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Barnett, S. M., and Loudon, R. (2015) Theory of radiation pressure on magneto–dielectric materials. New Journal of Physics, 17(6), 063027. Copyright © 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft Reproduced under Creative Commons License CC BY 3.0 Version: Published http://eprints.gla.ac.uk/107672/ Deposited on: 26 June 2015

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New J. Phys. 17 (2015) 063027 doi:10.1088/1367-2630/17/6/063027

PAPER

Theory of radiation pressure onmagneto–dielectric materials

StephenMBarnett1,3 andRodney Loudon2

1 School of Physics andAstronomy,University ofGlasgow,GlasgowG12 8QQ,UK2 Computer Science and Electronic Engineering, University of Essex, Colchester CO4 3SQ,UK3 Author towhomany correspondence should be addressed.

E-mail: stephen.barnett@glasgow.ac.uk and loudr@essex.ac.uk

Keywords: radiation pressure,magneto–dielectrics, opticalmomentum

AbstractWepresent a classical linear response theory for amagneto–dielectricmaterial and determine thepolariton dispersion relations. The electromagneticfieldfluctuation spectra are obtained andpolaritonsumrules for their optical parameters are presented. The electromagneticfield for systemswithmultiplepolariton branches is quantized in three dimensions andfield operators are converted to 1–dimensionalforms appropriate for parallel light beams.We show that thefield–operator commutation relationsagreewith previous calculations that ignored polariton effects. TheAbraham(kinetic) andMinkowski(canonical)momentumoperators are introduced and their corresponding single–photonmomenta areidentified. The commutation relations of these andof their angular analogues support the identification,in particular, of theMinkowskimomentumwith the canonicalmomentumof the light.We exploit theHeaviside–Larmor symmetry ofMaxwell’s equations to obtain, very directly, the Einsetin–Laub forcedensity for action on amagneto–dielectric. The surface and bulk contributions to the radiation pressureare calculated for the passage of anoptical pulse into a semi–infinite sample.

1. Introduction

At the heart of the problemof radiation pressure is the famous Abraham–Minkowski dilemma concerning thecorrect formof the electromagneticmomentum in amaterialmedium [1–4]; a problemwhich, despite of itslongevity, continues to attract attention [5–9]. The resolution of this dilemma lies is the identification of the twomomenta, due toAbrahamandMinkowski, with the kinetic and canonicalmomenta of the light, respectively[10]. It serves to indicate,moreover, why the differentmomenta are apparent in different physical situations [9].In this paper we shall be concernedwithmanifestations of opticalmomentum in radiation pressure onmediaand, in particular, onmagneto–dielectricmedia.

Most existingworkon the theoryof radiationpressure, as reviewed in [5–9], treatednon-magneticmaterials butthere has been recent progress in thepartial determinationof the effects ofmaterialmagnetization.The rise of interestinmetamaterials and inparticular thosewithnegative refractive index addsurgency to addressing this point [11].

We considermagneto–dielectricmaterials inwhich both the electric permittivity ε and themagneticpermeability μ are isotropic functions of the angular frequencyω. The quantum theory of the electromagneticfield for suchmedia is nowwell–developed, with analyses in the literature based onGreen’s functions [12, 13]and also onHopfield–likemodels based on coupling to a harmonic polarization andmagnetization field [14–17]. The approachwe shall adopt is toworkwith the elementary excitations within themedium, which arepolarition coupled–modes of the electromagnetic fieldwith the electric andmagnetic resonances. The classicallinear–response theory in section 2 allows direct calculations of the electric andmagnetic field–fluctuationspectra. The corresponding quantized field operators are expressed in terms of the polariton creation anddestruction operators andwe show that these satisfy the required standard commutation relations.We use theseto construct theMinkowski andAbrahammomentumoperators and also the corresponding angularmomenta,and showhow the commutators of thesewith the vector potential facilitate the interpretation of the rivalmomenta [10]. The known formof themagnetic Lorentz force is confirmed and used to explore themomentum

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transfer from light to a half–space sample and so provide the extension to permeablemedia of earlier work ondielectrics [18].

2. Classical theory

The original calculations byAbraham andMinkowski determined the electromagneticmomentum intransparentmedia, as has the greatmajority of the subsequent literature referenced here.We accordingly assumea real permittivity and permeability. These arise naturally from the dispersion relations in aHopfield–typemodel inwhich the electromagnetic field is coupled to harmonic polarization andmagnetization fieldsassociatedwith the hostmedium [19]. Including absorption losses presents no fundamental difficulties [12–17]but adds significantly to the technical nature of the analysis; a feature thatmight obscure the simplicity of theunderlying physics.

2.1. Linear responseThe relative permittivity and permeability of a (lossless)magneto–dielectric at angular frequencyωmay bewritten in the simple forms [20]

( )

( ) , (1)

e

ne

e

m

nm

m

1

L2 2

T2 2

1

L2 2

T2 2

e

m

∏

∏

ε ωω ωω ω

μ ωω ωω ω

=−−

=−−

=

=

where ne and nm are the numbers of electric andmagnetic dipole resonances in themedium, associatedwithlongitudinal and transverse frequencies indicated by the subscripts L andT. It follows that the values of thepermittivity and permeability at zero and infinite frequencies are

(0) , ( ) 1

(0) , ( ) 1, (2)

e

ne

e

m

nm

m

1

L2

T2

1

L2

T2

e

m

∏

∏

εωω

ε

μωω

μ

= ∞ =

= ∞ =

=

=

in accordwith the generalized Lyddane–Sachs–Teller relation [21]. The phase refractive index is

( ) ( ) ( ) , (3)pη ω ε ω μ ω=

as usual [22].We consider amagneto–dielectricmedium in the absence of free charges and currents, so that our electric

andmagnetic fields are governed byMaxwell’s equations in the form

t

t

DB

EB

HD

· 0· 0

. (4)

==

× = − ∂∂

× = ∂∂

The electromagnetic fields described by these propagate as transverse plane–waves, with an evolution describedby the complex factor tk rexp(i · i )ω− , with k k= ∣ ∣andω related by the appropriate dispersion relation. Thefour complex electric andmagnetic fields are related by [22]

D E P E

B H M H

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ), (5)

0 0

0 0⎡⎣ ⎤⎦

ω ε ω ω ε ε ω ω

ω μ ω ω μ μ ω ω

= + == + =

where P and M are, respectively, themedium’s polarization andmagnetization.Henceforth, for the sake ofbrevity, we omit explicit reference to the frequency fromour fields, permittivities and permeabilities.

