area d p/dt= 2S/c, which is equal to the force per unit area, F, exerted on the reflector. The force density of the light on a perfect reflector in free-space is thus given by F=EE The same expression may be derived by considering the Lorentz law F=q(E+vx B) in conjunction with the surface current density Js and the magnetic field H at the surface of the conductor. Here there are neither free nor(unbalanced) bound charges, and the motion of the conduction electrons constitutes a surface current density Js=gl. For time-harmonic fields, the force per unit area may thus be written F=‰2Real(J×B) In the case of a perfect conductor the magnitude of the surface current is equal to the magnetic field at the mirror surface, namely, Js=2H.(because VxH=J+ dp/dt; the factor of 2 arises from the interference between the incident and reflected beams where the two h- fields, being in-phase at the mirror surface, add up. )Since B=uH, one might conclude that F:=2uoHo. The factor of 2 in this formula, however, is incorrect because the magnetic field on the films surface, 2Ho, is assumed to exert a force on the entire s. The problem is that the field is 2Ho at the top of the mirror and zero just under the surface, say, below the skin-depth (Here we are using a limiting argument in which a good conductor, having a finite skin-depth, approaches an ideal conductor in the limit of zero skin-depth. ) Therefore, the average H-field through the"skin-depth"must be used in calculating the force, and this average is Ho not 2Ho The force per unit area thus calculated is F:=uHo =EEo, which is identical to the time rate of change of momentum of the incident beam derived in Eq (1). With 1.0 W/mm- of incident optical power, for example, the radiation pressure on the mirror will be 6.67 nN/mm Next, suppose the beam arrives on the mirror at an oblique angle 0, as in Fig. I(b); here the beam is assumed to be s-polarized. Compared to normal incidence, the component of the magnetic field H on the surface is now multiplied by cose, which requires the surface current density Js to be multiplied by the same factor(remember that s is equal to the magnetic field at the surface). The component of force density along the z-axis, Fs, is thus seen to have been reduced by a factor of cos"0. This result is consistent with the alternative derivation based on the time rate of change of the fields momentum in the z-direction, dp/dt, which is multiplied by cose in the case of oblique incidence. Since the beam has a finite diameter, its foot-print on the mirror is greater than that in the case of normal incidence by 1/cose. Thus the force density Fs, obtained by normalizing d p: /dt by the beams foot-print, is seen once again to be reduced by a factor of cos"0 Figure I(c)shows a p-polarized beam at oblique incidence on a mirror. The magnetic field component at the surface is 2Ho, which means that the surface current Js must also have the same magnitude as in normal incidence. We conclude that the force density on the mirror must be the same as that at normal incidence, namely, F: =EEo. The time rate of change of momentum in the z-direction, however, is similar to that in Fig. I(b), which means that the force density of normal incidence must have been multiplied by cos"0 in the case of oblique incidence. The two methods of calculating F: for p-light thus disagree by a factor of cos"0 The discrepancy is resolved when one realizes that, in addition to the magnetic force, an electric force is acting on the mirror in the opposite direction(-). This additional force pulls on the electric charges induced at the surface by El. Note that E1 2Eosine just above and El=0 just below the surface. The discontinuity in El gives the surface charge density as O= 2EECSine. The perpendicular E-field acting on these charges is the average of the fields just above and just below the surface, namely, E:=Eosine. The electric force density is thus F:=vReal(oE(e*)=EoEo'sine. The upward force on the charges thus reduces the #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5380
area d p/dt = 2S/c, which is equal to the force per unit area, F, exerted on the reflector. The force density of the light on a perfect reflector in free-space is thus given by F = εoEo 2 . (1) The same expression may be derived by considering the Lorentz law F = q (E + V × B) in conjunction with the surface current density Js and the magnetic field H at the surface of the conductor. Here there are neither free nor (unbalanced) bound charges, and the motion of the conduction electrons constitutes a surface current density Js = qV. For time-harmonic fields, the force per unit area may thus be written F = ½Real (Js × B*). (2) In the case of a perfect conductor the magnitude of the surface current is equal to the magnetic field at the mirror surface, namely, Js = 2Ho (because ∇ × H = J + ∂D/∂t; the factor of 2 arises from the interference between the incident and reflected beams where the two Hfields, being in-phase