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Effect of the surface integral in the torque equation of the electromagnetic angular momentum on the Faraday rotation
Optics Communications 284 (2011) 4248–4253
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Effect of the surface integral in the torque equation of the electromagnetic angular momentum on the Faraday rotation Osamu Yamashita Materials Science, Co., Ltd., 5-5-44, Minamikasugaoka, Ibaraki, Osaka 567-0046, Japan
a r t i c l e
i n f o
Article history: Received 3 January 2011 Received in revised form 29 April 2011 Accepted 13 May 2011 Available online 27 May 2011 Keywords: Electromagnetic angular momentum Torque equation Faraday rotation Surface integral Volume integral
a b s t r a c t A circularly polarized plane wave of infinite transverse extent (δ = ∞) has no spin angular momentum, while a realistic light does carry it. This paradox originates from the presence (δ = ∞) and absence (δ ≈ 0) of the surface integral in the total angular momentum J. The same holds for the torque equation of d J/dt, so that δ is also connected with the relative Faraday rotation angle ΘF/θF when a radius (a) of a cylindrical medium with optical activity is only a little larger than that (b) of light beam, where ΘF is the Faraday rotation angle and θF is the intrinsic Faraday rotation angle of a medium. It is shown here that it is possible to estimate δ for a realistic light from the drastic variation in ΘF/θF near b/a = 1. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The angular momentum (AM) carried by light can be characterized by the spin AM associated with circular polarization and the orbital AM associated with the spatial distribution of the wave. Both spin and orbital angular momenta of light beam have in fact been measured [1–4]. Theoretically, the spin and orbital angular momenta (S and L) of the radiation fields have been defined explicitly for free space and an isotropic medium [5–8]. The spin AM arises even from the transverse plane electromagnetic waves, but the orbital AM never appears in the plane electromagnetic waves. As is generally known, however, a circularly polarized plane wave of infinite transverse extent can have no spin AM [9]. However, only a quasiplane wave of finite transverse extent δ carries the spin AM whose direction is along the direction of propagation. As evident from the definition of the AM, the component of the AM in the direction of propagation must be zero, but it is non-zero actually. This paradox has been subject of discussion for a long time [9] and even recently [10–12]. Some ideas [8,10,13–15] were proposed to resolve this paradox, but it has not yet been settled. At present, one justifies these results by taking into account the fact that a detector placed in a plane wave causes gradients in this field [14]. The field can no longer be considered as a plane wave. In other words, any obstacle that absorbs the beam changes the electromagnetic field at the edges of the obstacle so that the field components in the direction of propagation are produced. Recently, another thought different from this was proposed by Stewart [15]. He
E-mail address: [email protected]. 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.05.031
took into account the effect of boundaries on the plane wave problem by decomposing the AM into three items of two volume integrals of a spin character and an orbital character and one surface integral and concluded that the contribution to the AM arises from the edges of the beam. This idea seems to be reasonable and consistent theoretically. In his paper [15], however, the contribution from the surface integral to the AM has not been expressed as a function of δ, where δ is the transverse extent of the plane wave. For this reason, it is shown here that the contribution from the surface integral to the AM of the plane wave can be expressed analytically as a function of δ, on the assumption that the intensity profile of the optical fields forms a flat plateau for r ≤ b and decreases according to exp [−(r − b)/δ] for r N b, where r is the radial distance from the center axis of beam and b is a radius of beam. The same is also applicable to the torque equation of AM. This torque equation links directly to the relative Faraday rotation angle (ΘF/θF) expressed as a function of δ, where ΘF is the Faraday rotation angle and θF is the intrinsic Faraday rotation angle of a medium. Of course, the transverse extent δ has not yet been measured for a realistic light, because there was no experimental procedure for measuring it. For this reason, we show here that it is possible theoretically to estimate δ from the measurement of the Faraday rotation angle when a is only a little larger than b, where a is a radius of an optically cylindrical medium. When this experiment was carried out successfully, the transverse extent of a realistic light is revealed so that the above paradox is resolved explicitly. The purpose of this study is to clarify the relation between the surface integrals in the torque equation for AM and the transverse extent δ or the relative Faraday rotation angle ΘF/θF, and to provide a new experimental procedure for measuring the transverse extent of a realistic light, resulting in the resolution of the traditional paradox.
O. Yamashita / Optics Communications 284 (2011) 4248–4253
current are absent [6,7]. In the radiation gage, therefore, the electric field E⍊ and magnetic induction field B are expressed by the vector potential A⍊ as [7]
2. Analysis 2.1. The definition of angular momentum for electromagnetic fields The Maxwell equations for the macroscopic electromagnetic fields in a medium are given by [16] ∇×E=−
∂B ; ∂t
ð1Þ
∂D ∇×H= + I; ∂t
ð2Þ
∇· D = ρ;
ð3Þ
∇· B = 0:
ð4Þ
It is assumed here that the medium is inhomogeneous and anisotropic but is free of dispersion. In these equations, the electric D and magnetic B induction fields are related to the field strengths E and H as Di = x, y, z = Σj = x, y, z εijEj and Bi = x, y, z = Σj = x, y, z μ ijHj where εij and μij represent the permittivity and permeability tensors, I is the electric current density, and ρ is the charge density. Let us consider an arbitrary volume τ inside the crystal filled by charges, described by the volume density ρ and current density I. Because of the interaction with the electromagnetic field, the charges experience a total mechanical torque dLc/dt given by [17] dLc = ∫τ dvr × ½ρE + ðI × BÞ; dt
B = ∇ × A⊥ ;
ð10Þ
and
where E⍊ and A⍊ are the transverse components of E and A. In other words, this is the same as the Coulomb gage for φ = 0. The transverse part A⊥ of the vector potential is thus gage invariant. We will hereafter treat D, E and A as D⍊, E⍊ and A⍊, respectively. By substituting Eq. (10) into Eq. (7) and applying partial integration, the angular momentum J [5–7] defined previously for the transverse electromagnetic field is separated into three parts as J = ∫τ ðD × AÞdv + ∫τ Σi = x; y; z Di ðr × ∇ÞAi dv
ð11Þ
+ ∫Σ ðA × r ÞðD·dsÞ = S + L + SΣ ; where the symbol ∇ is the gradient operator, and S, L and SΣ represent the volume integrals with spin and orbital characters and the surface integral, respectively, which are as follows, S = ∫τ ðD × AÞdv;
ð12Þ
L = ∫τ Σi = x; y; z Di ðr × ∇ÞAi dv
ð13Þ
and S Σ = ∫Σ ðA × rÞðD·dsÞ:
ð7Þ