Now suppose that external stimuli p and m with the same frequency are applied parallel to E and Hrespectively, so that the relations (5) become

D E p

B H m. (6)0

0 0

ε εμ μ μ

= += +

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New J. Phys. 17 (2015) 063027 SMBarnett andR Loudon

It is convenient to introduce 6–component field and stimulus variables defined to be

Z

V c

f E H

s p m

( , )

( , ), (7)0=

=

where Z0 0 0μ ε= is the usual free-space impedance andV is a suitably chosen sample volume. The energy ofinteraction between the field and the stimulus components is then

( )I f s V E p H m· · . (8)j

i i

1

6

0∑ μ= − = − +=

The solutions for thefield components obtained by substitution of (6) intoMaxwell’s equations can bewritten

f T s , (9)ij

ij j

1

6

∑==

where the linear responsematrix is

V

ck ck

ck ck

ck ck

ck ck

ck ck

ck ck

T1

Den

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

, (10)

z y

z x

y x

z y

z x

y x

0

2

2

2

2

2

2

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟

ε

ω μ ω ω

ω μ ω ωω μ ω ω

ω ω ω ε

ω ω ω εω ω ω ε

=

−

−−

−

−−

with the commondenominator

c kDen , (11)2 2 2εμω= −the zeroes of which give the dispersion relation for the field. The elements of thematrix T agreewith and extendpartial results obtained previously for non-magneticmedia [23, 24].

2.2. FieldfluctuationsThe frequency andwave–vector fluctuation spectra at zero temperature are obtained from theNyquist formula[23, 25]

( )f f TIm , (12)i j ijk,

π

=ω

where the required imaginary parts occur automatically inmagneto–dielectricmodels that include dampingmechanisms or, in the limit of zero damping, by the inclusion of a vanishingly small imaginary part inω. In thelatter case, which applies to ourmodel, wefind

kck

ck

cIm

1

Den 2. (13)

2

p p⎜ ⎟⎛⎝

⎞⎠

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

π δωη

δωη

→ − − +

The totalfluctuations are obtained by integration

f f f fV

f fkd d(2 )

d , (14)i j i j i jk0 0 3 ,

∫ ∫ ∫ω ωπ

= =ω ω

∞ ∞

where the frequencyω and the three–dimensional (3D)wave vector k are taken as continuous and independentvariables.

Consider plane–wave propagation parallel to the z-axis in a sample of length L and cross–sectional areaA.The frequency fluctuation spectra are then obtained from the one–dimensional (1D) version of (14) as

( )f fL

k T2

d Im . (15)i j ij2 0

∫π

=ω

∞

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New J. Phys. 17 (2015) 063027 SMBarnett andR Loudon

The contributions of the component x– and y–polarizedwaves give

( )

( )

(16)

E EA

B BA

D DA

H HA

4

4,

4,

4,

x y

y x

y x

y x

2 2 0

0

1 2

2 20 0

3 3 1 2

2 203 3

0

1 2

2 2 0

0

1 2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

ωπ

μ με ε

ωπ

ε εμ μ

ωπ

ε ε μ μ

ωπ

ε εμ μ

= =

= =

= =

= =

with the use of (10) and (13). TheE andH spectra, of course, have the usual relativemagnitude for amagneto–dielectricmaterial. The totalfluctuations are obtained by integration of the spectra overω as in (14).

2.3. PolaritonmodesThe combined excitationmodes of thematerial dipoles and the electromagnetic field are the polaritons [26].The transverse polaritons for amaterial of cubic symmetry, as assumed here, are twofold degeneratecorresponding to the two independent polarization directions. Their dispersion relation is determined by thepoles in the linear response function, the components of which are given in (10). The vanishing denominator(Den) gives

c k , (17)2 2 2p2 2εμω η ω= =

which is the desired transverse dispersion relation. For our relative permittivity and permeability, there aren n 1e m+ + transverse polariton frequencies for eachwave vector and these are independent of the direction ofk . They jointly involve all of the resonant frequencies in the electric permittivity and themagnetic permeability.It is convenient to enumerate the transverse polariton branches by a discrete index u, and there is no overlap infrequency ukω between the different branches. There are also n ne m+ longitudinalmodes at frequencies Le

ωand Lm

ω , independent of thewave vector. These contribute, for example, to Raman spectra at non-zeroscattering angles [20, 23, 27, 28] but they can be ignored here as they do not interact directly with a unidirectionaltransverse electromagnetic wave.

Subsequent calculations involve the polariton group index gη , defined in terms of the phase index pη :

( )c

kd

d

d d(18)g

pη

ω

ωη

ω= =

for ck/pη ω= . The two refractive indices satisfy a variety of sum rules over the branches, given by [29–31]

1

11 (19)

u u

u u

k

k

p

g

p g

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

∑

∑

η

η

η η

=

=

and by [32]

1

1, (20)

u u

u u

k

k

p g

p g

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

∑

∑

εη η

μη η

=

=

where all of the optical variables are evaluated at frequency ukω , so the sums run over every frequency ukω that isa solution of the dispersion relation (17) for a givenwave vector k .

3.Quantum theory

3.1. Field quantizationThe polaritons are bosonicmodes, quantized by standardmethods in terms of creation and destructionoperators with the commutation relation

4

New J. Phys. 17 (2015) 063027 SMBarnett andR Loudon

a a k kˆ , ˆ ( ) . (21)u u u uk k†

,⎡⎣ ⎤⎦ δ δ= − ′′ ′ ′

TheDirac andKronecker delta functions have their usual properties. The electromagnetic field quantizationderived previously for dispersive dielectricmedia [33] can be adapted for ourmagneto–dielectricmediumbyinsertion of polariton branch labels and by conversion to SI units and continuouswave vectors. It is convenienttowrite all of thefield operators in the form

t t tA r A r A rˆ ( , ) ˆ ( , ) ˆ ( , ), (22)( ) ( )= ++ −

where the second term is theHermitian conjugate of the first. The deduced formof the transverse (or Coulombgauge) vector potential operator is then given by

t a tA r k eˆ ( , )16

d ˆ ( )e , (23)u u

u

k

kk r

k( )

30

1 2

p g

1 2

i ·⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

∫∑π ε

μωη η

=+

where

a t a tˆ ( ) ˆ exp( i ). (24)u u uk k kω= −

The unit polarization vector ek is assumed to be the same for all of the polariton branches at wave vector k . Thetwo transverse polarizations for eachwave vector are not shown explicitly here but they are important for thesimplification ofmode summations [34] and they need to be kept inmind. For non-magneticmaterials, withμ=1, the vector potential (23) agrees with equation (5) from [35] and alsowith equation (A12) in [36] when thesummation over polariton branches is removed.

