A semi-infinite substrate of refractive index n is shown in Fig. 7, coated with a dielectric layer of index n and thickness d= n/ (4vn). This quarter-wave thick layer is a perfect anti- reflection coating that allows the incident beam into the semi-infinite medium with no reflection whatsoever. Conservation of energy requires the E-field to enter the semi-infinite medium with an amplitude E, =E/n, while the quarter-wave thickness of the coating layer shifts the phase of E, by 90 relative to that of the incident beam. The H-field in the semi- infinite medium is then H,=nE, /Zo=in EZo. Inside the coating layer, the counter propagating beams shown in Fig. 7 have the following distributions E2()=%(+1A)Eexp(12=/)+%(-1/)Eexp(-12xnn),(2la) HY(=)=v(1+n)Ho exp(i2iVn:/o+(1-Vn)Ho exp(-i2tvn=n) (21b) The force density, therefore, is given by F:=h2 Real (xB *)=v Real[-ioE(vE-DEXAoH " [r(n-1Nuoe, Imag(e,x h") -hear(n-1)/(nh)] sin( 4in -n)E 2 (22) The above force density must be integrated over the thickness of the coating layer, from ==0 to:=7/(4vn), to yield the total force per unit area exerted on the layer, namely F=-%(n-1)/n=E2 Considering that the incident momentum per unit time is d p /dt=2EE0, and that the time rate of change of the transmitted momentum into the semi-infinite medium is d p, /dt VE(E+ 1)E,=E(n+1)E n, it is clear that F:=d(pi-p )/dt, in other words,the upward force experienced by the coating layer is exactly equal to the time rate of change of the lights linear momentum upon crossing the layer. As a numerical example, consider the ase of a glass substrate of index n=2.0, coated with a quarter-wave thick layer of index Vn=1.414. At the incident power density of 1.0 W/mm, the computed net force on the coating layer is F:=-083 nN/mm 9. Homogeneous slab in free space The role of interference fringes in creating a magnetic Lorentz force is discussed in the present section. As far as we know, this topic has not been covered in the open literature Figure 8 shows a slab of thickness d and complex refractive index n+ ix= ve, surrounded by free-space and illuminated at normal incidence. The slab's(complex) reflection and transmission coefficients are denoted by rand t, respectively. The counter-propagating beams within the slab have E-field amplitudes En and E2; the total field distribution is given by x(==El exp(i2I vE:o)+E2 exp(-i2vE-/l, (24a) H,(=)=(NEE/)exp(i2nvE=/o)-(NEE2/Z )exp(-i2rvEi/o) (24b) Defining p=[(ve-1)/(vE+ D)lexp(i4Ive d/), the various parameters of the system of Fig. may be written as follows E1=2E0/(1+p)+vE(1-p E2=2pE/(+p)+VE(1-p #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5390A semi-infinite substrate of refractive index n is shown in Fig. 7, coated with a dielectric layer of index √n and thickness d = λo/(4√n). This quarter-wave thick layer is a perfect anti￾reflection coating that allows the incident beam into the semi-infinite medium with no reflection whatsoever. Conservation of energy requires the E-field to enter the semi-infinite medium with an amplitude |Et | = Eo/√n, while the quarter-wave thickness of the coating layer shifts the phase of Et by 90° relative to that of the incident beam. The H-field in the semi￾infinite medium is then Ht = nEt /Zo = i√n Eo/Zo. Inside the coating layer, the counter￾propagating beams shown in Fig. 7 have the following distributions: Ex ( z) = ½(1 + 1/√n ) Eo exp(i2π√n z/λo) + ½(1 − 1/√n ) Eo exp(−i2π√n z/λo), (21a) Hy ( z) = ½(1 + √n ) Ho exp(i2π√n z/λo) + ½(1 − √n ) Ho exp(−i2π√n z/λo). (21b) The force density, therefore, is given by Fz = ½ Real (Jx × By*) = ½ Real [−iω εo(√ε – 1)Ex × µ oHy*] = [π(n – 1)√µ oεo /λo] Imag (Ex × Hy*) = –½εo[π(n – 1)2 /(√n λo)] sin(4π√n z/λo)Eo 2 . (22) The above force density must be integrated over the thickness of the coating layer, from z = 0 to z = λo/(4√n), to yield the total force per unit area exerted on the layer, namely, Fz = –¼ [(n – 1)2 /n]εoEo 2 . (23) Considering that the incident momentum per unit time is d pi /d t = ½εoEo 2 , and that the time rate of change of the transmitted momentum into the semi-infinite medium is d pt /d t = ¼εo(ε + 1)|Et | 2 = ¼εo(n2 + 1)Eo 2 /n, it is clear that Fz = d( pi – pt) /d t; in other words, the upward force experienced by the coating layer is exactly equal to the time rate of change of the light’s linear momentum upon crossing the layer. As a numerical example, consider the case of a glass substrate of index n = 2.0, coated with a quarter-wave thick layer of index √n = 1.414. At the incident power density of 1.0 W/mm2 , the computed net force on the coating layer is Fz = −0.83 nN/mm2 . 9. Homogeneous slab in free space The role of interference fringes in creating a magnetic Lorentz force is discussed in the present section. As far as we know, this topic has not been covered in the open literature. Figure 8 shows a slab of thickness d and complex refractive index n + iκ = √ε, surrounded by free-space and illuminated at normal incidence. The slab’s (complex) reflection and transmission coefficients are denoted by r and t, respectively. The counter-propagating beams within the slab have E-field amplitudes E1 and E2; the total field distribution is given by Ex ( z) = E1 exp(i2π√ε z/λo) + E2 exp(−i2π√ε z/λo) (24a) Hy ( z) = (√ε E1/Zo) exp(i2π√ε z/λo) − (√ε E2 /Zo) exp(−i2π√ε z/λo) (24b) Defining ρ = [(√ε – 1) / (√ε + 1)]exp (i4π√ε d/λo), the various parameters of the system of Fig. 8 may be written as follows: E1 = 2Eo / [(1 + ρ) + √ε (1 – ρ)] (25a) E2 = 2ρEo / [(1 + ρ) + √ε (1 – ρ)] (25b) (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5390 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004