand. The expliit forms of the urrent for the harge arriers and the host are given in 4.6 and 4.10, respetively. The use of normal ordering in 4.4, indiated by the olons, ensures that vauum ontributions are exluded. The time-dependent fore operator is obtained by integration over the illuminated sample volume as :Fˆ dz:fˆz,t:. 4.5 :=A0 The fore operator vanishes for a transparent sample of infinite length, but nonzero radiation pressure ontributions arise in the presene of attenuation and at the surfaes of noninfinite samples 2. A. Momentum transfer to the harge arriers The polarization operator is expressed in terms of the eletri field operator 4.1 via the harge-arrier suseptibility from 3.3, with the result

6 LOUDON, BARNETT, AND BAXTER et al. Pˆ + xz,t = i 1/2 0 d A 0 3/2 1/2 â exp i nz. 4.6 The Lorentz fore-density operator 4.4, therefore, takes the form : fˆz C z,t:= d 1/2 2 2 A0 d0 n + n * â â expi in * n z, 4.7 with the onvention that the optial variables,, and n are evaluated at frequeny, while,, and n are evaluated at. Terms in the produts of two reation or two destrution operators are negleted as they do not ontribute to the momentum transfer of interest here. The expetation value of the fore density an be alulated for speifi states of the inident light, as in previous work 2,3, and this is done in Se. V. However, for omparison with experiment, we are mainly interested in the total transfer of momentum, represented by the time-integrated fore-density operator dt: fˆz C z,t:= d p â âexp z/l/l, A0 4.8 where 3.4 is used to express in terms of the power attenuation length l. The derivation above shows that the phase refrative index ontrols the fore density and is aordingly replaed by p in 4.8, whih an often be taken as the value at the mean frequeny of the inident light. A further integral over the entire illuminated sample gives the form of the operator that represents the total transfer of linear momentum to the harge arriers as dt:fˆz A0 dz C d p â z,t:=0 â. 4.9 This final expression is the same as the photon-number operator 4.3 but with an additional weighting fator. It shows that the oupling of the light to the harge arriers via the Lorentz fore results in a alulated momentum transfer of p / per photon, the same Minkowski value as found in the photon drag measurements 7. B. Momentum transfer to host semiondutor The propagation of an optial pulse through a transparent dieletri auses no transfer of momentum to the material, as a positive Lorentz fore in the leading part of the pulse is exatly balaned by a negative Lorentz fore in its trailing part 2. However, this balane is removed in the present problem beause of the attenuation of the light by its interation with the harge arriers. This auses the leading part of the pulse at a given time to be weaker than the trailing part and produes a net negative transfer of momentum to the bulk semiondutor. There is also a positive surfae ontribution that arises from the lak of balane between leading and trailing parts as the pulse passes through the integration utoff at z=0 into the ative region of the sample at z0. Here we alulate the ombined momentum transfer from these two effets and disuss their separate ontributions in Se. V. The host suseptibility from 3.3 provides a ontribution to the polarization operator with the time derivative Pˆ + xz,t = 1/2 0 d 4A 0 3/2 1/2 2 1â exp i nz The Lorentz fore-density operator 4.4 is, aordingly, : fˆz H i z,t:= d 1/2 2 1n 4 2 A0 d0 n * 2 1â â expi in * n z with the same onvention as in 4.7 The time-integrated fore density is expressed with the use of 3.4 as dt:fˆz H z,t:= d 2 p 1â â 2A0 exp z/l/ p l, 4.12 where the derivation again shows that the phase refrative index is involved. A further integration over the illuminated sample gives the operator that represents the total transfer of linear momentum to the host semiondutor as A0 dz dt:fˆz H z,t:= 0 d 2 p 1 2 p â â, 4.13 a negative quantity in the usual ase where p 1. The total transfer of momentum to harges and host is represented by the operator A0 dz dt:fˆz C z,t + fˆz H z,t: d 2 p +1 =0 2 p â â, 4.14 where 4.9 and 4.13 are used. The total linear momentum transferred to a general dieletri medium by narrow-band light that arries a single photon of energy 0 in the medium at z=0 an be obtained by onsiderations of energy and