The electric field, E, andmagnetic flux density or induction, B, field operators are readily obtained from thevector potential as

tt

t

a t

E r A r

k e

ˆ ( , ) ˆ ( , )

i16

d ˆ ( )e (25)u u

u

k

kk r

k

( ) ( )

30

1 2

p g

1 2

i ·⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

∫∑π ε

ωμη η

= − ∂∂

=

+ +

and

t t

a t

B r A r

k k e

ˆ ( , ) ˆ ( , )

i16

d ˆ ( )e . (26)u u

u

k

kk r

k

( ) ( )

30

1 2

p g

1 2

i ·⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

∫∑π ε

μωη η

= ×

= ×

+ +

The remainingfield operators, the displacement field D and themagnetic field H, follow from the quantumequivalents of (5) as

t a tD r k eˆ ( , ) i16

d ˆ ( )e (27)u u

u

k

kk r

k( )

30

1 2p

g

1 2

i ·⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

∫∑π ε

ωεη

η=+

and

t a tH r k k eˆ ( , ) i16

d1

ˆ ( )e . (28)u u

u

k

kk r

k( )

30

1 2

p g

1 2

i ·⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

∫∑π ε ωμη η

= ×+

The above four operator expressions are consistent with previouswork [33]. Thefield quantization for amagneto–dielectricmediumhas also been performed in a quantum–mechanical linear–response approach[12, 13] that assumes arbitrary complex forms for the permittivity and permeability. The resulting expressionsfor the E andB operators contain the polariton denominator (11) in their transverse parts. A theory along theselines has been further developed by amoremicroscopic approach inwhich the explicit forms of the electric andmagnetic susceptibilities are derived [14, 15].

3.2. Field commutation relationsThe validity of the field operators derived here (or at least their self–consistency) is demonstrated by theconfirmation that they satisfy the required equal–time commutation relations [34, 37]. Thus it follows from theforms of our operators, given above, that

5

New J. Phys. 17 (2015) 063027 SMBarnett andR Loudon

A D e e

e e

r r k

k

r r

ˆ ( ), ˆ ( )i

16d e e

i

8d e

i ( ), (29)

i j

u u

i j

i j

ij

k

k r r k r rk k

k r rk k

3

p

g

i ·( ) i ·( )

3i ·( )

⎡⎣ ⎤⎦⎛⎝⎜⎜

⎞⎠⎟⎟ ⎡⎣ ⎤⎦

∫

∫

∑π

η

η

πδ

− ′ = +

=

= − ′

− ′ ′−

− ′

⊥

with recognition that the two exponents in the first step give the same contributions and a crucial use of the firstsum rule from equation (19). The transverse delta function, r r( )ijδ − ′⊥ , in thefinal step [34, 37, 38] relies on theimplicit summation of the contributions of the two transverse polarizations. The common time t is omittedfrom the field operators in the commutators here and subsequently as it does not appear in thefinal results. Analternative canonical commutator is that for the vector potential and the electric field:

A E e er r k

r r

ˆ ( ), ˆ ( )i

8d e

i ( ), (30)

i j

u u

i j

ij

k

k r rk k0 3

p g

i ·( )⎡⎣ ⎤⎦⎛⎝⎜⎜

⎞⎠⎟⎟

∫∑επ

μη η

δ

− ′ =

= − ′

− ′

⊥

with the use of a sum rule from equation (20). It follows from the operator formof the first relation inequation (5) that the vector potential and the polarisztion commute:

A Pr rˆ ( ), ˆ ( ) 0. (31)i j⎡⎣ ⎤⎦′ =

This ismost satisfactory as the two operators are associatedwith properties of different physical entities: theelectromagnetic field and themedium respectively.We note that the commutator in equation (29) has beengiven previously [10] for non–magnetic dielectricmedia.

The remaining non–vanishing field commutators are

E B k

E H k

D B k

D H k

r r k

r r

r r k

r r

r r k

r r

r r k

r r

ˆ ( ), ˆ ( )8

d e

i ( )

ˆ ( ), ˆ ( )8

d1

e

i ( )

ˆ ( ), ˆ ( )8

d e

i ( )

ˆ ( ), ˆ ( )8

d e

i ( ), (32)

i j

u u

ijh h

ijh h

i j

u u

ijh h

ijh h

i j

u u

ijh h

ijh h

i j

u u

ijh h

ijh h

k

r r r

k

r r r

k

r r r

k

r r r

30 p g

i ·( )

0

30 0 p g

i ·( )

0 0

3

p

g

i ·( )

30 p g

i ·( )

0

⎡⎣ ⎤⎦⎛⎝⎜⎜

⎞⎠⎟⎟

⎡⎣ ⎤⎦⎛⎝⎜⎜

⎞⎠⎟⎟

⎡⎣ ⎤⎦⎛⎝⎜⎜

⎞⎠⎟⎟

⎡⎣ ⎤⎦⎛⎝⎜⎜

⎞⎠⎟⎟

∫

∫

∫

∫

∑

∑

∑

∑

π εμ

η ηϵ

εϵ δ

π μ ε η ηϵ

μ εϵ δ

π

η

ηϵ

ϵ δ

π με

η ηϵ

μϵ δ

′ =

= ′ − ′

′ =

= ′ − ′

′ =

= ′ − ′

′ =

= ′ − ′

− ′

− ′

− ′

− ′

wherewe have used the sum rules given in equations (19) and (20).Here ijhϵ is the familiar permutation symbol[39, 40] and the repeated index h is summed over the three cartesian coordinates x, y and z. Thefirst of theseresults generalizes a commutator previously derived forfields in vacuo [37].Note that together thesecommutation relations require that the polarization operator commutes with themagnetic field operator andthat themagnetization also commutes with the electric field operator:

P H

M E

r r

r r

ˆ ( ), ˆ ( ) 0

ˆ ( ), ˆ ( ) 0, (33)

i j

i j

⎡⎣ ⎤⎦⎡⎣ ⎤⎦

′ =

′ =

which is a physical consequence of the fact that the two pairs of operators in each commutator are associatedwith properties of different systems: themedium and the electromagnetic field.