in analogy to the interpretation of the linear momentum density (D × B) in a medium [6,16]. This definition is valid at least when the medium is linear, but not necessarily isotropic, in its response [16]. However, we do not enter here into the well-known question of the correct definition of momentum in media. 2.2. The angular momenta of radiation fields and their torque equations The electromagnetic fields can be separated into transverse and longitudinal fields, which have by definition a vanishing divergence and curl, respectively. The magnetic field is purely transverse, while the longitudinal electric induction field D∥ is given by the instantaneous Coulomb field arising from the charge density ρ, i.e., ∇·D∥ = ρ. The transverse electric induction field D⊥ thus describes the radiation part, which contains in fact the only real dynamical degrees of freedom of the field. The symbols || and ⊥ denote the components parallel and perpendicular to the optical axis, respectively. For D⊥, therefore, Eq. (3) may be rewritten as ∇· D⊥ = 0:
ð9Þ
ð14Þ
ð6Þ
where the angular momentum J of electromagnetic fields in a medium is defined as J = ∫τ dv½r × ðD × BÞ
∂A⊥ ∂t
E⊥ = −
ð5Þ
where Lc is the mechanical angular momentum of the (charged and neutral) particles. Substituting I and ρ from Eqs. (2) and (3) into Eq. (5), we get, after some manipulations, d ðL + J Þ = ∫τ dvfr × ½Eð∇ · DÞ–D × ð∇ × EÞ–B × ð∇ × H Þg; dt c
4249
ð8Þ
Generally, the radiation gage is defined as φ = 0 and ∇·A = 0 in the gage transformation, where φ is the scalar potential and A is the vector potential[16]. This gage is applicable to the transverse part of the electromagnetic fields and is often used when the charge and
When the radiation gage is employed, therefore, the angular momentum of J is gage independent. In this sense, the separation of S and L has a clear physical meaning [6,7], but we recognize that the identification of terms as spin and orbital momenta may not be unique in general. However, the orbital AM is ignored in this and subsequent subsections because we treat only the plane electromagnetic wave. When the electromagnetic field in an isotropic medium is composed of the plane wave of infinite extent, the surface integral of SΣ can be transformed to the volume integral of −∫τ (D × A)dv, so that S and SΣ cancel out, resulting in J = 0 [9]. When the electromagnetic field has infinitesimal extent, however, the surface integral of SΣ vanishes and only the volume integral of S survives, resulting in J = S. The surface integral thus varies significantly with changes in the magnitude of the transverse extent of the plane wave. Let us consider the plane waves of finite extent δ which form a light beam of radius b propagating along the z axis in a cylindrical medium of radius a, as shown in Fig. 1. Since it is difficult to calculate exactly the intensity profile of the optical fields, for simplicity, it is assumed here that the intensity profile of the optical fields forms a flat plateau for r ≤ b, while for r ≥ b, it is expressed by a function of exp [−(r–b)/δ], as shown in Fig. 1(b), where r is the radial distance from the center axis of beam and (r–b) is the distance from the edge of beam. It means that the intensity of the optical fields in the outside of beam core decreases exponentially and concentrically with an increase of (r–b). When the intensity of the optical fields at the distance δ from the beam edge is assumed to lower down to 1/e of that at the beam edge, δ represents the average transverse extent of the optical fields protruded slightly from the core edge, as evident from the integral value of ∫0∞e −r/δdr = δ. In practice, the intensity of the optical fields (including the vector potential A) would probably decrease rapidly near the edge of beam in a complicated manner. However, even this simple intensity profile is
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O. Yamashita / Optics Communications 284 (2011) 4248–4253
(a)
Next, the torque equations for S, L and SΣ are obtained by applying partial integration to the time derivatives of Eqs. (12)–(14) as
Cylindrical medium
∂S ∂ = ∫ dvðD × AÞ ∂t ∂t τ = ∫τ dv½ðE × DÞ+ ðH × BÞ–∫τ dv∇ðA·H Þ+∫τ dv½ðH·∇ÞA + ðA·∇ÞH ;
a
ð18Þ ∂L ∂ = ∫ dvΣDi ðr × ∇ÞAi ∂t ∂t τ
L
= –∫τ dvΣ½Bi ðr × ∇ÞHi –Di ðr × ∇ÞEi –∫τ dv½ðH·∇ÞA + ðA·∇ÞH
(b)
Intensity distribution
+ ∫Σ ½ðH · BÞðr × dsÞ + H ðA·dsÞ + ΣHi ðr × ∇ÞdF ð19Þ
1
and ∂S Σ = ∫Σ fðr × EÞðD·dsÞ + ðr × AÞ½∇· ðds × H Þg ∂t
exp[−(r−b)/δ] for r>b
b
b
0
= ∫Σ ðA ·H Þds + ∫Σ ½ðr × EÞðD·dsÞ + ðr × H ÞðB·dsÞ
r
–∫Σ ½ðH ·BÞðr × dsÞ + H ðA· dsÞ + ΣHi ðr × ∇ÞdF ;
(c) Beam
0
Cylindrical medium
a
b
z L
Fig. 1. (a) A cylindrical and optically active medium of a radius of a and a length of L. (b) Intensity profile of a plane wave that has passed through an aperture. The intensity of the beam is assumed to be constant for r≤b and to decrease exponentially with an increase of r for r N b, where r is the radial distance from the center axis of beam, δ is the transverse extent of a plane wave and b is a radius of beam. (c) Schematic diagram of a beam and a cylindrical medium for the measurement of the Faraday rotation angle, where the z axis runs through the center of them.
enough to study roughly the r-dependence of SΣ near the circumference of a cylindrical medium. The SΣ is dominated by the fields at the edge of a medium, as will be shown later. When the boundary of a medium overlaps locally with the edge of beam, therefore, SΣ at the boundary can be expressed in terms of a volume integral as S Σ = −∫τ ðD × AÞδðr–aÞdv;
ð15Þ
where δ(r–a) is the delta function. When the intensity of D and A is proportional to exp [−(r–b)/δ] for r N b, the cross product of (D × A) in Eq. (15) is expressed using an attenuation factor as (D × A) exp [− 2 (r–b)/δ]. Substituting this expression instead of (D × A) into Eq. (15) and integrating, SΣ is rewritten using S and δ as S Σ = −S exp½−2ða–bÞ = δ:
ð20Þ
ð16Þ
where i = x, y and z, dFi is the i-component of dF = (A × ds) and the symbol ∇ is a differential operator acting outside the brackets. These expressions change to some extent depending on the way of partial integration. The total torque equation of ∂J/∂t is derived from the summation of Eqs. (18)–(20). The first term on the right hand side of Eq. (18) arises from the anisotropy of the medium and vanishes when a medium has a rotational invariance around the z axis. The second term of ∇(A·H) represents the magnetic helicity [18] of the electromagnetic field caused by the anisotropy and inhomogeneity. This term is absent when a medium is isotropic and uniform, but it survives in an optically active medium possessing a rotational invariance, because the magnetic helicity term has no rotational invariance. The final term in Eq. (18) originates from both the anisotropy and the inhomogeneity and is the same as the second one in Eq. (19) but is opposite to it in sign. However, this term disappears for the transverse plane waves. The first and final terms on the right hand side of Eq. (19) describe the effect of the inhomogeneity of the medium, i.e., of the spatial dependence of the permittivity and permeability. The first term vanishes for the transverse plane wave. The final term in Eq. (19) is the same as the final term in Eq. (20) but opposite to it in sign. The first term on the right hand side of Eq. (20) is the surface integral of the magnetic helicity. The second term in Eq. (20) corresponds to the negative of the surface integral of the first term in Eq. (18) when δ is infinite. However, this term vanishes when a medium has a rotational invariance around the z axis, as described above. After all, for the transverse plane wave traveling along the z axis in an optically active medium, all the terms except the terms associated with the inner product of A and H vanish owing to the rotational invariance and the mutual cancelation, so that ∂J/∂t is expressed simply as ∂ S + SΣ ∂J ∂S = + ∫Σ dsðA·H Þ: = ∂t ∂t ∂t
ð21Þ
Since L is equal to zero for the transverse plane wave, J (=S + SΣ) is expressed for 0 b γ ≤ 1 as
The torque of ∂J/∂t will be connected with the Faraday rotation angle in the following subsection.
