RADIATION PRESSURE OF LIGHT PULSES AND PHYSICAL REVIEW E 73. 056604(2006) may be simplified depending on the circumstances. For ex ample, in ordinary dielectric materials we may neglect sec ond and higher order material dispersion terms, which elimi- nates second and higher order temporal derivatives. As example, in the spectral region of interest, which includes the near IR range(800-1200 nm), the dielectric function(ac- tual data)of Si3 N4 [24] may be written as Incident e(a)=37798+0178980+00408 Transmitted sing this approximately linear dielectric susceptibility model, indeed we have a=a[oE(51/a0=0.One may then estimate the second-order dispersion length, de Reflected fined as LB)+/"(l. where T, is incident pulse width, The result is Lp-2X10A(o -2 mm)for an incident, five wave-cycle pulse(-15 fs); 200-15010050 100I50 approximately 8 mm for a ten wave-cycle pulse; and 1 m for 100-wave cycle(-300 fs) pulses. In comparison, typical FIG. 1. A 100 fs pulse interacts with a 60 um thick Si3 N4 sub- multilayer stacks and substrates that we consider range from strate. Both E and H fields are shown as the pulse is partly trans- a few microns to a few tens of microns in thickness, and so mitted and partly reflected from both entry and exit interfaces. Out neglect of the second-order time derivative and beyond is side the structure the fields overlap, while inside pulse compression completely justified, even for pulses only a few wave cycles due to group velocity reduction and conservation of energy causes in duration the magnetic field to increase its amplitude with respect to the in In the frequency range and the materials that we are con- cident field sidering, assuming for the moment that u=y=l, in our aled coordinate system the simplified version of Eq (12)is D×B dg(,s,), minkowski i(eaHy+cc) (2 P(7)= 84, s, r)dy (21) +cC.+ These components may be used to calculate the angle of (e2:+c.) refraction [21]. To simplify matters further, for the moment we assume that the pulse is incident normal to the multilayer surface, i.e. Py(T)=0 at all times, and focus our attention on (19) the longitudinal component. Finally, assuming no frictional or other dissipative forces are present, conservation of mo- magnetic materials, Eq(15)also simpl f(r, t) X(r, t) Pstructure(a)=Pg-Ps(r) (22) P=P(7=0)=d厂=4,,7=0)d5 is the 4A2IiB(e'-1E' Hy-iB(e-1)E-H momentum initially carried by the pulse in free space, before t enters any medium. The force may then be calculated as the temporal derivative of the momentum in Eq (22). C4A, B(e-1)2H2-iB(e-1)EHz In this section we consider the interaction of I MW/cm Gaussian pulse of the type E,0,,T=0) =Eoe-(5-50)2+y yna, and similarly for the transverse mag (a-1)=2+(a-1)- (20) netic field, with a 60 um thick Si, N4 substrate, as depicted in Fig. 1. Choosing w- 20 corresponds to a l/e width of ap Having defined the relevant momentum densities in Eqs. proximately 100 fs in duration, but we note that the exact (12)and(13)above, the total momentum can then be easily temporal duration of the pulse is not crucial. The spatial calculated. In general one has two components, one longitu- extension(both longitudinal and transverse)of the pulse in dinal and one transverse, as follows [11] free space may be estimated from the figure at about 40 um 056604-5may be simplified depending on the circumstances. For ex￾ample, in ordinary dielectric materials we may neglect sec￾ond and higher order material dispersion terms, which elimi￾nates second and higher order temporal derivatives. As an example, in the spectral region of interest, which includes the near IR range
800–1200 nm, the dielectric function
ac￾tual data of Si3N4 24 may be written as 
˜ = 3.7798 + 0.178 98˜ + 0.044 08 ˜ .
18 Using this approximately linear dielectric susceptibility model, indeed we have = 
3 ˜ 
/
˜ 3  0 0. One may then estimate the second-order dispersion length, de- fined as LD
2  p 2 / k
˜ , where p is incident pulse width, and k
˜=
2 k/
˜ 2 . The result is LD
2 2103 r
or 2 mm for an incident, five wave-cycle pulse
15 fs; approximately 8 mm for a ten wave-cycle pulse; and 1 m for 100-wave cycle
300 fs pulses. In comparison, typical multilayer stacks and substrates that we consider range from a few microns to a few tens of microns in thickness, and so neglect of the second-order time derivative and beyond is completely justified, even for pulses only a few wave cycles in duration. In the frequency range and the materials that we are con￾sidering, assuming for the moment that
= =1, in our scaled coordinate system the simplified version of Eq.
12 is gMinkowski = D B 4c = 1 4c zˆ
ExH* y + c.c. + i 1 2

˜ H* y
Ex
+ c.c. + ¯  − 1 4c yˆ
ExH* z + c.c. + i 1 2

˜ H* z
Ex
+ c.c. + ¯ .
19 For nonmagnetic materials, Eq.
15 also simplifies to f
r,t = 1 c
P
r,t
t B
r,t = 1 4r zˆi
* − 1E* xHy − i
 − 1ExH* y + 
 − 1
Ex
H* y +
* − 1
E* x
Hy + ¯ = 1 4r yˆi
* − 1E* xHz − i
 − 1ExH* z + 
 − 1
Ex
H* z +
* − 1
E* x
Hz + ¯ .
20 Having defined the relevant momentum densities in Eqs.
12 and
13 above, the total momentum can then be easily calculated. In general one has two components, one longitu￾dinal and one transverse, as follows 11: P
= =− = d ˜ y=− ˜ y= g
˜y,, dy˜, P˜ y
= =− = d ˜ y=− ˜ y= g˜ y
˜y,, dy˜.
21 These components may be used to calculate the angle of refraction 21. To simplify matters further, for the moment we assume that the pulse is incident normal to the multilayer surface, i.e., P˜ y
=0 at all times, and focus our attention on the longitudinal component. Finally, assuming no frictional or other dissipative forces are present, conservation of mo￾mentum requires that the linear momentum imparted to the structure be given by Pstructure
= P 0 − P
,
22 where P 0=P
=0==− = d˜ y=− ˜ y= g
˜y ,, =0dy˜ is the total momentum initially carried by the pulse in free space, before it enters any medium. The force may then be calculated as the temporal derivative of the momentum in Eq.
22. A thick, uniform substrate In this section we consider the interaction of a 1 MW/cm2 Gaussian pulse of the type Ex
˜y ,, =0 =E0e−
−0 2 +y−2/w2 , and similarly for the transverse mag￾netic field, with a 60
m thick Si3N4 substrate, as depicted in Fig. 1. Choosing w20 corresponds to a 1/e width of ap￾proximately 100 fs in duration, but we note that the exact temporal duration of the pulse is not crucial. The spatial extension
both longitudinal and transverse of the pulse in free space may be estimated from the figure at about 40
m FIG. 1. A 100 fs pulse interacts with a 60
m thick Si3N4 sub￾strate. Both E and H fields are shown as the pulse is partly trans￾mitted and partly reflected from both entry and exit interfaces. Out￾side the structure the fields overlap, while inside pulse compression due to group velocity reduction and conservation of energy causes the magnetic field to increase its amplitude with respect to the in￾cident field. RADIATION PRESSURE OF LIGHT PULSES AND¼ PHYSICAL REVIEW E 73, 056604
2006 056604-5