6

New J. Phys. 17 (2015) 063027 SMBarnett andR Loudon

3.3. Parallel beams: 3D to 1D conversionFor parallel light beams, it is sometimes convenient toworkwith field operators defined for dependence on asingle spatial coordinate zwith a one–dimensional wave vector k. Thus, for a beamof cross–sectional areaA,conversion from3D to 1D is achieved bymaking the substitutions

Ak

Ak k

aA

a

k

k k

d4

d

( )4

( )

ˆ2

ˆ . (34)u kuk

2

2

∫ ∫π

δπ

δ

π

→

− ′ → − ′

→

The vector potential operator (23) for polarization parallel to the x-axis is converted in this way to

z tA

k a tA xˆ ( , )4

d ˆ ( )e ˜, (35)u ku

kukz( )

0

1 2

p g

1 2

i⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

∫∑πε

μωη η

=+

where x is the unit vector in the x-direction. The orthogonal degenerate polaritonmodes give a vector potentialparallel to y . The four field operators retain their forms given in equations (25) to (28) except for appropriatechanges in thefirst square-root factors and vector directions. The electric and displacement fields are parallel tothe x-axis, while themagnetic field and induction are parallel to the y-axis. The commutation relation for thecreation and destruction operators retains the form given in equation (21) butwith k replaced by k. For μ=1and a single–resonance dielectric, the vector potential (35) agrees with equation (3.25) in [31] and equation (9)in [29].

The single-coordinate vector potential can also be expressed as an integral over frequency bymeans of theconversions

kc

k kc

ac

a

d d

( ) ( )

ˆ ˆ . (36)ku

g

g

g

1 2⎛⎝⎜⎜

⎞⎠⎟⎟

∫ ∫ ωη

δη

δ ω ω

η

→

− ′ → − ′

→ ω

There are no overlaps in frequency between the different twofold–degenerate polariton branches and thesummation over u is accordingly removed from the vector potential, which becomes

( )z tA

a t z cA xˆ ( , ) d4

ˆ ( )exp i ˜, (37)( )

1 20

0

1 4

p⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

∫ ωπω

μ με ε

ωη= ω+

where

a t aˆ ( ) ˆ e . (38)ti=ω ωω−

This destruction operator and the associated creation operator satisfy the continuumcommutation relation

a aˆ , ˆ ( ). (39)†⎡⎣ ⎤⎦ δ ω ω= − ′ω ω′

Thefield operators obtained by conversion in this way of (25) to (28) are

( )

( )

( )

( )

z tA

a t z c

z tA

a t z c

z tA

a t z c

z tA

a t z c

E x

B y

D x

H y

ˆ ( , ) i d4

ˆ ( )exp i ˜

ˆ ( , ) i d4

( ) ( ) ˆ ( )exp i ˜

ˆ ( , ) i d4

( ) ( ) ˆ ( )exp i ˜

ˆ ( , ) i d4

ˆ ( )exp i ˜ . (40)

( )1 2

0

0

1 4

p

( )1 2

01 4

03 4

p

( )1 2

03 4

01 4

p

( )1 2

0

0

1 4

p

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜⎜

⎞⎠⎟⎟

∫

∫

∫

∫

ω ωπ

μ με ε

ωη

ω ωπ

ε ε μ μ ωη

ω ωπ

ε ε μ μ ωη

ω ωπ

ε εμ μ

ωη

=

=

=

=

ω

ω

ω

ω

+

+

+

+

There is again agreementwith previously defined expressions [29, 31] for μ=1.We can use thesefield operatorsto calculate the vacuumfluctuations and find

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New J. Phys. 17 (2015) 063027 SMBarnett andR Loudon

( )

( )

E z t E z t E z tA

E z t

B z t B z t B z tA

B z t

D z t D z t D z tA

D z t

H z t H z t H z tA

H z t

0 ˆ ( , ) 0 0 ˆ ( , ) ˆ ( , ) 0 d4

0 ˆ ( , ) 0

0 ˆ ( , ) 0 0 ˆ ( , ) ˆ ( , ) 0 d4

0 ˆ ( , ) 0

0 ˆ ( , ) 0 0 ˆ ( , ) ˆ ( , ) 0 d4

0 ˆ ( , ) 0

0 ˆ ( , ) 0 0 ˆ ( , ) ˆ ( , ) 0 d4

0 ˆ ( , ) 0 , (41)

x x x

y

y y y

x

x x x

y

y y y

x

2 ( ) ( ) 0

0

1 2

2

2 ( ) ( )0 0

3 3 1 2

2

2 ( ) ( )0 0

3 3 1 2

2

2 ( ) ( ) 0

0

1 2

2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

∫

∫

∫

∫

ω ωπ

μ με ε

ω ωπ

ε εμ μ

ω ωπ

μ με ε

ω ωπ

ε εμ μ

= =

=

= =

=

= =

=

= =

=

+ −

+ −

+ −

+ −

in exact agreementwith results obtained in section 2.2.We shallmake use of the 1Dfield operators derived here to calculate the force exerted by a photon on a

magneto–dielectricmedium, butfirst return to the full 3D description to investigate the electromagneticmomentum.

4.Momentumoperators

4.1.Minkowski andAbrahamThemuch debated Abraham–Minkowski dilemma ismost simply stated as a question: which of two eminentlyplausiblemomentumdensities, D B× and c E H2 ×− , is the true or preferred value [7, 9]?We amplify upon theanswer given in [10],making special reference to the effects of amagneto–dielectricmedium.