at the mirror surface, add up.) Since B = µ oH, one might conclude that Fz = 2µ oHo 2 . The factor of 2 in this formula, however, is incorrect because the magnetic field on the film’s surface, 2Ho, is assumed to exert a force on the entire Js. The problem is that the field is 2Ho at the top of the mirror and zero just under the surface, say, below the skin-depth. (Here we are using a limiting argument in which a good conductor, having a finite skin-depth, approaches an ideal conductor in the limit of zero skin-depth.) Therefore, the average H-field through the “skin-depth” must be used in calculating the force, and this average is Ho not 2Ho. The force per unit area thus calculated is Fz = µ oHo 2 = εoEo 2 , which is identical to the time rate of change of momentum of the incident beam derived in Eq. (1). With 1.0 W/mm2 of incident optical power, for example, the radiation pressure on the mirror will be 6.67 nN/mm2 . Next, suppose the beam arrives on the mirror at an oblique angle θ, as in Fig. 1(b); here the beam is assumed to be s-polarized. Compared to normal incidence, the component of the magnetic field H on the surface is now multiplied by cosθ, which requires the surface current density Js to be multiplied by the same factor (remember that Js is equal to the magnetic field at the surface). The component of force density along the z-axis, Fz, is thus seen to have been reduced by a factor of cos2 θ. This result is consistent with the alternative derivation based on the time rate of change of the field’s momentum in the z-direction, dpz/dt, which is multiplied by cosθ in the case of oblique incidence. Since the beam has a finite diameter, its foot-print on the mirror is greater than that in the case of normal incidence by 1/cosθ. Thus the force density Fz, obtained by normalizing d pz/d t by the beam’s foot-print, is seen once again to be reduced by a factor of cos2 θ. Figure 1(c) shows a p-polarized beam at oblique incidence on a mirror. The magnetic field component at the surface is 2Ho, which means that the surface current Js must also have the same magnitude as in normal incidence. We conclude that the force density on the mirror must be the same as that at normal incidence, namely, Fz = εoEo 2 . The time rate of change of momentum in the z-direction, however, is similar to that in Fig. 1(b), which means that the force density of normal incidence must have been multiplied by cos2 θ in the case of oblique incidence. The two methods of calculating Fz for p-light thus disagree by a factor of cos2 θ. The discrepancy is resolved when one realizes that, in addition to the magnetic force, an electric force is acting on the mirror in the opposite direction (−z). This additional force pulls on the electric charges induced at the surface by E⊥. Note that E⊥= 2Eosinθ just above and E⊥ = 0 just below the surface. The discontinuity in E⊥ gives the surface charge density as σ = 2εoEosinθ. The perpendicular E-field acting on these charges is the average of the fields just above and just below the surface, namely, Ez (eff) = Eosinθ. The electric force density is thus Fz = ½Real (σ Ez (eff)*) = εoEo 2 sin2 θ. The upward force on the charges thus reduces the (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5380 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004
downward force on the current, leading to a net F: =&Eo(1-sin"0), which is the same as that in the case of normal incidence multiplied by cos"8 The charge density o=2EE sine exp(i2Tx sine/? ple is produced the spatial variations of the current density J,= 2Hoexp(i2Tx SinA/o).Conservation of charge requires V J+ dp/dt=0, which, for time-harmonic fields, reduces to a@o=0 Considering that H.= EdZo and 2nc/Mo, it is readily seen that the above J, and o satisfy the required conservation law Note: The separate contributions of charge and current to the radiation pressure discussed in this section were originally discussed by Max Planck in his 1914 book, The Theory of Heat Radiation [10]. Our brief reconstruction of his arguments here is intended to facilitate the following discussion of electromagnetic force and momentum in dielectric media 4. Semi-infinite dielectric This section presents the core of the argument that leads to a new expression for the momentum of light inside dielectric media. Loudon [6, 7] has presented a similar argument in his quantum mechanical treatment of the problem. Although Loudons final result comes close to ours, there are differences that can be traced to his neglect of the mechanism of photon entry from the free-space into the dielectric medium H E1=(1+p)E H=(1-r)H n+IK Fig. 2. A linearly-polarized plane wave is normally incident on the surface of a sem medium of complex dielectric constant E. The Fresnel reflection coefficient at the surfac Shown are the e- and H-field magnitudes for the incident, reflected and transmitted bean Figure 2 shows a linearly-polarized plane wave at normal incidence on the flat surface of a semi-infinite dielectric. The incident E-and H-fields have magnitudes Eo and Ho= Eo/Zo Assuming a beam cross-sectional area of unity (A=1.0m"), the time rate of flow of momentum onto the surface is 2EE0, of which a fraction rF is reflected back. The net rate of change of linear momentum, which must be equal to the force per unit area exerted on the surface, is thus F:=hE(1+IrP)E. We assume that the mediums dielectric constant g is #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5381
