SCALORA et al PHYSICAL REVIEW E 73. 056604(2006) shown in Eq.(12)are usually more than sufficient to accu- rately describe the dynamics, even for very short pulses(a F(O drf(r, t) few wave cycles in duration), because typical dispersion lengths may be on the order of meters, as we will see below 1/ dD(r, t) dE(r, t) The expression for the force density function, Eq (1), in the B absence of free charges and free currents, may be written as {(V)×H×Bdr (16) f(r, t)=phone +c(V×M)×B In deriving Eq.(16)from Eq. (15), we have assumed that the E(V·E)+ tCl( ar magnetic permeability is approximately real and constant to show the basic contributions, including a surface term when +cV×B-cV×H×B the magnetic permeability is discontinuous. We will general (14) ize this expression later when we deal with negative index materials We have made use of the usual constitutive relationsh Next, substituting Eqs. 8)and (11)into Maxwells equa between the fields, namely D=E+4P and B=H+4M. tions yields the following coupled differential equations Equation(14)includes a Coulomb contribution from bound [21-23] and magnetic current densities, in order to allow application ae. a' d8. a a8 to magnetically active materials. The Coulomb term shown 4丌ar24m2ar3 may be expressed in a variety of ways. For example, using the first of Eqs. (11), and by using the condition V.D=0,one =iBlE(s)E-H, sin 8-H, cos 0+-+-, can show that, in the absence of absorption(e=e), the Cou lomb term takes the form ame E(ve. E)+e/r de nl +i Ar as-24 7,+ E|+… =i所()H-Ecos]- The presence of higher order terms is implied. The form 4丌a24m iven in Eq. (14)thus suggests that there is a Coulomb con- ribution if (i) the incident field has a TM-polarized compo- (17) nent;(ii) scattering generally occurs from a three- =i以(9)H2-Exin+ dimensional structure with complex topology that generates other field polarizations; and (iii) if the field has curvature in all three dimensions. Under some circumstances, one may d@E(5)] F[oE(E) ignore the Coulomb contribution, for example by consider- ing TE modes using our Eqs.(8), which lead directly to V.E=dE (, z, t)/dx=0. This is a sufficient condition that may be easily satisfied in problems that exploit one-or two- 到,y=画 dimensional symmetries, as we do here. It should be appar ent, however, that more complicated topologies and/or the consideration of TM-polarized incident fields are in need and the prime symbol denotes differentiation with respect to the general approach afforded by Eq. (14). In light of the frequency. B; is the angle of incidence. The following scaling previous discussion, we will first examine the case of TE- has been adopted: 5=z/Ar y=y/Ar T=ct/Ar B=2o, and polarized incident pulses, and in the last section of the manu- @=o/or where A, =I um is conveniently chosen as the ref- script we will briefly discuss results that concern a TM- erence wavelength. We note that nonlinear effects may be polarized pulse that traverses a single, ordinary dielectric taken into account by adding a nonlinear polarization to the interface. Therefore,for TE-polarized waves, Eq. (14)re- right-hand sides of Eqs.(17), as shown in Ref. [23],for duces to example As we pointed out after the constitutive relation Eq.(9) r(r,) 4m( CVd dE(r,)×B the development that culminates with Eqs.(17)assumes that (15) the medium is isotropic, a restriction that can be removed should the need arise, without impacting the relative simplic Using Maxwell's equations, the total force can then be cal- ity of the approach or method of solution. Beyond this fact culated as Eqs.(17)do not contain any other approximations, but they 056604-4shown in Eq.
12 are usually more than sufficient to accu￾rately describe the dynamics, even for very short pulses
a few wave cycles in duration, because typical dispersion lengths may be on the order of meters, as we will see below. The expression for the force density function, Eq.
1, in the absence of free charges and free currents, may be written as f
r,t = boundE + 1 c 
P
t + c

M B = 1 4 E

· E + 1 4c 
D
t −
E
t  + c
B − c
H B.
14 We have made use of the usual constitutive relationships between the fields, namely D=E+4P and B=H+4M. Equation
14 includes a Coulomb contribution from bound charges, and contributions from bound dielectric polarization and magnetic current densities, in order to allow application to magnetically active materials. The Coulomb term shown may be expressed in a variety of ways. For example, using the first of Eqs.
11, and by using the condition
·D=0, one can show that, in the absence of absorption
* , the Cou￾lomb term takes the form 1 4− E

 · E + 0E


  0 · E + ¯  1 + 1  

  0 0 + ¯ . The presence of higher order terms is implied. The form given in Eq.
14 thus suggests that there is a Coulomb con￾tribution if
i the incident field has a TM-polarized compo￾nent;
ii scattering generally occurs from a three￾dimensional structure with complex topology that generates other field polarizations; and
iii if the field has curvature in all three dimensions. Under some circumstances, one may ignore the Coulomb contribution, for example by consider￾ing TE modes using our Eqs.
8, which lead directly to
·E=
Ex
y ,z,t/
x0. This is a sufficient condition that may be easily satisfied in problems that exploit one- or two￾dimensional symmetries, as we do here. It should be appar￾ent, however, that more complicated topologies and/or the consideration of TM-polarized incident fields are in need of the general approach afforded by Eq.
14. In light of the previous discussion, we will first examine the case of TE￾polarized incident pulses, and in the last section of the manu￾script we will briefly discuss results that concern a TM￾polarized pulse that traverses a single, ordinary dielectric interface. Therefore, for TE-polarized waves, Eq.
14 re￾duces to f
r,t = 1 4c c
B −
E
r,t
t  B.
15 Using Maxwell’s equations, the total force can then be cal￾culated as F
t = volume dr3 f
r,t = volume  1 4

D
r,t
t −
E
r,t
t  Bdr3 + 1 4 volume 


H Bdr3.
16 In deriving Eq.
16 from Eq.
15, we have assumed that the magnetic permeability is approximately real and constant to show the basic contributions, including a surface term when the magnetic permeability is discontinuous. We will general￾ize this expression later when we deal with negative index materials. Next, substituting Eqs.
8 and
11 into Maxwell’s equa￾tions yields the following coupled differential equations 21–23: 
Ex
+ i 
4
2 Ex
2 −  242
3 Ex
3 + ¯ = i
Ex − Hz sin i − Hy cos i  +
Hz
˜y +
Hy
 ,
Hy
+ i
4
2 Hy
2 − 242
3 Hy
3 + ¯ = i

Hy − Ex cos i  −
Ex
 ,
Hz
+ i
4
2 Hz
2 − 242
3 Hz
3 + ¯ = i

Hz − Ex sin i  +
Ex
˜y .
17 Here  = 
˜ 

˜  0 , 
= 
2 ˜ 

˜ 2  0 , = 
˜


˜  0 ,
= 
2 ˜


˜ 2  0 , and the prime symbol denotes differentiation with respect to frequency. i is the angle of incidence. The following scaling has been adopted: =z/r, ˜y=y /r, =ct/r, =2˜, and ˜ =/r, where r=1
m is conveniently chosen as the ref￾erence wavelength. We note that nonlinear effects may be taken into account by adding a nonlinear polarization to the right-hand sides of Eqs.
17, as shown in Ref. 23, for example. As we pointed out after the constitutive relation Eq.
9, the development that culminates with Eqs.
17 assumes that the medium is isotropic, a restriction that can be removed should the need arise, without impacting the relative simplic￾ity of the approach or method of solution. Beyond this fact, Eqs.
17 do not contain any other approximations, but they SCALORA et al. PHYSICAL REVIEW E 73, 056604
2006 056604-4