Two rival forms for the electromagnetic energy–momentum tensors were derived byMinkowski [1] andAbraham [2–4]. Their original formulations considered electromagnetic fields inmoving bodies, but it sufficesfor our purposes to set thematerial velocity equal to zero. These results continue to hold for themagneto–dielectricmedia of interest to us. The two formulations differ principally in their expressions for theelectromagneticmomentum andwe consider here the respective quantized versions of these.

TheMinkowskimomentum is quite difficult tofind in the paper [1], but it can be deduced from theexpressions for other quantities. Its quantized form is represented by the operator

t t t

a t a t a t a t

a t a t a t a t

G rD r B r

k

k

ˆ ( ) d ˆ ( , ) ˆ ( , )

2d ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( )

ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( ) , (42)

u u u u

u u u u

u u u u

k k

k k k k

k k k k

M

,

p

g

1 2

p g

1 2

† †

† †

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟ ⎡⎣

⎤⎦

∫

∫∑ωεη

ημ

ωη η

= ×

= +

+ +

′ ′

′ ′

− ′ − ′

where equations (26) and (27) have been used. The diagonal part of thismomentumoperator is

( )a a a aG k kˆ 1

2d ˆ ˆ ˆ ˆ . (43)

u u

u u u u

k

k k k kdiagM p

g

† †⎛⎝⎜⎜

⎞⎠⎟⎟ ∫∑

η

η= +

The fact that k cpη ω= leads us to identify a single–photonmomentum

pc

(44)

uk

(M)

p2

g

⎛⎝⎜⎜

⎞⎠⎟⎟

ω η

η=

with the individual polaritonmode uk . This formof theMinkowski single-photonmomentumhas been derivedpreviously [41] for a single-resonance non-magneticmaterial. There are good reasons, however as we shall seebelow, not to assign this value to theMinkowskimomentum.

The corresponding formof the Abrahammomentumoperator, proportional to the Poynting vector forenergyflow, is [2–4]

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New J. Phys. 17 (2015) 063027 SMBarnett andR Loudon

tc

t t

a t a t a t a t

a t a t a t a t

G rE r H r

k

k

ˆ ( )1

d ˆ ( , ) ˆ ( , )

2d

1ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( )

ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( ) , (45)

u u u u

u u u u

u u u u

k k

k k k k

k k k k

A

2

, p g

1 2

p g

1 2

† †

† †

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟ ⎡⎣

⎤⎦

∫

∫∑ ωμη η ωμη η

= ×

= +

+ +

′ ′

′ ′

− ′ − ′

which differs from the corresponding expression for tG ( )M

only in the two square–root factors that occur in thefield operators (25) and (28). The diagonal part of thismomentumoperator is

( )a a a aG k kˆ 1

2d

1ˆ ˆ ˆ ˆ , (46)

u u

u u u u

k

k k k kdiagA

p g

† †⎛⎝⎜⎜

⎞⎠⎟⎟ ∫∑

η η= +

so that the associated single-photonmomentum is

pc

1, (47)

uk

Ag

⎛⎝⎜⎜

⎞⎠⎟⎟

ωη

=

which is the usual Abraham value.The subscriptM appears in brackets in equation (44) because an alternative form for theMinkowski

momentum, given by

pc

, (48)M pω η=

is observed in experiments sufficiently accurate to distinguish between the phase and group refractive indices,particularly the submergedmirrormeasurements of Jones and Leslie [42]. Note that the difference between thetwo candidatemomenta, pA and pM, can be very large and even, the case ofmediawith a negative refractiveindex, point in opposite directions [43]. The resolution of the apparent conflict between the two forms ofMinkowskimomentum is discussed in section 4.3.

4.2. Vector potential–momentumcommutatorsIt is instructive to evaluate the commutators of the twomomentumoperators with the vector potential.We cando this either by using the expressions for the fields in terms of the polariton creation and destruction operatorsor,more directly, by using thefield commutation relations (29) and (30), together with the fact that the vectorpotential commutes with both themagnetic field and the induction.

For theMinkowskimomentumwefind

A G A D B

B

A

A A

A

r r r r r

r r r r

r r r r

r r r r r

r

ˆ ( ), d ˆ ( ), ˆ ( ) ˆ ( )

i d ( ) ˆ ( )

i d ( ) ˆ ( )

i ˆ ( ) i d ( ) ˆ ( )

i ˆ ( ), (49)

i j jkl i k l

jkl ik l

jkl lmn ik m n

j i ik k j

j i

M⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦

∫∫

∫∫

ϵ

ϵ δ

ϵ ϵ δ

δ

= ′ ′ ′

= − ′ − ′ ′

= − ′ − ′ ′ ′

= − + ′ − ′ ′ ′

= −

⊥

⊥

⊥

wherewe have used the summation convention so that repeated indices are summed over the three cartesiandirections. It should also be noted that the remaining fields will have the same formof commutation relationwith theMinkowskimomentum, for example for the electric fieldwe have

E G Er rˆ ( ), ˆ i ˆ ( ). (50)i j j iM⎡

⎣⎢⎤⎦⎥ = −

This follows directly from the relationships between thesefields and the vector potential together with the factthat the vector potential commutes with the polarization and themagnetization.

For the Abrahammomentumwe can exploit our calculation for theMinkowskimomentum tofind thecommutator

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New J. Phys. 17 (2015) 063027 SMBarnett andR Loudon

( )

A G A E H

A E B M

A M

r r r r r

r r r r r

r r r r r

ˆ ( ), ˆ d ˆ ( ), ˆ ( ) ˆ ( )

d ˆ ( ), ˆ ( ) ˆ ( ) ˆ ( )

i ˆ ( ) i d ( ) ˆ ( ). (51)

i j jkl i k l

jkl i k l l

j i jkl ik l

A0 0

0 0

0

⎡⎣⎢

⎤⎦⎥ ⎡⎣ ⎤⎦

⎡⎣ ⎤⎦

∫∫

∫

ϵ ε μ

ϵ ε μ

ϵ δ μ

= ′ ′ ′

= ′ ′ ′ − ′

= − + ′ − ′ ′⊥

The second term,with its integration over themagnetization,means that the commutator depends on both afield property, the vector potential, and amediumproperty, themagnetization [10].