7 RADIATION PRESSURE AND MOMENTUM TRANSFER IN momentum onservation. When 0 0, as assumed here, the result is 3 total momentum transfer = 0 2 p +1, p onsistent with V. TIME-DEPENDENT FORCES FOR SINGLE-PHOTON PULSE The details of the transfer of momentum to the harge arriers and host rystal an be further investigated with the assumption of inident light in the form of a single-photon pulse. The optial state is defined by 25,26 1 = dâ 0, where 0 is the vauum state, and these states satisfy â1 = The normalized funtion desribes the spetrum of the photon pulse and a onvenient hoie is the narrow-band Gaussian of spatial length L, = 2 1/4exp L2 L , /L The narrow spetrum ensures that an often be set equal to the entral frequeny 0. The peak of the pulse defined in this way passes through the oordinate z=0 and into the ative region of the sample at time t=0. A. Momentum transfer to the harge arriers The expetation value of the Lorentz fore-density operator 4.7 for the single-photon pulse is obtained straightforwardly with use of the property 5.2 as 1:fˆz C z,t:1 = d 1/2 2 2 A0 d0 n + n * * expi in * n z. 5.4 The frequeny integrals are now to be evaluated without the prior integration over time in 4.8. It is useful to approximate the integrand for frequenies and in the viinity of 0. With use of 3.4 and the inequalities in 3.2 and 5.3, standard Taylor expansions give and n + n * 2 0 p 2 0 = p 2 /l n * n i/l + g, where the group refrative index is defined by g = = p Straightforward integration now gives 1: fˆz C z,t:1 = 2 p 0 AlL exp z l 2 gz 2 2 L The expetation value of the fore operator defined in 4.5 is obtained by integration over z and, with use of the standard Gaussian integral and the definition of the omplementary error funtion, the result is 1:Fˆ z C :1 = p 0 2 g l exp t g l + 2 L2 8 2 g l erf 2t L + L 2 2g l. 5.9 The fore expetation value learly vanishes in the absene of the attenuation aused by the harge arriers when l. Two limiting ases are of interest. For a pulse that is muh shorter than the attenuation length, 5.9 redues approximately to 1:Fˆ z C :1 = p 0 t 2t exp, L/ p l. 2 g l g lerf L 5.10 The value of the omplementary error funtion inreases from 0 to 2 over a time of order L/ as the pulse enters the ative region of the sample at z0 and the fore subsequently deays over a longer time of order g l/ as the pulse is attenuated. For a pulse that is muh longer than the attenuation length, we use the asymptoti form 27 erf x exp x 2 /x to redue 5.9 to :Fˆ z C :1 = 2 p 0 exp 22 t 2 L L 2, L/ p l The time dependene is now entirely determined by the freespae pulse shape. Note that integration of 5.10 or 5.12 produes a total momentum transfer to the harge arriers of dt1:fˆ z C :1 = p 0 /, 5.13 in agreement with the expetation value of 4.9 for a singlephoton narrow-band pulse. B. Momentum transfer to the host semiondutor The expetation value of the Lorentz fore-density operator 4.11 for the host semiondutor is obtained in a similar manner to that used for the harge arriers. We need the additional Taylor expansion

8 LOUDON, BARNETT, AND BAXTER et al. 2 1n n * 2 1 i 2 p 1/l + 2 p g + g 2 p, 5.14 and the resulting single-photon pulse expetation value is 1:fˆz H z,t:1 = 2 0 p AL 2 p 1 2l + 2 p g + g 2 p gz 22 exp z l 2 gz 2 2 L 2, L analogous to 5.8. The orresponding expetation value of the fore operator 4.5 is 1:Fˆ z H :1 = 0 2 p g p l p 1 gexp t g l + 2 L2 8 2 g l 2t erf L + L 2 1 L 2 p g + g 2 p exp 22 t 2 L 2, g l analogous to the form of the harge-arrier fore in 5.9. It is again instrutive to onsider two limiting ases. For a pulse that is muh shorter than the attenuation length, 5.16 redues approximately to 1:Fˆ z H :1 = 0 2 p g p l p 1 gexp t erf 2t L exp 22 t 2 g l 2 1 L 2 p g + g 2 p L 2, L/ p l Integration over the time produes a total momentum transfer to the host semiondutor of dt1:fˆ z H :1 = 0 1 g p + p p g, 5.18 where the two terms in the main braket are the ontributions of the orresponding terms in In the opposite limit of a pulse that is muh longer than the attenuation length, use of the asymptoti form 5.11 leads to 1:Fˆ z H :1 = p 1 exp 22 t 2 2 p L L 2, L/ p l, 5.19 FIG. 1. Time dependene of the fores on harge arriers and host for a semiondutor whose attenuation length is 10 times the single-photon pulse length in the medium. The bulk fore on