downward force on the current, leading to a net Fz = εoEo 2 (1 − sin2 θ), which is the same as that in the case of normal incidence multiplied by cos2 θ. The charge density σ = 2εoEosinθ exp(i2πx sinθ/λo) in the above example is produced by the spatial variations of the current density Js = 2Hoexp(i2πx sinθ/λo). Conservation of charge requires ∇ · J + ∂ρ/∂t = 0, which, for time-harmonic fields, reduces to ∂Js/∂ x − iω σ = 0. Considering that Ho = Eo/Zo and ω = 2πc/λo, it is readily seen that the above Js and σ satisfy the required conservation law. Note: The separate contributions of charge and current to the radiation pressure discussed in this section were originally discussed by Max Planck in his 1914 book, The Theory of Heat Radiation [10]. Our brief reconstruction of his arguments here is intended to facilitate the following discussion of electromagnetic force and momentum in dielectric media. 4. Semi-infinite dielectric This section presents the core of the argument that leads to a new expression for the momentum of light inside dielectric media. Loudon [6,7] has presented a similar argument in his quantum mechanical treatment of the problem. Although Loudon’s final result comes close to ours, there are differences that can be traced to his neglect of the mechanism of photon entry from the free-space into the dielectric medium. Fig. 2. A linearly-polarized plane wave is normally incident on the surface of a semi-infinite medium of complex dielectric constant ε. The Fresnel reflection coefficient at the surface is r. Shown are the E- and H-field magnitudes for the incident, reflected, and transmitted beams. Figure 2 shows a linearly-polarized plane wave at normal incidence on the flat surface of a semi-infinite dielectric. The incident E- and H-fields have magnitudes Eo and Ho = Eo/Zo. Assuming a beam cross-sectional area of unity (A = 1.0m2 ), the time rate of flow of momentum onto the surface is ½εoEo 2 , of which a fraction | r |2 is reflected back. The net rate of change of linear momentum, which must be equal to the force per unit area exerted on the surface, is thus Fz = ½εo(1 + |r |2 )Eo 2 . We assume that the medium’s dielectric constant ε is Ho Eo Et = (1 + r) Eo Ht = (1 − r) Ho X Z -rHo rEo n + iκ = √ε (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5381 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004
not purely real, but has a small imaginary part. The complex refractive index of the material is and the reflection coefficient is r=(1-VE V(+ve) Inside the dielectric, the E-field is E(=)=Er exp(inve:o), where E,=(1+r)Eo,the field is H(E)=vE(E, /Z exp(inve:/o), the D-field is D(E)=E()+P()=EEE(),and the dipolar current density is J)=-ioP()=-ioE(E-1)E(), where @=2If= 2nc/o is the optical frequency. The force per unit volume is thus given by F:=%2 Real(xB")=y(2/o)Real [-ive"(E-1)) EolE, Pexp(-4x-/o %(2)(n2+x2+1) EoE, Fexp(-4rx2) The total force per unit surface area is obtained by integrating the above F: from==0 to The multiplicative coefficient x disappears after integration, and the force per unit area becomes F:=4(+K+1)EE, I. Upon substitution for E, and r, this expression for F: turns out to be identical to that obtained earlier based on momentum considerations We now let x-0 and write the radiation force per unit surface area of the dielectric as F:=v4(n+ 1)EE,F(A similar trick has been used by R. Loudon in his calculation of the photon momentum inside dielectrics [7].)Considering that H,=nE,/Zo, one may also write F=V4 EoE, F+4 HolH, P. This must be equal to the rate of the momentum entering the medium at ==0. Since the speed of light in the medium is c/n, the momentum density(per unit volume)within the dielectric may be expressed as follor P:=7(n+ I)nEeR /c= v4(E+ l)EoJE, BrI Equation(4), the fundamental expression for the momentum density of plane waves in dielectrics, may also be written as p= 4(DXB)+4(Ex Hye. Historically, there has been a dispute as to whether the proper form for the momentum density of light in dielectrics is Minkowski's DxB or Abraham's Ex H/c2 [5]. The above discussion leads to the conclusion that neither form is appropriate; rather, it is the average of the two that yields most plausible expression for P In the limit when 2-1, the two terms in the expression for p become identical, and the familiar form for the free-space, P=S/c2, emerges Replacing D with EE+ Pand B with uH, we obtain P=y4(Px B)+ExH)c,which shows the separate contributions to a plane-wave's momentum density by the medium and by the radiation field. The mechanical momentum of the medium. 