4.3. InterpretationThe identification of theMinkowski andAbrahamphotonmomenta respectively with the electromagneticcanonical and kineticmomenta has been proposed in the past [41, 44–46], but rigorously proven onlymorerecently [7, 10].We need add here only a few brief remarks.

The commutation relation (49) satisfied by theMinkowskimomentumoperator resembles the familiarcanonical commutator of the particlemomentumoperator,

F p Fr r( ), ˆ i ( ), (52)i j j i⎡⎣ ⎤⎦ =

where F r( ) is an arbitrary vector function of position. Thus, analogously to its particle counterpart, theMinkowskimomentumoperator for the electromagnetic field generates a spatial translation, in this case of thevector potential and the electric andmagnetic fields. The operator therefore indeed represents the canonicalmomentumof the field and it is the observedmomentum in experiments thatmeasure the displacement of abody embedded in amaterial host, as has been seen for amirror immersed in a dielectric liquid [42], for thetransfer ofmomentum to charge carriers in the photon drag effect [47] and in the recoil of an atom in a host gas[48]. The simple spatial derivative that occurs on the right of equation (49) shows that themeasured single–photonmomentum should have theMinkowski form in (48) and not that given in equation (44).

The kineticmomentumof amaterial body is the simple product of itsmass and velocity. The formof theAbraham single-photonmomentum in equation (47) is verified by thought experiments of the Einstein-boxvariety [49, 50]. These use the principle of uniformmotion of the centre ofmass-energy as a single-photon pulsepasses through a transparent dielectric slab and they reliably produce the Abrahammomentum. Thecalculations remain validwith no essentialmodificationswhen the slab ismade from amagneto–dielectricmaterial.

The totalmomentum includes contributions both from the opticalfield and themedium throughwhich itpropagates. The relationship between the kinetic and canonicalmomenta for themedium ismost readilyobtained by considering first a single atom. For an atomwith electric dipole operator d, the kinetic andcanonicalmomentumoperators differ by the Röntgen term associatedwith transforming the fields into theatom’s rest–frame [34, 44, 51, 52]

d Bˆ ˆ ˆ ˆ . (53)atomkinetic

atomcanonicalπ π= + ×

If the atomhas, in addition, amagnetic dipolemoment operator m then this relationship becomes [46]

cd B E mˆ ˆ ˆ ˆ 1 ˆ ˆ . (54)atom

kineticatomcanonical

2π π= + × + ×

Onadding the contributions from all of the atoms the dipole terms lead to a polarization P and amagnetizationM. Hencewe canwrite the total kineticmomentumofmedium in the form

cr P B r E Mˆ ˆ d ˆ ˆ 1

d ˆ ˆ . (55)mediumkinetic

mediumcanonical

2∫ ∫Π Π= + × + ×

Combining thesewith ourMinkowski andAbrahammomenta for the opticalfieldwe find that the totalmomentum is unique:

G Gˆ ˆ ˆ ˆ . (56)mediumcanonical M

mediumkinetic AΠ Π+ = +

It is this totalmomentum that is conserved in the interactions between electromagnetic waves andmaterialmedia.

4.4. AngularmomentumThe electromagnetic field carries not only energy and linearmomentumbut also angularmomentum and it isnatural to introduce angularmomenta derived from theMinkowski andAbrahammomenta in the forms

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New J. Phys. 17 (2015) 063027 SMBarnett andR Loudon

c

J r r D r B r

J r r E r H r

ˆ d ˆ ( ) ˆ ( )

ˆ 1d ˆ ( ) ˆ ( ) . (57)

M

A

2

⎡⎣ ⎤⎦⎡⎣ ⎤⎦

∫∫

= × ×

= × ×

A careful analysis of a light beam carrying angularmomentum entering a dielectricmedium shows that, incontrast with the linearmomentum, theMinkowski angularmomentum is the same inside and outside themedium, but that the Abrahamangularmomentum is reduced in comparison to its free–space value by theproduct of the phase and group indices [53]. The analogue of the Einstein–box argument suggests that lightcarrying angularmomentum entering amedium exerts a torque on it, inducing a rotation on propagationthrough it. An object imbedded in the host, however,may be expected to experience the influence of the sameangularmomentum as in free space and, indeed, this is what is seen in experiment [54].

The canonical orMinkowski angularmomentum should be expected to induce a rotation of theelectromagnetic fields, which requires both a rotation of the coordinate and also of the direction of the field. Therequirement to provide both of these transformations provides a stringent test of the identification of theMinkowski and canonicalmomenta. It is convenient tofirst rewrite theMinkowski linearmomentumdensity ina new form:

( )

D A D A

D A D A

D Bˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ , (58)

ij i j j j i

j i j j j i

⎡⎣ ⎤⎦

× = −

= −

wherewe have used the firstMaxwell equation, D 0j j = .We can insert this form into our expression for JM

and, on performing an integration by parts and discarding a physically–unimportant boundary termwe find

D AJ r r D Aˆ d ˆ ( ) ˆ ˆ ˆ , (59)j jM ⎡⎣ ⎤⎦∫= × + ×

which, in the absence of themedium reduces to the formobtained byDarwin [55, 56].It is tempting, even natural, to associate the two contributions in the integrandwith the orbital and spin

angularmomentum components of the total angularmomentum. This is indeed reasonable, but it should benoted that neither part alone is a true angularmomentum [57–59]. It is instructive to consider the commutation

relationwith a single component of the angularmomentum and so consider the operator J· ˆMθ :

r J r A A

r A A

A( ), · ˆ i · ( ) ˆ i ( ˆ )

i · ( ) ˆ ˆ . (60)

M⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦⎡⎣ ⎤⎦

θθ θ

θ θ

= − × + ×

= − × − ×

⊥ ⊥

The orbital and spin parts rotate, as far as is possible given the constraints of transversality, the amplitude anddirection of the potential [57–59]. The combination of both of these gives the required transformation. Thecommutator (60) gives the first order rotation of the vector potential about an axis parallel to θ through the smallangle θ, as the canonical angularmomentum should.

5.Magnetic Lorentz force

It remains to determine the radiation pressure due to a lightfield on ourmagneto–dielectricmedium. Tocomplete this taskwe adopt themethod used previously of evaluating the force exerted by a single-photon plane-wave pulse normally incident on themedium [18, 60]. Before we can complete the calculation, however, weneed to determine a suitable form for the electromagnetic force density.