J = S f1–exp½−2að1–γÞ = δg;
ð17Þ
2.3. The torque equation expressed as a function of transverse extent δ
where γ = b/a. This expression gives a sufficient explanation for the presence and absence of AM which are dependent on the magnitude of δ.
As shown in Fig. 2, when the boundary of an optically active (cylindrical) medium overlaps partially with the edge of beam, the surface integral of the inner product of optical fields is given by the
Intensity of optical fields
O. Yamashita / Optics Communications 284 (2011) 4248–4253
4251
Faraday
exp [−(r−b)/δ] for r>b 1
ϕ ϕr
exp [−(a−b)/δ]
=L
x 0
rotation
θF=(ϕr–ϕ )/2
ba
r
Propagation axis Fig. 2. Intensity of optical fields drawn as a function of r in the inside and outside of the beam core, where r is the radial distance from the center axis of beam, a and b are radiuses of a cylindrical medium and a beam core, respectively.
inner product at the circumference of a cylindrical medium. Therefore, the surface integral of (A·H) at r = a (≥b) can be expressed by the volume integral in the same way as Eq. (15) as ∫Σ dsðA·H Þ = ∫τ ∇ðA·H Þδðr–aÞdv;
∂S exp½−2að1–γÞ = δ; ∂t
ð23Þ
where γ = b/a. This representation is convenient and useful to express approximately the surface integral using the volume integral when a plane wave has finite δ, as will be shown later. Substituting Eq. (23) into Eq. (21), the resultant torque of ∂J/∂t is expressed for 0 b γ ≤ 1 as ∂J ∂S = f1– exp½−2að1–γÞ = δÞg: ∂t ∂t
=0
Fig. 3. Schematic diagram of Faraday rotation in an optically active medium along the z axis of propagation. When φr = φ‘, no Faraday rotation occurs, while when φr ≠ φ‘, the resultant field undergoes a net Faraday rotation θF.
ð22Þ
where ∇(A·H) is a product of a factor of exp [−2(r–b)/δ)] and −∂S/∂t. When a plane wave has finite δ, therefore, Eq. (22) is rewritten for a ≥ b as ∫Σ dsðA⋅H Þ = −
y
ð24Þ
This is just the time derivative of Eq. (17). It indicates that ∂J/∂t arises from the difference between the volume integral (within a medium) and the surface integral (at the circumference of a cylindrical medium) of (A∙H) whose gradient with respect to z gives the magnetic helicity [18]. In order to clarify the relation between ∂J/∂t and γ, let us consider a linearly polarized plane wave of infinitesimal, finite and infinite extents (δ) passing along the z axis through an optically active (cylindrical) medium. When 0 b γ ≤ 1, ∂SΣ/∂t vanishes and only the volume integral of ∂S/∂t survives for δ ≈ 0, while for δ ≈ ∞, ∂SΣ/∂t and ∂S/∂t cancel out, resulting in ∂J/∂t =0. When δ has an intermediate value between infinitesimal and infinite extents, however, ∂SΣ/∂t becomes almost equal to −∂S/∂t near γ = 1, so that ∂J/∂t decreases significantly near γ = 1. It is thus found that J and ∂J/∂t are dominated by the optical fields at the circumference of a cylindrical medium.
Faraday effect. It is in general a magneto-optical effect due to the interaction between light and magnetic field in a medium [19,20]. The rotation angle of the plane of polarization is proportional to the intensity of magnetic field along the progress direction of light. In physics, the Faraday effect is a result of ferromagnetic resonance when the permittivity of a material is represented by a tensor with the offdiagonal elements, such as a tensor for an optically active medium. The Faraday rotation is connected with ∂J/∂t or ∂S/∂t for the transverse plane waves, because it is generally caused by a torque exerting on the AM in a medium. When b b a in Fig. 2, the torque equation of ∂S/∂t is expressed for a cylindrical medium of length L along the z axis as (see Appendix) ∂Sz E H 2 = −∫τ dv∇z ðA·H Þ = 0 0 sin½ðkr −k‘ ÞLπb ; ω ∂t
ð25Þ
where ∇z is the scalar differential operator with respect to z, and kr and k‘ are the wave vectors for the right and left circularly polarized waves, respectively, which are represented by kr = ω(εr μ r) 1/2 and k‘ = ω(ε‘ μ ‘) 1/2. When two circularly polarized waves with wavelength λ propagate from z = 0 to L in such a medium, the phase difference θ between them is derived from Eq. (25) as θ = (kr − k ‘) L = 2πL(nr − n‘)/λ (see Appendix), where c is the velocity of light in a medium, and nr [=c(εr μ r) 1/2] and n‘ [=c(ε‘ μ ‘) 1/2] are the refractive indices for the right and left circularly polarized waves, respectively. The Faraday rotation angle θF is equal to one-half of the phase angle difference θ, so that it is expressed as [21] θF =
πLðnr –n‘ Þ : λ
ð26Þ
3. Results and discussion 3.1. The relationship between the torque equation and Faraday rotation angle As shown in Fig. 1(c), when the linearly polarized monochromatic light propagating along the z axis entered an optically active medium, it splits to the right and left circularly polarized waves in a medium, resulting in a phase difference θ between the two (right and left) circularly polarized waves. After coming out from the crystal, these waves will be recombined to a linearly polarized light with the plane of polarization rotated by θF from the plane of polarization of an incident polarized one, as shown in Fig. 3. This phenomenon is referred to as the
This is the well-known formula of the Faraday rotation angle obtained from the phenomenological theory [19]. 3.2. γ-dependence of Faraday rotation angle for plane waves of finite, intermediate and infinite δ When two circularly polarized plane waves of finite δ are traveling along the z axis in an optically active medium, ∂Jz/∂t is rewritten for 0 b γ ≤ 1 by substituting Eq. (25) into Eq. (24) as ∂Jz E H 2 = 0 0 f1– exp½−2að1–γÞ = δgsinð2θF Þπb ; ω ∂t
ð27Þ
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O. Yamashita / Optics Communications 284 (2011) 4248–4253
where θF is the intrinsic Faraday rotation angle of a medium. If L is adjusted so that θF becomes smaller than 0.2 rad, Eq. (27) is approximated as ∂Jz E0 H0 E H 2 2 2θF f1–exp½−2að1–γÞ = δgπb = 0 0 2ΘF πb ; ≈ ω ω ∂t
ð28Þ
