Abstract

The electromagnetic plane-wave force on homogeneous materials having the possible set of constitutive parameter signs is evaluated. The force remains positive in all cases with loss, but a negative force results with gain. A negative force is shown to occur for an evanescent field when there is no component of the electric field in that direction. Both forces have significant magnitude and hence should be observable in experiments.

© 2012 Optical Society of America

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Figures (5)

Fig. 1.
Fig. 1.

(a) Material dispersion obtained using μ = μ loss = 1 + 2 ( 1.05 2 ω 2 i 0.2 ω ) 1 and ϵ = ϵ loss = 1 + 2 ( 0.95 2 ω 2 i 0.2 ω ) 1 . (b) Corresponding time-averaged normalized electromagnetic plane-wave force density calculated by the exact numerical procedure. The dotted–dashed line is Case (i) with μ = ϵ = ϵ loss , the solid line is Case (ii) with μ = 1 , ϵ = ϵ loss , and the dotted line is Case (iii) with μ = μ loss , ϵ = ϵ loss . The electric field temporal modulation is a Gaussian defined in Eq. (19), and the carrier frequency is ω 0 = 1 . The circles show the force obtained for Case (ii) ( μ = 1 , ϵ = ϵ loss ) using the analytical result in Eq. (9).

Fig. 2.
Fig. 2.

(a) Material dispersion obtained using μ = μ gain = 1 0.6 ( 0.95 2 ω 2 i 0.2 ω ) 1 and ϵ = ϵ gain = 1 0.6 ( 1.05 2 ω 2 i 0.2 ω ) 1 . (b) Corresponding time-averaged normalized electromagnetic plane-wave force density calculated using the exact numerical procedure, and the analytical expression in Eq. (9), shown by circles. The dotted–dashed line is Case (i) with μ = ϵ = ϵ gain , the solid line is Case (ii) with μ = 1 , ϵ = ϵ gain , and the dotted line is Case (iii) with μ = μ gain , ϵ = ϵ gain . The electric field modulation is given by Eq. (19), and the carrier frequency is ω 0 = 1 .

Fig. 3.
Fig. 3.

Three normalized force density components and the total normalized force density in Eq. (9), calculated using μ = μ loss = 1 + 2 ( 1.05 2 ω 2 i 0.2 ω ) 1 and ϵ = ϵ loss = 1 + 2 ( 0.95 2 ω 2 i 0.2 ω ) 1 . The electric field is given in Eq. (5) with carrier frequency ω 0 = 1 , and the various curves show the influence of changing σ in Eq. (19): σ = 2 9 π (dotted–dashed), σ = 2 8 π (dashed), and σ = 2 7 π (solid). The total temporal support for the calculation was fixed at 2 13 π , with N = 2 20 sample points.

Fig. 4.
Fig. 4.

Three normalized force density components and the total normalized force density in Eq. (9), calculated for μ = 1 + 2 ( 1.05 2 ω 2 i γ ω ) 1 and ϵ = 1 + 2 ( 0.95 2 ω 2 i γ ω ) 1 . The curves show the effect of varying the linewidth γ : γ = 0.6 (dotted–dashed), γ = 0.4 (dashed), and γ = 0.2 (solid). The electric field modulation is given by Eq. (19) with σ = 2 8 π , and the carrier frequency is ω 0 = 1 .

Fig. 5.
Fig. 5.

(a) Numerical electric field solution [ | E y ( z ) | ] from an FEM simulation for a TE plane wave ( E y , H x , H z ). (b) Corresponding numerical time-averaged force density f z . The dashed line is for a lossy material with ϵ loss ( ω 0 ) = 4 + i 1 , and the dotted line is for a material with gain having ϵ gain ( ω 0 ) = 4 i 1 , and for both, H z = 0 and k x = 0 . The solid line is the case of an evanescent field in a medium with ϵ ( ω 0 ) = 4 and for k = x ^ k x + z ^ k z and k x = 1.1 k . The circles in (b) show the analytic f z , evaluated using f z = μ 0 ϵ 0 C 1 e 2 ( t , z ) with e ( t , z ) = | E y ( ω 0 , z ) | / π and C 1 from Eq. (10). The frequency is ω = 2 π × 10 8 ( λ = 3 m ), and μ = μ 0 .