5.1.Heaviside–Larmor symmetryMaxwell’s equations in the absence of free charges and currents (4) exhibit the so–calledHeaviside–Larmorsymmetry [61, 62], with the forms that are invariant under rotational duality transformations given by [22]

Z

Z

Z

Z

E E H

H H E

D D B

B B D

cos sin

cos sin

cos sin

cos sin , (61)

0

01

01

0

ξ ξξ ξξ ξξ ξ

= ′ + ′= ′ − ′

= ′ + ′= ′ − ′

−

−

for any value of ξ andwhereZ0 is again the impedance of free space. It is readily verified that the fourMaxwellequations are converted to the same set of equations in the primedfields. The various physical properties of theelectromagnetic fieldmust also be unchanged by the transformations [63].We note, in particular, that the

Minkowski andAbrahammomentumoperators, GMand G

Agiven in equations (42) and (45) and also the usual

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New J. Phys. 17 (2015) 063027 SMBarnett andR Loudon

expressions for the electromagnetic field energy density and Poynting vector are all invariant under thetransformation (61).

The standard formof the Lorentz force law in a non–magnetic dielectric is [64]

f P E P H( · ) ˙ , (62)L0 μ= + ×

with terms proportional to the electric polarization and its time derivative3. For amagneto–dielectricmediumweneed to add the force due to themagnetization and to do so in amanner that gives a force density that isinvariant under theHeaviside–Larmor transformation. It follows from the transformation (61) that thepolarization andmagnetization are similarly transformed:

c

c

P P M

M M P

cos sin

cos sin . (63)

1ξ ξξ ξ

= ′ + ′= ′ − ′

−

The required formof the force density, satisfying theHeaviside–Larmor symmetry is

f P E P H M H M E( · ) ˙ ( · ) ˙ . (64)EL0 0 0 0 μ μ ε μ= + × + − ×

The invariance of this expression under theHeavisde–Larmor transformation is easily shown. This formof theforce density was derived by Einstein and Laub over 100 years ago [65] and there have since been severalindependent re–derivations of it [66–71]. Thefinal term in equation (64) has been given special attention in[72], where it is treated as amanifestation of the so–called ‘hiddenmomentum’ given by M E0 0ε μ × . This lineof thought has attracted a series of publications, with several listed on page 616 of [22]. Omission of thefinalterm leads to difficulties, not the least of which is the identification of amomentumdensity that does not satisfytheHeaviside–Larmor symmetry [73, 74]4. The above derivation of the Einstein–Laub force density shows howthemagnetic terms follow from the polarization terms by simple symmetry arguments and so provides a newperspective on the complete force density. It is interesting to note,moreover, that the force density appropriatefor a dielectricmedium (62)may be obtained by consideration of the action on the individual dipolesmaking upthemedium [64] and that themost direct way to arrive at the Einstein–Laub force density is to obtain themagnetization part by treating a collection ofGilbertianmagnetic dipoles [81].

It is also shown in the original paper [65] that the classical formof the Abrahammomentum and the classicalforce density satisfy the conservation condition

tt tG r f r( ) d ( , ). (65)A EL∫∂

∂= −

The integration is taken over all space and the relation is valid forfields that vanish at infinity. This equality of therate of change of the Abrahammomentumof the light tominus the total Einstein–Laub force on themedium, orrate of change ofmaterialmomentum, is as expected on physical grounds and further underlines theidentification of theAbrahammomentumwith the kineticmomentumof the light [7, 10]. Equation (65) issimply an expression of the conservation of totalmomentum.

5.2.Momentum transfer to a half-spacemagneto-dielectricWeconsider a single-photon pulse normally incident from free space at z 0< on theflat surface of a semi-infinitemagneto–delectric that fills the half space z 0> . The pulse is assumed to have a narrow range offrequencies centred on 0ω . Its amplitude and reflection coefficients, the same as in classical theory, are [22]

R

T

1

1

2

1, (66)

ε με μ

ε μ

= −−+

=+

where all of the optical parameters depend on the frequency. The damping parameters in the imaginary parts ofε and μ, as they occur inR andT, are assumed negligible but they should be sufficient for the attenuation length

3This is usually written in terms of themagnetic induction as [64]

f P E P B( · ) ˙ .L = + ×

In amagneticmedium, however, we need to distinguish between H0μ and B. That it should be the former that appears in the force densityfollows on consideration of the screening effect of surroundingmagnetic dipoles in themedium, inmuch the sameway as electric dipolesscreen the electric field in amedium.4It is by nomeans straightforward to obtain the Einstein–Laub force density, in particular the final hidden–momentm related term, from

themicroscopic Lorentz force law and it has been suggested, for this reason, that the latter is incorrect [75]. A relativistic treatment, however,reveals that the required hidden–momentum contribution arises quite naturally from the Lorentz force law [76–80].

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New J. Phys. 17 (2015) 063027 SMBarnett andR Loudon

( )c

( )2 Im

(67)ℓ ωω εμ

=

to give complete absorption of the pulse over its semi–infinite propagation distance in themedium.Themomentum transfer to themedium as awhole is entirely determined by the free-space single-photon

momenta, before and after reflection of the pulse, as

( )Rc

Tc

12

, (68)2 0 2

p

0⎛⎝⎜

⎞⎠⎟

ω εμ

ε μη

ω+ = +

with the small imaginary parts of the optical parameters again ignored. It is often useful to re–express themomentum transfer in terms of a single transmitted photonwith energy 0ω at z 0= +. This quantity isobtained by removal of the transmitted fraction of the pulse energy, given by the bracketed termon the right of(68), as

pc2

1

2, (69)total

p

0⎛⎝⎜

⎞⎠⎟

ε μη

ω εμ

με

= + = +

in agreementwith previouswork [69, 82]. The remainder of the section considers the separation of this totaltransfer ofmomentum to themagneto–dielectric into its surface and bulk contributions.