because of sin (2θF) ≈2θF, where ΘF = θF{1 − exp[−2a(1− γ)/δ]}. This ΘF is the resultant Faraday rotation angle dependent on γ. It indicates that it is possible to estimate δ experimentally by measuring ΘF/θF as a function of γ near γ = 1, because ΘF/θF varies drastically with only a little change in γ. Here we propose a new experimental procedure for observing the transverse extent δ of a realistic light by measuring the Faraday rotation angle. Let us consider a monochromatic light beam of radius b with wavelength λ which propagates in a cylindrical (optically active) medium of radius a. The relative Faraday rotation angle ΘF/θF was plotted as a function of γ (=b/a) for various values of δ in Fig. 4, to clarify the γ-dependence of ΘF/θF. When δ has infinite extent, ΘF/θF is equal to 0 for γ ≤ 1, as denoted by a thick solid line in the figure. When δ has infinitesimal extent, however, ΘF/θF is equal to 1, except γ = 1, while at γ = 1, it takes a value of 0, as denoted by a dashed line. If δ has intermediate extent, the smaller δ, the steeper the slope of ΘF/θF near γ = 1, as denoted by thin curves in the figure. The γ-dependence of ΘF/θF for a realistic light would show an abrupt decrease near γ = 1, owing to the abrupt increase in the negative of the contribution from the surface integral to the volume integral in J/dt. Of course, ΘF/θF has no γdependence in the conventional measuring system in which γ is usually smaller than 0.5. In a realistic light, the optical field is expected to vanish quickly at infinite in the transverse x, y-plane [22]. In addition δ is also guessed to be not more than 100 Å in a medium from the experimental result that the photoluminescence is emitted from a 5 nm thick GaAs active layer surrounded by two confinement layers of Al 0.3Ga 0.7As with a thickness of 30 nm [23]. δ may possibly be equivalent to the minimum radius of a waveguide surrounded by the confinement layers through which a plane wave of λ can pass. δ might also be connected closely with λ; for example, δ is proportional to λ. In any case we have a strong interest in how much value δ has and whether δ varies with changes in λ. Since the presence and absence of the surface integrals in J are physically the same as those in dJ/dt, the measurement of the γ-dependence of ΘF/θF should result in the resolution of the paradox. In estimating δ experimentally for a realistic light, ΘF needs to measure precisely and carefully as a function of γ, particularly near γ=1, by changing the radius b of light beam without dispersing, i.e., by constricting and expanding an aperture thought which a well collimated light passes, as shown in Fig. 1(c). In addition, the experiment system
should be adjusted so that a well collimated plane wave is never disturbed and dispersed, because only a slight disturbance of the plane wave has a serious influence on the γ-dependence of ΘF, particularly near γ= 1. The relative Faraday rotation angle ΘF/θF is obtained by dividing ΘF by the intrinsic Faraday rotation angle θF of a medium. In any case, the experimental values of ΘF/θF would probably fall on the calculated curve as a function of γ from the relation ΘF/θF ={1− exp[−2a(1− γ)/δ]} when the experiment was performed successfully and δ was optimized. When it was carried out successfully, the transverse extent of a realistic light is first revealed and the paradox can be resolved. 4. Conclusions The presence and absence of the angular momentum for a circularly polarized plane wave are related with the absence and presence of the surface integral in J, respectively. The same also holds for the torque equation of dJ/dt. The absence and presence of the surface integrals in dJ/dt are thus connected with the presence and absence of the Faraday rotation in an optically active medium, respectively. When a plane wave has infinite extent, J and dJ/dt vanish for 0 b γ ≤ 1. When it has infinitesimal extent, however, both J and dJ/dt survive, except γ = 1. For 0 b γ ≤ 1, the torque of dJ/dt thus leads directly to the Faraday rotation; the Faraday rotation is not induced for a linearly polarized incident plane wave of infinite extent, while it occurs for that of finite or infinitesimal extent. When a plane wave has small extent, therefore, the relative Faraday rotation angle ΘF/θF shows an abrupt decrease near γ = 1. It indicates that it is possible to estimate δ experimentally for a realistic light from the drastic variation in ΘF/θF near γ = 1. When the experiment was done well, therefore, the transverse extent of a realistic light is first revealed and the paradox would be resolved and settled. Finally, we hope that the experiment proposed here will be carried out in the near future. Appendix A It is desirable to choose a representation that most conveniently expresses the relationship between the Faraday effect and the ε and μ tensors. Here we consider the incident plane wave to be a superposition of equal amplitude right and left circularly polarized waves. The complex ε and μ tensors for an optically active medium with threefold or higher rotational symmetry about the z axis has the following form [20] 0
1 ε0 −iε1 0 ½ε = @+ iε1 ε0 0 A 0 0 εz and
: Infinitesimal extent : Intermediate extent : Infinite extent δ
0
( (
μ −iκ ½μ = @+iκ μ 0 0
1
ðA:1Þ
Θ /θ F F
where the symbols ε0, ε1, μ and κ are adopted to conform to accepted conventions. When the linearly polarized incident wave propagating along the z axis entered such a medium, it splits to the right and left circularly polarized waves in a medium. Of course it never splits in a medium without the off-diagonal elements of the ε and μ tensors. The E and H fields for two monochromatic plane waves propagating along the z axis can be expressed as
0.5
0
1 0 0 A μz
0.98
γ (=b/a)
1
iðωt−krzÞ
E = ðE0 = 2Þ½ðe
iðωt−krzÞ
Fig. 4. The relative Faraday rotation angle ΘF/θF calculated as a function of γ (= b/a) for various values of δ, where ΘF = θF{1 − exp[− 2a(1 − γ)/δ)]} and θF is the intrinsic Faraday rotation angle of a medium. The symbols ○ and ● show that the dashed line takes a value of 0 at γ = 1.