Equations (25)

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f = · ( T e + T d + T m ) t ( G e + G d + G m ) = · t + ρ E + J × μ 0 H + P · E + μ 0 M · H + P t × μ 0 H μ 0 M t × ϵ 0 E + · ( v P ) × μ 0 H + · ( v μ 0 M ) × ϵ 0 E + v × [ ( P · ) μ 0 H ] v × [ ( μ 0 M · ) ϵ 0 E ] ,
P ( t ) = ϵ 0 4 π [ ϵ ( ω ) 1 ] χ E ( ω ) E ( ω ) e i ω t d ω + c . c . ,
M ( t ) = 1 4 π [ μ ( ω ) 1 ] χ M ( ω ) H ( ω ) e i ω t d ω + c . c . ,
f = P t × μ 0 H μ 0 M t × ϵ 0 E .
E ( t ) = e ^ E ( t ) = e ^ e ( t ) cos ( ω 0 t ) ,
H ( t ) = h ^ H ( t ) = h ^ 4 π u ( ω ) E ( ω ) e i ω t d ω + c . c . ,
f = e ^ × h ^ f ,
f = μ 0 H P t + μ 0 ϵ 0 E M t .
f n f c 2 = ( C 1 + D 1 ) e 2 ( t ) f n ( 1 ) + ( C 2 + D 2 ) e ( t ) e ( t ) t f n ( 2 ) + C 3 [ e ( t ) t ] 2 f n ( 3 ) ,
C 1 ω 0 2 [ u ( ω 0 ) χ E ( ω 0 ) u ( ω 0 ) χ E ( ω 0 ) ] , C 2 1 2 [ u ( ω 0 ) ( ω χ E ) ω | ω 0 + u ( ω 0 ) ( ω χ E ) ω | ω 0 ω 0 χ E ( ω 0 ) u ω | ω 0 ω 0 χ E ( ω 0 ) u ω | ω 0 ] , C 3 1 2 [ u ω | ω 0 ( ω χ E ) ω | ω 0 u ω | ω 0 ( ω χ E ) ω | ω 0 ] , D 1 ω 0 2 v = ω 0 2 [ u ( ω 0 ) χ M ( ω 0 ) + u ( ω 0 ) χ M ( ω 0 ) ] , D 2 1 2 ( ω v ) ω | ω 0 = 1 2 [ u ( ω 0 ) ( ω χ M ) ω | ω 0 u ( ω 0 ) ( ω χ M ) ω | ω 0 + ω 0 χ M ( ω 0 ) u ω | ω 0 ω 0 χ M ( ω 0 ) u ω | ω 0 ] .
f n = E 0 2 ( C 1 + D 1 ) = E 0 2 ω 0 2 [ u ( χ E + χ M ) u ( χ E χ M ) ] = E 0 2 ω 0 2 [ u ( ϵ + μ ) u ( ϵ μ ) ] = E 0 2 ω 0 2 | η | 2 [ η ( ϵ + μ ) + η ( ϵ μ ) ] .
H x P y t = E 0 2 ω 0 ϵ 0 2 [ u z ( ω 0 ) χ E ( ω 0 ) u z ( ω 0 ) χ E ( ω 0 ) ] ,
H z P y t = E 0 2 ω 0 ϵ 0 2 [ u x ( ω 0 ) χ E ( ω 0 ) u x ( ω 0 ) χ E ( ω 0 ) ] .
f n ( μ = 1 ) = z ^ E 0 2 ω 0 2 u z χ E ,
= z ^ E 0 2 2 k z μ 0 ( ϵ 1 ) .
H y P x t = E x 0 2 ω 0 ϵ 0 2 [ u z ( ω 0 ) χ E ( ω 0 ) u z ( ω 0 ) χ E ( ω 0 ) ] ,
H y P z t = E z 0 2 ω 0 ϵ 0 2 [ u x ( ω 0 ) χ E ( ω 0 ) u x ( ω 0 ) χ E ( ω 0 ) ] ,
f n ( μ = 1 ) = z ^ E x 0 2 ω 0 2 ϵ 0 2 2 k z ϵ ( ϵ 1 ) .
e ( t ) = exp [ t 2 2 σ 2 ] .
μ loss = 1 + 2 ( 1.05 2 ω 2 i 0.2 ω ) ,
ϵ loss = 1 + 2 ( 0.95 2 ω 2 i 0.2 ω ) .
μ gain = 1 0.6 ( 0.95 2 ω 2 i 0.2 ω ) ,
ϵ gain = 1 0.6 ( 1.05 2 ω 2 i 0.2 ω ) .
μ = 1 + 2 ( 1.05 2 ω 2 i γ ω ) ,
ϵ = 1 + 2 ( 0.95 2 ω 2 i γ ω ) ,

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