Previous calculations [18, 60] of the radiation pressure on a semi–infinite dielectric weremade by evaluationof the Lorentz force on thematerial. Thismethod is generalized here for amagneto–dielectricmedium. For atransverse plane–wave pulse propagated parallel to the z-axis, with electric andmagnetic fields parallel to the xand y axes respectively, the relevant component of the Einstein–Laub force from (64) has the quantized form

fP

tH

c

M

tE

c

E

tH E

H

t

ˆˆ

ˆ 1 ˆˆ

1( 1)

ˆˆ ˆ ( 1)

ˆ. (70)

z

EL

0 2

2

⎡⎣⎢

⎤⎦⎥

μ

ε μ

= ∂∂

− ∂∂

= − ∂∂

+ − ∂∂

The 1Dfield operators from section 3.3, given in (40), are appropriate here. The single–photon pulse isrepresented by the state

a1 d ( ) ˆ ( ) 0 , (71)†∫ ω ξ ω ω=

where 0∣ ⟩ is the vacuum state. This single-photon state is normalized if we require our function ( )ξ ω to satisfy

d ( ) 1. (72)2∫ ω ξ ω =

The states, from their construction, satisfy

a ( ) 1 ( ) 0 . (73)ω ξ ω=

A simple choice for the pulse amplitude is

L

c

L

c( )

2exp

( )

4(74)

2

2

1 4 20

2

2

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥ξ ω

πω ω

= −−

with c L 0ω≪ . The narrowness of the spectrumof this pulsemeans thatω can often be set equal to 0ω .The radiation pressure of themagneto–dielectric is obtained by evaluation of the expectation value of the

Einstein–Laub force (70) for the single–photon pulse. The normal-order part of the force operator, indicated bycolons, is used to eliminate unwanted vacuumcontributions. A calculation similar to that carried out in [18] fora dielectricmediumgives

f z tAL

c

Lt

z

c

c

Lt

z

c

z

1 : ˆ ( , ): 12

( ) 24

exp2

, (75)

z

EL 0

p

p p 2 g

2

2 g

2

⎜ ⎟

⎜ ⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎧⎨⎩⎡⎣ ⎤⎦

⎛⎝

⎞⎠

⎫⎬⎭⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

ωπ η

ε μℓ

η ε μ η η

ηℓ

≈ +

− + − −

× − − −

where the real ε, μ, and their derivatives in the group velocity are evaluated at frequency 0ω and their relativelysmall imaginary parts survive only in the attenuation lengthℓ. It is readily verified by integration over t then zthat this expression regenerates the totalmomentum transfer in equation (69). The force on the entirematerialat time t is

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New J. Phys. 17 (2015) 063027 SMBarnett andR Loudon

F t A z f z t

ct

L

L ct

L

c t

L

L

1 : ˆ ( ): 1 d 1 : ˆ ( , ): 12

2erfc 2

4exp

( ) 2exp

2

8, (76)

z z

EL

0

EL 0

p g

p

g g g

g p 2 2

2

2

g2 2

⎛⎝⎜⎜

⎞⎠⎟⎟

⎧⎨⎪⎩⎪

⎡

⎣⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤

⎦⎥⎥

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎫⎬⎪⎭⎪

∫ ωπ η η

η

η ℓπ

η ℓ η ℓ

η ε μ η

η ℓ

= ≈

× − + −

++ −

− −

∞

with appropriate approximations neglecting small terms in the exponents for the long attenuation–lengthregimewith L ℓ≪ . This expression, with error function and exponential contributions, has the same overallstructure as found in other radiation pressure problems [18, 60]. The two terms in the large bracket arerespectively the bulk and surface contributions, with time–integrated values

t F t p

c c

d 1 : ˆ ( ): 1

1

2

1. (77)

zEL

total

0

g

bulk

0

p g

surface

⎛⎝⎜⎜

⎞⎠⎟⎟

∫ω

ηω ε μ

η η

=

= + + −

−∞

∞

This is again in agreement with equation (69) and it also reduces to equation (5.21) of [18] for a non-magneticmaterial, where themomentum transfer can bewritten entirely in terms of the phase and group indices. Thesimple Abrahamphotonmomentum again represents that available for transfer to the bulkmaterial, once thetransmitted part of the pulse has cleared the surface, while themore complicated surfacemomentum transferdepends on both ε and μ, togetherwith their functional forms embodied in the phase and group refractiveindices.

We conclude this section by noting that theHeaviside–Larmor symmetry retains a presence in all of theforces and force densities obtained here, in that their forms are unchanged if we interchange, everywhere, therelative permittivity and permeability.

6. Conclusion

Muchof the content of the paper presents the generalizations tomagneto–dielectrics of results previouslyestablished for non–magneticmaterials with μ=1. The classical linear response theory in section 2 allows directcalculation of the electric andmagnetic field–fluctuation spectra. The elementary excitations for amediumwithmultiple electric andmagnetic resonances are the polaritons, whose phase and group velocities obey generalizedsum rules formagneto–dielectrics [32].

The quantum theory in section 3 introduces electromagnetic field operators based on themultiple–branchpolariton creation and destruction operators. It is shown that the vacuum fluctuations of the quantized electricandmagnetic fields reproduce the spectra obtained fromour classical linear–response theory. The generalizedfield operators are shown to satisfy the same required canonical commutation relations as their simplercounterparts that hold in vacuum.

TheMinkowski andAbraham electromagneticmomentumoperators are introduced in section 4 and theirassociated single-photonmomenta are identified. The commutators of thesemomentum and the vector-potential operators, previously calculated, rely on the canonical commutation relations and, through these, onthe polariton sum rules. An extension to angularmomentum, both canonical and kinetic, is achieved byintroducing angularmomentumdensities that are the cross product of the position and theMinkowski andAbrahammomentumdensities respectively.

Throughout ourworkwe are guided by theHeaviside–Larmor symmetry between electric andmagneticfields.We show, in section 5, that application of this symmetry leads directly to the Einstein–Laub force density[65]. Our final result identifies the surface and bulk contributions in the force on a semi–infinitemagneto–dielectric for the transmission of a single-photon pulse through its surface.

Acknowledgments

We thankColin Baxter for his early contributions to the full-blown theory presented in the current paper. Thisworkwas supported by the Engineering and Physical Sciences ResearchCouncil (EPSRC) under grant numberEP/I012451/1, by theRoyal Society and the byWolfson Foundation.

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