H = ðH0 = 2Þ½ðe
iðωt−k‘zÞ
+e
iðωt−k‘zÞ
−e
iðωt−krzÞ
Þe1 −ðe
iðωt−krzÞ
Þe1 +ðe
iðωt−k‘zÞ
−e
Þe2 ;
iðωt−k‘zÞ
+e
Þe2 ;
ðA:2Þ ðA:3Þ
where E0 and H0 are the amplitude of electric and magnetic fields which are related to each other by the relations H0/E0 = (ε‘/μ ‘)1/2 = (εr/μ r)1/2
O. Yamashita / Optics Communications 284 (2011) 4248–4253
and the subscripts “r” and “‘” refer to the right and left circularly polarized waves. The right and left circularly polarized waves describe a circle in the x-y plane that is traced out clockwise and counterclockwise, respectively, when the plane is viewed from the negative z direction, as shown in Fig. 3. In a medium with the rotational symmetry around the optical axis, the electric D and magnetic B induction fields are expressed using the ε and μ tensors of Eq. (A.1) as D = Σj = x ; y εx j Ej e1 + εyj Ej e2 ;
ðA:4Þ
B = Σj = x ; y μx j Hj e1 + μ yj Hj e2 :
ðA:5Þ
The products of ε and μ tensors for the right and left circularly polarized waves are obtained as the eigenvalues of the [ε]·[μ] tensor, as follows ½ε · ½μ =
εr μr 0 ; 0 ε‘ μ ‘
ðA:6Þ
where εr = (ε0 + ε1), ε‘ = (ε0 − ε1), μ r = (μ + κ) and μ ‘ = (μ − κ), and εzμz is dropped because it is presumed that there is no active signal field component in the z direction [20]. The vector potential A is obtained by the time integration of E after substituting Eq. (A.2) into Eq. (9) as
½
A = iE0 = ð2ωÞ
h i iðωt−krzÞ iðωt−k‘zÞ iðωt−krzÞ iðωt−k‘zÞ e1 − e e2 ; e +e −e
ðA:7Þ
where kr = ω(εrμ r) 1/2 and k‘ = ω(ε‘ μ ‘) 1/2. Substituting Eqs. (A.3) and (A.7) into the second term in Eq. (18), ∂Sz/∂t is obtained for a cylindrical medium with radius a and length L along the z axis as ∂Sz E H 2 = −∫τ dv∇z ðA ·H Þ = 0 0 sin½ðkr −k‘ ÞLπb ; ω ∂t
ðA:8Þ
4253
where the radius b of beam is assumed to be equal to or less than a. Here the inner product of vectors was calculated using the conjugate complex to obtain their real parts.
References [1] R.A. Beth, Phys. Rev. 50 (1936) 115. [2] E. Santamato, B. Daino, M. Romagnoli, M. Settemlre, Y.R. Shen, Phys. Rev. Lett. 57 (1986) 2423. [3] H. He, M.E.J. Friese, N.R. Heckenberg, H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75 (1995) 826. [4] M.E.J. Friese, J. Enger, H. Rubinsztein-Dunlop, N.R. Heckenberg, Phys. Rev. A 54 (1996) 593. [5] S.J. van Enk, G. Nienhuis, Opt. Commun. 94 (1992) 147. [6] S.J. van Enk, G. Nienhuis, Opt. Commun. 112 (1994) 225. [7] S.J. van Enk, G. Nienhuis, J. Mod. Opt. 41 (1994) 963. [8] S.M. Barnett, L. Allen, Opt. Commun. 110 (1994) 670. [9] W. Heitler, The Quantum Theory of Radiation, Clarendon, Oxford, 1954. [10] R.I. Khrapko, Am. J. Phys. 69 (2001) 405. [11] V.B. Yurchenko, Am. J. Phys. 70 (2002) 568. [12] L. Allen, M.J. Padgett, Am. J. Phys. 70 (2002) 567. [13] H.C. Ohanian, Am. J. Phys. 54 (1986) 500. [14] J.W. Simmons, M.J. Guttman, Waves and Photons: A Modern Introduction to Light, Addison-Wesley, Reading, MA, 1970. [15] A.M. Stewart, Eur. J. Phys. 26 (2005) 635. [16] J.D. Jackson, Classical Electrodynamics, John Wiley and Sons, New York, 1975. [17] A. Ciattoni, G. Cincotti, C. Palma, Phys. Rev. E 67 (2003) 036618-1-10. [18] P.K. Kotiuga, J. Appl. Phys. 63 (1988) 3360. [19] H.S. Bennett, E.A. Stern, Phys. Rev. 137 (1965) A448. [20] G.E. Dionne, G.A. Allen, P.R. Haddad, C.A. Ross, B. Lax, Lincoln Lab. J. 15 (2005) 323. [21] S. Wittekock, T.J.A. Popma, J.M. Robertson, P.F. Bongers, Phys. Rev. B 12 (1975) 2777. [22] E. Santamato, Fortschr. Phys. 52 (2004) 1141. [23] J.S. Kim, M. Kawabe, N. Koguchi, J. J. Appl. Phys. 43 (2004) L103.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Effect of the surface integral in the torque equation of the electromagnetic angular momentum on the Faraday rotation Osamu Yamashita Materials Science, Co., Ltd., 5-5-44, Minamikasugaoka, Ibaraki, Osaka 567-0046, Japan
a r t i c l e
i n f o
Article history: Received 3 January 2011 Received in revised form 29 April 2011 Accepted 13 May 2011 Available online 27 May 2011 Keywords: Electromagnetic angular momentum Torque equation Faraday rotation Surface integral Volume integral
a b s t r a c t A circularly polarized plane wave of infinite transverse extent (δ = ∞) has no spin angular momentum, while a realistic light does carry it. This paradox originates from the presence (δ = ∞) and absence (δ ≈ 0) of the surface integral in the total angular momentum J. The same holds for the torque equation of d J/dt, so that δ is also connected with the relative Faraday rotation angle ΘF/θF when a radius (a) of a cylindrical medium with optical activity is only a little larger than that (b) of light beam, where ΘF is the Faraday rotation angle and θF is the intrinsic Faraday rotation angle of a medium. It is shown here that it is possible to estimate δ for a realistic light from the drastic variation in ΘF/θF near b/a = 1. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The angular momentum (AM) carried by light can be characterized by the spin AM associated with circular polarization and the orbital AM associated with the spatial distribution of the wave. Both spin and orbital angular momenta of light beam have in fact been measured [1–4]. Theoretically, the spin and orbital angular momenta (S and L) of the radiation fields have been defined explicitly for free space and an isotropic medium [5–8]. The spin AM arises even from the transverse plane electromagnetic waves, but the orbital AM never appears in the plane electromagnetic waves. As is generally known, however, a circularly polarized plane wave of infinite transverse extent can have no spin AM [9]. However, only a quasiplane wave of finite transverse extent δ carries the spin AM whose direction is along the direction of propagation. As evident from the definition of the AM, the component of the AM in the direction of propagation must be zero, but it is non-zero actually. This paradox has been subject of discussion for a long time [9] and even recently [10–12]. Some ideas [8,10,13–15] were proposed to resolve this paradox, but it has not yet been settled. At present, one justifies these results by taking into account the fact that a detector placed in a plane wave causes gradients in this field [14]. The field can no longer be considered as a plane wave. In other words, any obstacle that absorbs the beam changes the electromagnetic field at the edges of the obstacle so that the field components in the direction of propagation are produced. Recently, another thought different from this was proposed by Stewart [15]. He
E-mail address: [email protected]. 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.05.031
took into account the effect of boundaries on the plane wave problem by decomposing the AM into three items of two volume integrals of a spin character and an orbital character and one surface integral and concluded that the contribution to the AM arises from the edges of the beam. This idea seems to be reasonable and consistent theoretically. In his paper [15], however, the contribution from the surface integral to the AM has not been expressed as a function of δ, where δ is the transverse extent of the plane wave. For this reason, it is shown here that the contribution from the surface integral to the AM of the plane wave can be expressed analytically as a function of δ, on the assumption that the intensity profile of the optical fields forms a flat plateau for r ≤ b and decreases according to exp [−(r − b)/δ] for r N b, where r is the radial distance from the center axis of beam and b is a radius of beam. The same is also applicable to the torque equation of AM. This torque equation links directly to the relative Faraday rotation angle (ΘF/θF) expressed as a function of δ, where ΘF is the Faraday rotation angle and θF is the intrinsic Faraday rotation angle of a medium. Of course, the transverse extent δ has not yet been measured for a realistic light, because there was no experimental procedure for measuring it. For this reason, we show here that it is possible theoretically to estimate δ from the measurement of the Faraday rotation angle when a is only a little larger than b, where a is a radius of an optically cylindrical medium. When this experiment was carried out successfully, the transverse extent of a realistic light is revealed so that the above paradox is resolved explicitly. The purpose of this study is to clarify the relation between the surface integrals in the torque equation for AM and the transverse extent δ or the relative Faraday rotation angle ΘF/θF, and to provide a new experimental procedure for measuring the transverse extent of a realistic light, resulting in the resolution of the traditional paradox.