4P xB arises from the interaction between the induced polarization density P and the light's B-field. The contribution of the radiation field. ExH/C2 has the same form. S/c. as the momentum density of electromagnetic radiation in free space. Since P=E(E-D)E, the mechanical momentum density may be written as 4PxB=/dE-1)S/c. For a dilute medium having refractive index n= l, the coefficient of S/c in the above formula reduces to vE-1=n-I which leads to the expression(n-1)S/c derived in [5] for the mechanical momentum of dilute gases. The physical basis for the separation of the momentum density into electromagnetic and mechanical contributions will be further elaborated in Section 12 Note: In a recent paper [151, Obukhov and Hehl argue, as we do here, that the correct interpretation of the electromagnetic momentum in dielectric media must be based on the standard form of the Lorentz force, taking into account both free and bound charges and currents. In their discussion of the case of normal incidence from vacuum onto a semi-infinite dielectric, however, they neglect to account for the mechanical momentum imparted to the dielectric medium. As a result, they find only the electromagnetic part of the momentum density;their Eq( 27)is in fact identical to ExH/c, where E and H are evaluated inside the dielectric. In contrast, our approach in the present section, which involves the introduction of a small(but non-zero)K, followed by an integration of the feeble magnetic Lorentz force over the infinite thickness of the dielectric ensures that the mechanical momentum of the #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5382
not purely real, but has a small imaginary part. The complex refractive index of the material is n + iκ = √ε , and the reflection coefficient is r = (1 − √ε )/(1 + √ε ). Inside the dielectric, the E-field is E(z) = Et exp(i2π√ε z/λo), where Et = (1 + r )Eo, the Hfield is H(z) = √ε (Et /Zo)exp(i2π√ε z/λ0), the D-field is D(z) = εoE(z) + P(z) = εoεE(z), and the dipolar current density is J(z) = −iω P(z) = −iω εo(ε − 1)E(z), where ω = 2πf = 2πc/λo is the optical frequency. The force per unit volume is thus given by Fz = ½ Real (J × B*) = ½(2π/λo) Real [−i√ε*( ε − 1)] εo|Et |2 exp(−4πκ z/λo) = ½(2π/λo) (n2 + κ2 + 1)κ εo|Et |2 exp(−4πκ z/λo). (3) The total force per unit surface area is obtained by integrating the above Fz from z = 0 to ∞. The multiplicative coefficient κ disappears after integration, and the force per unit area becomes Fz = ¼ (n2 + κ2 + 1) εo|Et |2 . Upon substitution for Et and r, this expression for Fz turns out to be identical to that obtained earlier based on momentum considerations. We now let κ → 0 and write the radiation force per unit surface area of the dielectric as Fz = ¼ (n2 + 1)εo|Et |2 . (A similar trick has been used by R. Loudon in his calculation of the photon momentum inside dielectrics [7].) Considering that Ht = nEt /Zo, one may also write Fz = ¼ εo|Et |2 + ¼ µo|Ht |2 . This must be equal to the rate of the momentum entering the medium at z = 0. Since the speed of light in the medium is c/n, the momentum density (per unit volume) within the dielectric may be expressed as follows: pz = ¼ (n2 + 1) nεo|Et |2 /c = ¼(ε + 1)εo|Et Bt |. (4) Equation (4), the fundamental expression for the momentum density of plane waves in dielectrics, may also be written as p = ¼ (D × B ) + ¼ (E × H )/c2 . Historically, there has been a dispute as to whether the proper form for the momentum density of light in dielectrics is Minkowski’s ½ D × B or Abraham’s ½ E × H /c2 [5]. The above discussion leads to the conclusion that neither form is appropriate; rather, it is the average of the two that yields the most plausible expression for p. In the limit when ε → 1, the two terms in the expression for p become identical, and the familiar form for the free-space, p = S/c2 , emerges. Replacing D with εoE + P and B with µ oH, we obtain p = ¼(P × B) + ½(E × H )/c2 , which shows the separate contributions to a plane-wave’s momentum density by the medium and by the radiation field. The mechanical momentum of the medium, ¼P × B, arises from the interaction between the induced polarization density P and the light’s B-field. The contribution of the radiation field, ½ E × H /c 2 , has the same form, S/c 2 , as the momentum density of electromagnetic radiation in free space. Since P = εo(ε − 1)E, the mechanical momentum density may be written as ¼P × B = ½(ε − 1)S/c2 . For a dilute medium having refractive index n ≈ 1, the coefficient of S/c2 in the above formula reduces to ½(ε − 1) ≈ n – 1, which leads to the expression (n − 1)S/c2 derived in [5] for the mechanical momentum of dilute gases. The physical basis for the separation of the momentum density into electromagnetic and mechanical contributions will be further elaborated in Section 12. Note: In a recent paper [15], Obukhov and Hehl argue, as we do here, that