the harges CB is given by 5.10, while the bulk HB and surfae HS fores on the host are given by the two terms in The fore is in units 0 /2l, the time is in units L/ 2, and p = g =4. with a time variation idential to that of the harge-arrier fore in Further integration of 5.19 over time produes a total momentum transfer of dt1:fˆ z H :1 = 0 p p, 5.20 idential to the summed value in 5.18 and in agreement with the expetation value of 4.13 for a single-photon narrow-band pulse. The total transfer of momentum to the ombined system of harge arriers and host semiondutor obtained by summation of 5.13 and 5.20 agrees with C. Disussion A more omplete understanding of the detailed proesses of momentum transfer from light to harge arriers and host semiondutor is provided by onsideration of some limiting ases. i L/ p ld. These onditions ensure that the passage of the pulse into the ative region of the sample ours in a time short ompared to the duration of its subsequent attenuation in the bulk material and that none of the light reahes the far boundary of the sample. The surfae and bulk ontributions to the Lorentz fore, desribed at the beginning of Se. IV B, are learly separated in this ase. They are haraterized by a time dependene similar to that of the Gaussian pulse itself for the surfae ontribution and an exponential fall-off over the attenuation time g l/ for the bulk ontribution. The time-dependent fore 5.10 on the harge arriers shows only a bulk ontribution, while the fore 5.17 on the host shows both bulk and surfae ontributions given, respetively, by the two terms in the large braket. Illustrations of the time dependenes of these three ontributions are shown in Fig. 1. The total time-integrated fore obtained from 5.13 and 5.18 is

9 RADIATION PRESSURE AND MOMENTUM TRANSFER IN dt1:fˆ z H + Fˆ z C :1 = 0 2 p p g = 0 host surfae 2 p +1 2 p surfae 1 g + 1 g p host bulk + 1 g bulk. + p harge bulk 5.21 Combination of the harge arrier and host momentum transfers to the bulk thus produes the Abraham value of 0 / g, in ontrast to the Minkowski value of p 0 / produed by the harge arriers alone. This Abraham value represents the total momentum available for transfer from the pulse to the bulk material one it has fully entered the ative region of the sample, at the relatively very short time of order L/. A total bulk momentum transfer of 0 / was found previously 2 for a semi-infinite dispersionless dieletri in the form of a single-omponent optial system, without the separation into and ontributions, whih ours in the photon drag effet. ii ldl/ p. The ondition that none of the light reahes the far boundary of the sample is retained, but now the pulse is muh longer than both the attenuation length and the sample thikness. This is the regime of the photon drag measurements 7. The time-dependent fores on the harge arriers and host semiondutor, given by 5.12 and 5.19, respetively, are ompletely determined by the Gaussian pulse shape. The momentum transfer per photon to the harge arriers retains the Minkowski value of p 0 /, and the total transfer to the host retains the value given in 5.20, but there is no longer the separation into surfae and bulk ontributions shown in The proportionality of the harge arrier fore in 5.12 to the pulse intensity is reprodued in the indued voltage generated aross the semiondutor sample by the photon drag effet. The measured voltage forms the basis for the photon drag detetor, a devie that measures infrared pulse profiles 22. iii L/ p Dl. Conditions in whih the attenuation length is muh greater than the sample thikness have been treated in earlier work 2,3, This limit is not overed by the present alulations on the photon drag effet, but the results are relevant to the value of the total momentum transfer derived here. With negligible attenuation over the sample thikness D, all of the inident light is either refleted or transmitted. The surfae fore mehanism desribed at the beginning of Se. IV B ontinues to operate, but the positive fore generated as the pulse enters the medium at z=0 is at least partially ompensated by a negative fore generated by the same mehanism as the pulse leaves the medium at z =D. The two fores exatly anel for a slab of dieletri with antirefletion oatings 3,29 when the passage of the pulse shifts an initially stationary slab to