O. Yamashita / Optics Communications 284 (2011) 4248–4253
current are absent [6,7]. In the radiation gage, therefore, the electric field E⍊ and magnetic induction field B are expressed by the vector potential A⍊ as [7]
2. Analysis 2.1. The definition of angular momentum for electromagnetic fields The Maxwell equations for the macroscopic electromagnetic fields in a medium are given by [16] ∇×E=−
∂B ; ∂t
ð1Þ
∂D ∇×H= + I; ∂t
ð2Þ
∇· D = ρ;
ð3Þ
∇· B = 0:
ð4Þ
It is assumed here that the medium is inhomogeneous and anisotropic but is free of dispersion. In these equations, the electric D and magnetic B induction fields are related to the field strengths E and H as Di = x, y, z = Σj = x, y, z εijEj and Bi = x, y, z = Σj = x, y, z μ ijHj where εij and μij represent the permittivity and permeability tensors, I is the electric current density, and ρ is the charge density. Let us consider an arbitrary volume τ inside the crystal filled by charges, described by the volume density ρ and current density I. Because of the interaction with the electromagnetic field, the charges experience a total mechanical torque dLc/dt given by [17] dLc = ∫τ dvr × ½ρE + ðI × BÞ; dt
B = ∇ × A⊥ ;
ð10Þ
and
where E⍊ and A⍊ are the transverse components of E and A. In other words, this is the same as the Coulomb gage for φ = 0. The transverse part A⊥ of the vector potential is thus gage invariant. We will hereafter treat D, E and A as D⍊, E⍊ and A⍊, respectively. By substituting Eq. (10) into Eq. (7) and applying partial integration, the angular momentum J [5–7] defined previously for the transverse electromagnetic field is separated into three parts as J = ∫τ ðD × AÞdv + ∫τ Σi = x; y; z Di ðr × ∇ÞAi dv
ð11Þ
+ ∫Σ ðA × r ÞðD·dsÞ = S + L + SΣ ; where the symbol ∇ is the gradient operator, and S, L and SΣ represent the volume integrals with spin and orbital characters and the surface integral, respectively, which are as follows, S = ∫τ ðD × AÞdv;
ð12Þ
L = ∫τ Σi = x; y; z Di ðr × ∇ÞAi dv
ð13Þ
and S Σ = ∫Σ ðA × rÞðD·dsÞ:
ð7Þ
in analogy to the interpretation of the linear momentum density (D × B) in a medium [6,16]. This definition is valid at least when the medium is linear, but not necessarily isotropic, in its response [16]. However, we do not enter here into the well-known question of the correct definition of momentum in media. 2.2. The angular momenta of radiation fields and their torque equations The electromagnetic fields can be separated into transverse and longitudinal fields, which have by definition a vanishing divergence and curl, respectively. The magnetic field is purely transverse, while the longitudinal electric induction field D∥ is given by the instantaneous Coulomb field arising from the charge density ρ, i.e., ∇·D∥ = ρ. The transverse electric induction field D⊥ thus describes the radiation part, which contains in fact the only real dynamical degrees of freedom of the field. The symbols || and ⊥ denote the components parallel and perpendicular to the optical axis, respectively. For D⊥, therefore, Eq. (3) may be rewritten as ∇· D⊥ = 0:
ð9Þ
ð14Þ
ð6Þ
where the angular momentum J of electromagnetic fields in a medium is defined as J = ∫τ dv½r × ðD × BÞ
∂A⊥ ∂t
E⊥ = −
ð5Þ
where Lc is the mechanical angular momentum of the (charged and neutral) particles. Substituting I and ρ from Eqs. (2) and (3) into Eq. (5), we get, after some manipulations, d ðL + J Þ = ∫τ dvfr × ½Eð∇ · DÞ–D × ð∇ × EÞ–B × ð∇ × H Þg; dt c
4249
ð8Þ
Generally, the radiation gage is defined as φ = 0 and ∇·A = 0 in the gage transformation, where φ is the scalar potential and A is the vector potential[16]. This gage is applicable to the transverse part of the electromagnetic fields and is often used when the charge and
When the radiation gage is employed, therefore, the angular momentum of J is gage independent. In this sense, the separation of S and L has a clear physical meaning [6,7], but we recognize that the identification of terms as spin and orbital momenta may not be unique in general. However, the orbital AM is ignored in this and subsequent subsections because we treat only the plane electromagnetic wave. When the electromagnetic field in an isotropic medium is composed of the plane wave of infinite extent, the surface integral of SΣ can be transformed to the volume integral of −∫τ (D × A)dv, so that S and SΣ cancel out, resulting in J = 0 [9]. When the electromagnetic field has infinitesimal extent, however, the surface integral of SΣ vanishes and only the volume integral of S survives, resulting in J = S. The surface integral thus varies significantly with changes in the magnitude of the transverse extent of the plane wave. Let us consider the plane waves of finite extent δ which form a light beam of radius b propagating along the z axis in a cylindrical medium of radius a, as shown in Fig. 1. Since it is difficult to calculate exactly the intensity profile of the optical fields, for simplicity, it is assumed here that the intensity profile of the optical fields forms a flat plateau for r ≤ b, while for r ≥ b, it is expressed by a function of exp [−(r–b)/δ], as shown in Fig. 1(b), where r is the radial distance from the center axis of beam and (r–b) is the distance from the edge of beam. It means that the intensity of the optical fields in the outside of beam core decreases exponentially and concentrically with an increase of (r–b). When the intensity of the optical fields at the distance δ from the beam edge is assumed to lower down to 1/e of that at the beam edge, δ represents the average transverse extent of the optical fields protruded slightly from the core edge, as evident from the integral value of ∫0∞e −r/δdr = δ. In practice, the intensity of the optical fields (including the vector potential A) would probably decrease rapidly near the edge of beam in a complicated manner. However, even this simple intensity profile is
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(a)
Next, the torque equations for S, L and SΣ are obtained by applying partial integration to the time derivatives of Eqs. (12)–(14) as
Cylindrical medium
∂S ∂ = ∫ dvðD × AÞ ∂t ∂t τ = ∫τ dv½ðE × DÞ+ ðH × BÞ–∫τ dv∇ðA·H Þ+∫τ dv½ðH·∇ÞA + ðA·∇ÞH ;
a
ð18Þ ∂L ∂ = ∫ dvΣDi ðr × ∇ÞAi ∂t ∂t τ
L
= –∫τ dvΣ½Bi ðr × ∇ÞHi –Di ðr × ∇ÞEi –∫τ dv½ðH·∇ÞA + ðA·∇ÞH
(b)
Intensity distribution
+ ∫Σ ½ðH · BÞðr × dsÞ + H ðA·dsÞ + ΣHi ðr × ∇ÞdF ð19Þ
1
and ∂S Σ = ∫Σ fðr × EÞðD·dsÞ + ðr × AÞ½∇· ðds × H Þg ∂t
exp[−(r−b)/δ] for r>b
b
b
0
= ∫Σ ðA ·H Þds + ∫Σ ½ðr × EÞðD·dsÞ + ðr × H ÞðB·dsÞ
r
–∫Σ ½ðH ·BÞðr × dsÞ + H ðA· dsÞ + ΣHi ðr × ∇ÞdF ;
(c) Beam
0
Cylindrical medium
a
b
z L
Fig. 1. (a) A cylindrical and optically active medium of a radius of a and a length of L. (b) Intensity profile of a plane wave that has passed through an aperture. The intensity of the beam is assumed to be constant for r≤b and to decrease exponentially with an increase of r for r N b, where r is the radial distance from the center axis of beam, δ is the transverse extent of a plane wave and b is a radius of beam. (c) Schematic diagram of a beam and a cylindrical medium for the measurement of the Faraday rotation angle, where the z axis runs through the center of them.