the correct interpretation of the electromagnetic momentum in dielectric media must be based on the standard form of the Lorentz force, taking into account both free and bound charges and currents. In their discussion of the case of normal incidence from vacuum onto a semi-infinite dielectric, however, they neglect to account for the mechanical momentum imparted to the dielectric medium. As a result, they find only the electromagnetic part of the momentum density; their Eq. (27) is in fact identical to ½ E × H /c 2 , where E and H are evaluated inside the dielectric. In contrast, our approach in the present section, which involves the introduction of a small (but non-zero) κ, followed by an integration of the feeble magnetic Lorentz force over the infinite thickness of the dielectric, ensures that the mechanical momentum of the (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5382 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004
expression for the momentum density, which is missing from Obukhov and Hehl's Eq. 74> medium is properly taken into consideration. This yields the term 4 (Px B)in our 5. Oblique incidence with s-polarized light To the best of our knowledge, the momentum of light at oblique incidence has not been discussed previously. This is an extremely important case, since it reveals the existence of a lateral radiation pressure at the edges of the beam within a dielectric medium; consistency with the results of Section 4 simply demands the existence of such lateral pressures. We analyze the case of s-polarization here, leaving a discussion of p-polarized light at oblique incidence for the next section. The two cases turn out to be fundamentally different, although both retain the expression for momentum density derived in the case of normal incidence Figure 3 shows the case of oblique incidence with s-polarized light at the interface between the free-space and a dielectric medium. Again, we assume that the dielectric constant E is complex, allowing it to approach a real number only after calculating the total force by ntegrating through the thickness of the medium. Inside the medium, the E- and H-field distributions are Er (x, =)=(1+r3)E exp[i2T(x sine +NvE-sin e nol (5a) Hx(x,=)=ve-sin0 Err(x, syZ HI: (x, =)=-sine Ety(x,=)Zo Here rs=(cose-Ve-sin 0)/(cos0+VE-sin'e)is the Fresnel reflection coefficient for s- light. Since there are no free charges inside the medium(nor on its surface), the only relevant force here is the magnetic Lorentz force on the dipolar current density Jy(x, = -ioEE-DErv(, =). Following the same procedure as before, we find the net force components along the x-and z-axes to be r X E1=(1+rs)E。 n+ik=ve medium of(complex)dielectric constant E The Fresnel reflection coefficient is denoted by rs #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5383
medium is properly taken into consideration. This yields the term ¼(P × B) in our last expression for the momentum density, which is missing from Obukhov and Hehl’s Eq. (27). 5. Oblique incidence with s-polarized light To the best of our knowledge, the momentum of light at oblique incidence has not been discussed previously. This is an extremely important case, since it reveals the existence of a lateral radiation pressure at the edges of the beam within a dielectric medium; consistency with the results of Section 4 simply demands the existence of such lateral pressures. We analyze the case of s-polarization here, leaving a discussion of p-polarized light at oblique incidence for the next section. The two cases turn out to be fundamentally different, although both retain the expression for momentum density derived in the case of normal incidence. Figure 3 shows the case of oblique incidence with s-polarized light at the interface between the free-space and a dielectric medium. Again, we assume that the dielectric constant ε is complex, allowing it to approach a real number only after calculating the total force by integrating through the thickness of the medium. Inside the medium, the E- and H-field distributions are Et y (x, z) = (1 + rs)Eo exp[i2π(x sinθ + z√ε − sin2 θ )/λo] (5a) Ht x (x, z) = √ε − sin2 θ Et y (x, z)/Zo (5b) Ht z (x, z) = −sinθ Et y (x, z)/Zo (5c) Here rs = (cosθ − √ε – sin2 θ ) / (cosθ + √ε – sin2 θ ) is the Fresnel reflection coefficient for slight. Since there are no free charges inside the medium (nor on its surface), the only relevant force here is the magnetic Lorentz force on the dipolar current density Jy(x, z) = −iω εo(ε − 1)Et y (x, z). Following the same procedure as before, we find the net force components along the x- and z-axes to be Fig. 3. Obliquely incident s-polarized plane wave arrives at the surface of a semi-infinite medium of (complex) dielectric constant ε. The Fresnel reflection coefficient is denoted by rs. Ho Eo θ θ′ Ht X Z rs Eo rsHo Et = (1 + rs)Eo n + iκ = √ε (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5383 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004