a new stationary position, with no permanent transfer of momentum. However, appliation of momentum onservation to the state of the system when the pulse lies within the slab 30 produes an optial momentum of 0 / g per photon, in agreement with the Abraham value derived here for the total momentum transfer. More ompliated behavior ours in the presene of refletion at the slab surfaes 2,28. The results for a transparent slab emphasize the qualitative differene between the reversibility of the surfae fore and the irreversibility of the bulk fore. VI. CONCLUSIONS The main results of the alulations are the expressions 4.9 and 4.13 for the transfers of momentum from light to the harge arriers and host semiondutor in the photon drag effet and the expressions 5.9 and 5.16 for the respetive time-dependent fores produed by a single-photon pulse. The alulated transfer of linear momentum to the harge arriers in 4.9 agrees with the measurements 7 in assigning the Minkowski value of p / per photon. The transfer of momentum to the host semiondutor in 4.13 is essentially fixed by momentum onservation, whih requires the total momentum transfer per photon to have the value given in The momentum transfers alulated in Se. IV are valid for all forms of inident light that satisfy the overall ondition 1 disussed in Se. III. Additional information on the detailed proesses of momentum transfer is provided by the alulations of Se. V for an inident single-photon pulse, and the signifiane of the results is disussed in Se. V C. The main additional feature emerges for pulses muh shorter than the attenuation length, when the momentum transfer to the host semiondutor an be separated into surfae and bulk ontributions, as in 5.21; the total bulk momentum transfer in this ase, harges plus host, equals the Abraham value of 0 / g. The pratial devie of the photon drag detetor relies on the opposite limit of pulses muh longer than the attenuation length, when the time-dependent voltage generated by the fores on the harge arriers mimis the optial pulse shape. Our method of alulation, based on evaluation of the Lorentz fore, has the advantage of providing results for the momentum transfers from light to marosopi media measurable in experiments. A great deal of previous work, well reviewed in 1, is onerned with the identifiation of the momentum arried by the photon in dieletri media, often viewed as a onflit between the Abraham and Minkowski expressions, with one or the other or neither regarded as the orret answer for the system in question. We have argued in Se. II that there is no onflit between the Abraham, Minkowski, and other formulations of the eletromagneti stress tensor. Furthermore, we have shown in Ses. IV and V that there is no unique expression for the momentum transfer from light to matter in the photon drag effet but that the Abraham and Minkowski values an both, in priniple, be observed by appropriate measurements. Another fous of previous work that is untouhed by our method of alulation is the devision of total momentum between eletromagneti and material ontributions, and we an make no omment on this aspet of the problem. Our speifi results for the photon drag system harmonize with some previous, more general disussions of photon momentum. Thus, Gordon 16 finds the Abraham form

10 LOUDON, BARNETT, AND BAXTER et al. 0 / g for the true field momentum and the Minkowski form p /, identified as the pseudomomentum, for the momentum transfer to material objets in dieletri media. Joye 31 presents a lassial partile desription of the radiation pressure problem, with the Abraham form for the energy-arrying momentum that determines the displaement of a slab and the Minkowski form for the impulsive momentum that determines the momentum transfer to a refletor immersed in the medium. MIntyre 32 onsiders more general forms of wave and matter, partly in the ontext of fluid mehanis, and reahes onlusions similar to those in 16. Nelson 33 finds the Abraham form for the eletromagneti field in nonmagneti media and the Minkowski form for the wave momentum that enters wave-vetor onservation relations. Very reently, Garrison and Chiao 34 have given a quantum theory of the eletromagneti momentum in a dispersive dieletri, finding that the Abraham form determines the rigid aeleration of a dieletri, while their