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SJOERD STALLINGA PHYSICAL REVIEW E 73. 026606(2006) dod(o[(O,F)-oF+F E]8aB(56) a ja+agaB=0 (60) where the total angular momentum density and angular mo- The nondispersive contributions to the momentum density mentum flux density are given by and momentum flux density are recognized as the Minkow ski momentum density and the Minkowski stress tensor, re spectively. The momentum balance equations for the two dodt(o)EaB FBd, Fr (62) Division of the total angular momentum into field and matter (58) contributions, and into dispersive and nondispersive contri- where the Minkowski force density I' is given by (41). This butions are completely analogous to the linear momentum force density is approximately zero when dispersion may be case. A division into spin and orbital parts can be developed neglected. In that case the total momentum may be approxi- along the lines of Refs. [20,21] but will not be pursued here ated by the nondispersive (Minkowski) momentum. In general, however, the dispersive terms need to be taken into IV WAVE PACKETS account. The importance of including dispersive contribu tions has also been stressed by Nelson [3] and Garrison and Certain interesting features of the auxiliary field model Chiao [4] become apparent when studying wave packets In this sec- tion, one-dimensional propagation along the z axis of a lin- early polarized wave packet is considered. Then the electric C Angular momentum field only has a nonzero x component given by Our treatment of angular momentum will be brief, as it is quite similar to the case of linear momentum treated previ E(z,) ously. The angular momentum quantities are simply found 2r (2, @exp(-ior) from the linear momentum quantities by taking the cross product with the position vector. where An issue frequently popping up in discussions about an- lar momentum conservation is the symmetry, or lack of it, E(z, 0)=E(o)explik(o)z (64) of the stress tensor. It appears that dispersion can result in an asymmetric stress tensor, although the medium is isotropic. Here E(o)=E(-o), because E(z, t) is real, and where the This can be seen as follows. The dielectric displacement D at (magnitude of) the wave vector is given by a specific time t depends on the electric field E at all previ ous times. The electric field at these times t'<t is not nec- essarily oriented in the same direction as the electric field at k()=[n(o)+ik(o) (65) time t. It follows that D and e at time t are not necessarily parallel, implying that the stress tensor defined by Eq (50)is with n(a) the refractive index and K(o) the absorption coef- generally asymmetric. This seemingly points to nonconser- ficient. These are related to the dielectric function by vation of angular momentum. however. it turns out that antisymmetric part of the stress tensor can be expressed as a E(o)=[n()+ik(o)]2 time derivative of a quantity, which may be interpreted as contributing to the internal angular momentum so that the real and imaginary parts can be written as EaByPBE dwd(o)EaB FBa, Fy+oFy) E(a)=n(a)2-k(a)2 (67) o) =2n(o)k(o) (68) dodt(o)EaB FBd, F The other field components of interest are The internal angular momentum contribution depends on the cross product of the material field and the time derivative of he material field. It follows that this contribution is only e(o)E(z, )exp(-iat), (69) nonzero if the orientation of the fields changes with time Dz)=80」27 which corroborates the qualitative argument given previ ously. An alternative, equally valid, way of dealing with this asymmetry is to absorb it into a redefinition of the linear B(,1)=oH,(2) momentum density Conservation of angular momentum is thus expressed by (70) 026606-6T DS = − 0 0 dˆt F 2 − 2 F2 + F · E. 56 The nondispersive contributions to the momentum density and momentum flux density are recognized as the Minkowski momentum density and the Minkowski stress tensor, respectively. The momentum balance equations for the two parts are t g ND + T ND = − f , 57 t g DS + T DS = f , 58 where the Minkowski force density f is given by 41. This force density is approximately zero when dispersion may be neglected. In that case the total momentum may be approximated by the nondispersive Minkowski momentum. In general, however, the dispersive terms need to be taken into account. The importance of including dispersive contributions has also been stressed by Nelson 3 and Garrison and Chiao 4. C. Angular momentum Our treatment of angular momentum will be brief, as it is quite similar to the case of linear momentum treated previously. The angular momentum quantities are simply found from the linear momentum quantities by taking the cross product with the position vector. An issue frequently popping up in discussions about angular momentum conservation is the symmetry, or lack of it, of the stress tensor. It appears that dispersion can result in an asymmetric stress tensor, although the medium is isotropic. This can be seen as follows. The dielectric displacement D at a specific time t depends on the electric field E at all previous times. The electric field at these times tt is not necessarily oriented in the same direction as the electric field at time t. It follows that D and E at time t are not necessarily parallel, implying that the stress tensor defined by Eq. 50 is generally asymmetric. This seemingly points to nonconservation of angular momentum. However, it turns out that the antisymmetric part of the stress tensor can be expressed as a time derivative of a quantity, which may be interpreted as contributing to the internal angular momentum T = PE = 20 0 dˆ Ft 2 F + 2 F = t 20 0 dˆ Ft F . 59 The internal angular momentum contribution depends on the cross product of the material field and the time derivative of the material field. It follows that this contribution is only nonzero if the orientation of the fields changes with time, which corroborates the qualitative argument given previously. An alternative, equally valid, way of dealing with this asymmetry is to absorb it into a redefinition of the linear momentum density. Conservation of angular momentum is thus expressed by tj + M = 0, 60 where the total angular momentum density and angular momentum flux density are given by j = rg + 20 0 dˆ Ft F , 61 M = r T. 62 Division of the total angular momentum into field and matter contributions, and into dispersive and nondispersive contributions are completely analogous to the linear momentum case. A division into spin and orbital parts can be developed along the lines of Refs. 20,21, but will not be pursued here. IV. WAVE PACKETS Certain interesting features of the auxiliary field model become apparent when studying wave packets. In this section, one-dimensional propagation along the z axis of a linearly polarized wave packet is considered. Then the electric field only has a nonzero x component given by Exz,t = − d 2 E ˆz,exp− it, 63 where E ˆz, = E ˆexpikz. 64 Here E ˆ=E ˆ− * , because Exz,t is real, and where the magnitude of the wave vector is given by k = n + i c , 65 with n the refractive index and the absorption coef- ficient. These are related to the dielectric function by ˆ = n + i2, 66 so that the real and imaginary parts can be written as ˆr = n 2 − 2, 67 ˆi = ˆ = 2n. 68 The other field components of interest are Dxz,t = 0 − d 2 ˆE ˆz,exp− it, 69 Byz,t = 0Hyz,t = − d 2 k E ˆz,exp− it, 70 SJOERD STALLINGA PHYSICAL REVIEW E 73, 026606 2006 026606-6