enough to study roughly the r-dependence of SΣ near the circumference of a cylindrical medium. The SΣ is dominated by the fields at the edge of a medium, as will be shown later. When the boundary of a medium overlaps locally with the edge of beam, therefore, SΣ at the boundary can be expressed in terms of a volume integral as S Σ = −∫τ ðD × AÞδðr–aÞdv;
ð15Þ
where δ(r–a) is the delta function. When the intensity of D and A is proportional to exp [−(r–b)/δ] for r N b, the cross product of (D × A) in Eq. (15) is expressed using an attenuation factor as (D × A) exp [− 2 (r–b)/δ]. Substituting this expression instead of (D × A) into Eq. (15) and integrating, SΣ is rewritten using S and δ as S Σ = −S exp½−2ða–bÞ = δ:
ð20Þ
ð16Þ
where i = x, y and z, dFi is the i-component of dF = (A × ds) and the symbol ∇ is a differential operator acting outside the brackets. These expressions change to some extent depending on the way of partial integration. The total torque equation of ∂J/∂t is derived from the summation of Eqs. (18)–(20). The first term on the right hand side of Eq. (18) arises from the anisotropy of the medium and vanishes when a medium has a rotational invariance around the z axis. The second term of ∇(A·H) represents the magnetic helicity [18] of the electromagnetic field caused by the anisotropy and inhomogeneity. This term is absent when a medium is isotropic and uniform, but it survives in an optically active medium possessing a rotational invariance, because the magnetic helicity term has no rotational invariance. The final term in Eq. (18) originates from both the anisotropy and the inhomogeneity and is the same as the second one in Eq. (19) but is opposite to it in sign. However, this term disappears for the transverse plane waves. The first and final terms on the right hand side of Eq. (19) describe the effect of the inhomogeneity of the medium, i.e., of the spatial dependence of the permittivity and permeability. The first term vanishes for the transverse plane wave. The final term in Eq. (19) is the same as the final term in Eq. (20) but opposite to it in sign. The first term on the right hand side of Eq. (20) is the surface integral of the magnetic helicity. The second term in Eq. (20) corresponds to the negative of the surface integral of the first term in Eq. (18) when δ is infinite. However, this term vanishes when a medium has a rotational invariance around the z axis, as described above. After all, for the transverse plane wave traveling along the z axis in an optically active medium, all the terms except the terms associated with the inner product of A and H vanish owing to the rotational invariance and the mutual cancelation, so that ∂J/∂t is expressed simply as ∂ S + SΣ ∂J ∂S = + ∫Σ dsðA·H Þ: = ∂t ∂t ∂t
ð21Þ
Since L is equal to zero for the transverse plane wave, J (=S + SΣ) is expressed for 0 b γ ≤ 1 as
The torque of ∂J/∂t will be connected with the Faraday rotation angle in the following subsection.
J = S f1–exp½−2að1–γÞ = δg;
ð17Þ
2.3. The torque equation expressed as a function of transverse extent δ
where γ = b/a. This expression gives a sufficient explanation for the presence and absence of AM which are dependent on the magnitude of δ.
As shown in Fig. 2, when the boundary of an optically active (cylindrical) medium overlaps partially with the edge of beam, the surface integral of the inner product of optical fields is given by the
Intensity of optical fields
O. Yamashita / Optics Communications 284 (2011) 4248–4253
4251
Faraday
exp [−(r−b)/δ] for r>b 1
ϕ ϕr
exp [−(a−b)/δ]
=L
x 0
rotation
θF=(ϕr–ϕ )/2
ba
r
Propagation axis Fig. 2. Intensity of optical fields drawn as a function of r in the inside and outside of the beam core, where r is the radial distance from the center axis of beam, a and b are radiuses of a cylindrical medium and a beam core, respectively.
inner product at the circumference of a cylindrical medium. Therefore, the surface integral of (A·H) at r = a (≥b) can be expressed by the volume integral in the same way as Eq. (15) as ∫Σ dsðA·H Þ = ∫τ ∇ðA·H Þδðr–aÞdv;
∂S exp½−2að1–γÞ = δ; ∂t
ð23Þ
where γ = b/a. This representation is convenient and useful to express approximately the surface integral using the volume integral when a plane wave has finite δ, as will be shown later. Substituting Eq. (23) into Eq. (21), the resultant torque of ∂J/∂t is expressed for 0 b γ ≤ 1 as ∂J ∂S = f1– exp½−2að1–γÞ = δÞg: ∂t ∂t
=0
Fig. 3. Schematic diagram of Faraday rotation in an optically active medium along the z axis of propagation. When φr = φ‘, no Faraday rotation occurs, while when φr ≠ φ‘, the resultant field undergoes a net Faraday rotation θF.
ð22Þ
where ∇(A·H) is a product of a factor of exp [−2(r–b)/δ)] and −∂S/∂t. When a plane wave has finite δ, therefore, Eq. (22) is rewritten for a ≥ b as ∫Σ dsðA⋅H Þ = −
y
ð24Þ
This is just the time derivative of Eq. (17). It indicates that ∂J/∂t arises from the difference between the volume integral (within a medium) and the surface integral (at the circumference of a cylindrical medium) of (A∙H) whose gradient with respect to z gives the magnetic helicity [18]. In order to clarify the relation between ∂J/∂t and γ, let us consider a linearly polarized plane wave of infinitesimal, finite and infinite extents (δ) passing along the z axis through an optically active (cylindrical) medium. When 0 b γ ≤ 1, ∂SΣ/∂t vanishes and only the volume integral of ∂S/∂t survives for δ ≈ 0, while for δ ≈ ∞, ∂SΣ/∂t and ∂S/∂t cancel out, resulting in ∂J/∂t =0. When δ has an intermediate value between infinitesimal and infinite extents, however, ∂SΣ/∂t becomes almost equal to −∂S/∂t near γ = 1, so that ∂J/∂t decreases significantly near γ = 1. It is thus found that J and ∂J/∂t are dominated by the optical fields at the circumference of a cylindrical medium.