anonial momentum, equivalent to the Minkowski form as defined by us, determines the transfer to immersed objets. These authors also use marosopi field quantization of the lassial expression to derive a form of Minkowski momentum that differs from ours by an additional fator p / g, not detetable in the photon drag experiments of 7. By ontrast, Mansuripur 35,36 has onsidered experiments sensitive to the total momentum transfer 4.15, equal to the arithmeti mean of the Abraham and Minkowski forms. The best measurements of the momentum transfer to an immersed objet are those of Jones and Leslie 37. They suspended a highly refleting mirror in a range of dieletri liquids and observed the Minkowski transfer of p / per photon. The measurements were suffiiently aurate to establish the ourrene of the phase refrative index p in the momentum transfer, and not the group value g. The system is somewhat similar to that in the photon drag effet, with the optial properties of the mirror determined by its extintion oeffiient and those of the host liquid by its real refrative index p. A Lorentz fore alulation 2 again provides an expression for the momentum transfer in agreement with experiment. It was shown that the liquid takes up a momentum transfer equal to the differene between the Abraham and Minkowski values, as given by the host bulk term in 5.21, although the theory in 2 did not distinguish phase and group refrative indies. A similar interpretation of the mirror experiments has been given by Mansuripur 35. On the basis of these two examples, it appears that the total momentum transfer to bulk material, free of any boundary or surfae effets, has the Abraham value of 0 / g but that the transfer to an attenuating subsystem within the bulk material has the Minkowski value of p /. The photon drag system treated here has a remarkable ombination of properties, in providing bases for both the pratial devie of the photon drag detetor and the theoretial understanding of momentum transfer from light to matter. The separate ontributions of the two system omponents to the real and imaginary parts of the omplex refrative index in the photon drag effet provide a uniquely simple division of the momentum transfers to the host semiondutor unmeasured and harge arriers measured, respetively. Note added in proof: The Minkowski momentum transfer to the harge arriers in the photon drag effet has been derived by an alternative treatment 38. ACKNOWLEDGMENTS We thank Andy Walker for enouragement and Maurie Kimmitt for muh advie on the proedures and interpretation of the photon drag experiments. Work at the University of Strathlyde was supported by the UK Engineering and Physial Sienes Researh Counil. 1 I. Brevik, Phys. Rep. 52, R. Loudon, J. Mod. Opt. 49, R. Loudon, Phys. Rev. A 68, A. M. Danishevskii, A. A. Kastalskii, S. M. Ryvkin, and I. D. Yaroshetskii, Sov. Phys. JETP 31, A. F. Gibson, M. F. Kimmitt, and A. C. Walker, Appl. Phys. Lett. 17, A. F. Gibson and S. Montasser, J. Phys. C 8, A. F. Gibson, M. F. Kimmitt, A. O. Koohian, D. E. Evans, and G. F. D. Levy, Pro. R. So. London, Ser. A 370, M. Abraham, Rend. Cir. Mat. Palermo 28, 11909; 30, H. Minkowski, Nahr. Ges. Wiss. Goettingen, Math.-Phys. Kl., ; Math. Ann. 68, A. Einstein and J. Laub, Ann. Phys. 26, ; 26, R. Peierls, Pro. R. So. London, Ser. A 347, ; 355, R. Peierls, More Surprises in Theoretial Physis Prineton University Press, Prineton, 1991 pp J. D. 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11 RADIATION PRESSURE AND MOMENTUM TRANSFER IN 25 K. J. Blow, R. Loudon. S. J. D. Phoenix, and T. J. Shepherd, Phys. Rev. A 42, R. Loudon, The Quantum Theory of Light, 3rd ed. Oxford University Press, Oxford, H. B. Dwight, Tables of Integrals and other Mathematial Data, 4th ed. Mamillan, New York, M. S. Kim, L. Allen, and R. Loudon, Phys. Rev. A 50, see also the Appendix of M. Padgett, S. M. Barnett, and R. Loudon, J. Mod. Opt. 50, R. Loudon, Fortshr. Phys. 52, W. B. Joye, Am. J. Phys. 43, M. E. MIntyre, J. Fluid Meh. 106, D. F. Nelson, Phys. Rev. A ; R. Loudon, L. Allen, and D. F. Nelson, ibid. 55, J. C. Garrison and R. Y. Chiao, Phys. Rev. A 70, M. Mansuripur, Opt. Express 12, M. Mansuripur, A. R. Zakharian and J. M. Moloney, Opt. Express 13, R. V. Jones and B. Leslie, Pro. R. So. London, Ser. A 360, M. Mansuripur, Opt. Express 13,