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Radiation pressure and the linear momentum of light in dispersive dielectric media

Masud Mansuripur

Open Access Open Access

Abstract

We derive an exact expression for the radiation pressure of a quasi-monochromatic plane wave incident from the free space onto the flat surface of a semi-infinite dielectric medium. In order to account for the total optical momentum (incident plus reflected) that is transferred to the dielectric, the mechanical momentum acquired by the medium must be added to the rate of flow of the electromagnetic momentum (the so-called Abraham momentum) inside the dielectric. We confirm that the electromagnetic momentum travels with the group velocity of light inside the medium. The photon drag effect in which the photons captured in a semiconductor appear to have the Minkowski momentum is explained by analyzing a model system consisting of a thin absorptive layer embedded in a transparent dielectric.

©2005 Optical Society of America

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Figures (2)
Fig. 1.
Fig. 1. A light beam consisting of two equal-amplitude plane-waves of slightly differing frequencies, f 1 and f 2, is normally incident on a semi-infinite dielectric of refractive index n(f). The beam is linearly polarized, having its E-field along the x-axis and H-field along the y-axis. While a fraction of the beam is reflected at the surface, the remainder enters the dielectric, penetrating it at the group velocity Vg =c/n+n – f). Here f = 1 2 ( f 1 + f 2 ) is the center frequency, and n ′=dn/df ; both n and n ′ are evaluated at the center frequency.

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Fig. 2.
Fig. 2. A thin absorptive layer of thickness d and complex refractive index n+iκ is embedded in a transparent, homogeneous dielectric medium of refractive index n (same n as the real part of the complex index of the absorptive layer). A monochromatic plane-wave, having vacuum wavelength λ o=c/f, E-field amplitude E o, and H-field amplitude H o=nE o/Z o, is normally incident on the absorbing layer. The layer’s (amplitude) reflection and transmission coefficients are ρ and τ, respectively. Each absorbed photon of energy hf transfers the equivalent of its Minkowski momentum nhf/c to the absorbing layer.

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Equations (25)

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p = 1 4 Real ( E × H * ) ⁄ c 2 + 1 4 Real ( D × B * ) .
p = 1 2 Real ( E × H * ) ⁄ c 2 + 1 4 Real ( P × B * ) .
E x ( z , t ) = E o sin { 2 π f 1 [ ( z ⁄ c ) − t ] } − E o sin { 2 π f 2 [ ( z ⁄ c ) − t ] }
H y ( z , t ) = ( E o ⁄ Z o ) sin { 2 π f 1 [ ( z ⁄ c ) − t ] } − ( E o ⁄ Z o ) sin { 2 π f 2 [ ( z ⁄ c ) − t ] }
< S z ( z , t ) > = E o 2 ⁄ Z o
d q z ⁄ d t = { 1 + 1 2 ∣ ( 1 − n 1 ) ⁄ ( 1 + n 1 ) ∣ 2 + 1 2 ∣ ( 1 − n 2 ) ⁄ ( 1 + n 2 ) ∣ 2 } ε o E o 2 .
d q z ⁄ d t = 2 [ ( n 2 + 1 ) ⁄ ( n + 1 ) 2 ] ε o E o 2 ,
E x ( z , t ) = [ 2 E o ⁄ ( n 1 + 1 ) ] sin { 2 π f 1 [ ( n 1 z ⁄ c ) − t ] } − [ 2 E o ⁄ ( n 2 + 1 ) ] sin { 2 π f 2 [ ( n 2 z ⁄ c ) − t ] }
H y ( z , t ) = { 2 n 1 E o ⁄ [ Z o ( n 1 + 1 ) ] } sin { 2 π f 1 [ ( n 1 z ⁄ c ) − t ] }
− { 2 n 2 E o ⁄ [ Z o ( n 2 + 1 ) ] } sin { 2 π f 2 [ ( n 2 z ⁄ c ) − t ] }
< S z ( z , t ) > = 1 2 { [ 4 n 1 ⁄ ( n 1 + 1 ) 2 ] + [ 4 n 2 ⁄ ( n 2 + 1 ) 2 ] } E o 2 ⁄ Z o .
d q z ⁄ d t = p z V g = 4 n ε o E o 2 ⁄ [ ( n + n ′ f ) ( n + 1 ) 2 ] .
J x ( z , t ) = ∂ P x ( z , t ) ⁄ ∂ t = − 4 π f 1 ( n 1 − 1 ) ε o E o cos { 2 π f 1 [ ( n 1 z ⁄ c ) − t ] }
+ 4 π f 2 ( n 2 − 1 ) ε o E o cos { 2 π f 2 [ ( n 2 z ⁄ c ) − t ] } .
F z ( t ) ⁄ ( 2 ε o E o 2 ) = { [ n 1 n 2 − ( n 1 f 2 − n 2 f 1 ) ⁄ ( n 2 f 2 − n 1 f 1 ) ] ⁄ [ ( n 1 + 1 ) ( n 2 + 1 ) ] } { 1 − cos [ 2 π ( f 2 − f 1 ) t ] }
− { ( n 2 − n 1 ) ( f 2 − f 1 ) ⁄ [ ( n 1 + 1 ) ( n 2 + 1 ) ( n 1 f 1 + n 2 f 2 ) ] } cos [ 4 π ( n 2 − n 1 ) f 1 f 2 t ⁄ ( n 2 f 2 − n 1 f 1 ) ]
+ { [ n 1 n 2 − ( n 1 f 2 + n 2 f 1 ) ⁄ ( n 1 f 1 + n 2 f 2 ) ] ⁄ [ ( n 1 + 1 ) ( n 2 + 1 ) ] } cos [ 2 π ( f 1 + f 2 ) t ]
− 1 2 [ ( n 1 − 1 ) ⁄ ( n 1 + 1 ) ] cos ( 4 π f 1 t ) − 1 2 [ ( n 2 − 1 ) ⁄ ( n 2 + 1 ) ] cos ( 4 π f 2 t ) .
< F z > = ( 1 ⁄ T ) ∫ 0 T F z ( t ) d t = 2 ε o E o 2 [ n 1 n 2 − ( n 1 f 2 − n 2 f 1 ) ⁄ ( n 2 f 2 − n 1 f 1 ) ] ⁄ [ ( n 1 + 1 ) ( n 2 + 1 ) ] .
< F z > = 2 { n 2 − [ ( n − n ′ f ) ( n + n ′ f ) ] } ε o E o 2 ⁄ ( n + 1 ) 2 .
p z V g + < F z > = 2 [ ( n 2 + 1 ) ⁄ ( n + 1 ) 2 ] ε o E o 2 .
ρ = − [ 1 + i ( κ ⁄ 2 n ) ] ( 2 π κ d ⁄ λ o ) ,
τ = 1 − ( 2 π κ d ⁄ λ o ) + i [ 2 n − ( κ 2 ⁄ n ) ] π d ⁄ λ o .
γ = 1 2 ( 1 − ∣ ρ ∣ 2 − ∣ τ ∣ 2 ) n E o 2 ⁄ Z o = ( 2 π n κ d ⁄ λ o ) E o 2 ⁄ Z o .
< F z > = ( 2 π n 2 κ d ⁄ λ o ) ε o E o 2 .
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