Faraday effect. It is in general a magneto-optical effect due to the interaction between light and magnetic field in a medium [19,20]. The rotation angle of the plane of polarization is proportional to the intensity of magnetic field along the progress direction of light. In physics, the Faraday effect is a result of ferromagnetic resonance when the permittivity of a material is represented by a tensor with the offdiagonal elements, such as a tensor for an optically active medium. The Faraday rotation is connected with ∂J/∂t or ∂S/∂t for the transverse plane waves, because it is generally caused by a torque exerting on the AM in a medium. When b b a in Fig. 2, the torque equation of ∂S/∂t is expressed for a cylindrical medium of length L along the z axis as (see Appendix) ∂Sz E H 2 = −∫τ dv∇z ðA·H Þ = 0 0 sin½ðkr −k‘ ÞLπb ; ω ∂t
ð25Þ
where ∇z is the scalar differential operator with respect to z, and kr and k‘ are the wave vectors for the right and left circularly polarized waves, respectively, which are represented by kr = ω(εr μ r) 1/2 and k‘ = ω(ε‘ μ ‘) 1/2. When two circularly polarized waves with wavelength λ propagate from z = 0 to L in such a medium, the phase difference θ between them is derived from Eq. (25) as θ = (kr − k ‘) L = 2πL(nr − n‘)/λ (see Appendix), where c is the velocity of light in a medium, and nr [=c(εr μ r) 1/2] and n‘ [=c(ε‘ μ ‘) 1/2] are the refractive indices for the right and left circularly polarized waves, respectively. The Faraday rotation angle θF is equal to one-half of the phase angle difference θ, so that it is expressed as [21] θF =
πLðnr –n‘ Þ : λ
ð26Þ
3. Results and discussion 3.1. The relationship between the torque equation and Faraday rotation angle As shown in Fig. 1(c), when the linearly polarized monochromatic light propagating along the z axis entered an optically active medium, it splits to the right and left circularly polarized waves in a medium, resulting in a phase difference θ between the two (right and left) circularly polarized waves. After coming out from the crystal, these waves will be recombined to a linearly polarized light with the plane of polarization rotated by θF from the plane of polarization of an incident polarized one, as shown in Fig. 3. This phenomenon is referred to as the
This is the well-known formula of the Faraday rotation angle obtained from the phenomenological theory [19]. 3.2. γ-dependence of Faraday rotation angle for plane waves of finite, intermediate and infinite δ When two circularly polarized plane waves of finite δ are traveling along the z axis in an optically active medium, ∂Jz/∂t is rewritten for 0 b γ ≤ 1 by substituting Eq. (25) into Eq. (24) as ∂Jz E H 2 = 0 0 f1– exp½−2að1–γÞ = δgsinð2θF Þπb ; ω ∂t
ð27Þ
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O. Yamashita / Optics Communications 284 (2011) 4248–4253
where θF is the intrinsic Faraday rotation angle of a medium. If L is adjusted so that θF becomes smaller than 0.2 rad, Eq. (27) is approximated as ∂Jz E0 H0 E H 2 2 2θF f1–exp½−2að1–γÞ = δgπb = 0 0 2ΘF πb ; ≈ ω ω ∂t
ð28Þ
because of sin (2θF) ≈2θF, where ΘF = θF{1 − exp[−2a(1− γ)/δ]}. This ΘF is the resultant Faraday rotation angle dependent on γ. It indicates that it is possible to estimate δ experimentally by measuring ΘF/θF as a function of γ near γ = 1, because ΘF/θF varies drastically with only a little change in γ. Here we propose a new experimental procedure for observing the transverse extent δ of a realistic light by measuring the Faraday rotation angle. Let us consider a monochromatic light beam of radius b with wavelength λ which propagates in a cylindrical (optically active) medium of radius a. The relative Faraday rotation angle ΘF/θF was plotted as a function of γ (=b/a) for various values of δ in Fig. 4, to clarify the γ-dependence of ΘF/θF. When δ has infinite extent, ΘF/θF is equal to 0 for γ ≤ 1, as denoted by a thick solid line in the figure. When δ has infinitesimal extent, however, ΘF/θF is equal to 1, except γ = 1, while at γ = 1, it takes a value of 0, as denoted by a dashed line. If δ has intermediate extent, the smaller δ, the steeper the slope of ΘF/θF near γ = 1, as denoted by thin curves in the figure. The γ-dependence of ΘF/θF for a realistic light would show an abrupt decrease near γ = 1, owing to the abrupt increase in the negative of the contribution from the surface integral to the volume integral in J/dt. Of course, ΘF/θF has no γdependence in the conventional measuring system in which γ is usually smaller than 0.5. In a realistic light, the optical field is expected to vanish quickly at infinite in the transverse x, y-plane [22]. In addition δ is also guessed to be not more than 100 Å in a medium from the experimental result that the photoluminescence is emitted from a 5 nm thick GaAs active layer surrounded by two confinement layers of Al 0.3Ga 0.7As with a thickness of 30 nm [23]. δ may possibly be equivalent to the minimum radius of a waveguide surrounded by the confinement layers through which a plane wave of λ can pass. δ might also be connected closely with λ; for example, δ is proportional to λ. In any case we have a strong interest in how much value δ has and whether δ varies with changes in λ. Since the presence and absence of the surface integrals in J are physically the same as those in dJ/dt, the measurement of the γ-dependence of ΘF/θF should result in the resolution of the paradox. In estimating δ experimentally for a realistic light, ΘF needs to measure precisely and carefully as a function of γ, particularly near γ=1, by changing the radius b of light beam without dispersing, i.e., by constricting and expanding an aperture thought which a well collimated light passes, as shown in Fig. 1(c). In addition, the experiment system
should be adjusted so that a well collimated plane wave is never disturbed and dispersed, because only a slight disturbance of the plane wave has a serious influence on the γ-dependence of ΘF, particularly near γ= 1. The relative Faraday rotation angle ΘF/θF is obtained by dividing ΘF by the intrinsic Faraday rotation angle θF of a medium. In any case, the experimental values of ΘF/θF would probably fall on the calculated curve as a function of γ from the relation ΘF/θF ={1− exp[−2a(1− γ)/δ]} when the experiment was performed successfully and δ was optimized. When it was carried out successfully, the transverse extent of a realistic light is first revealed and the paradox can be resolved. 4. Conclusions The presence and absence of the angular momentum for a circularly polarized plane wave are related with the absence and presence of the surface integral in J, respectively. The same also holds for the torque equation of dJ/dt. The absence and presence of the surface integrals in dJ/dt are thus connected with the presence and absence of the Faraday rotation in an optically active medium, respectively. When a plane wave has infinite extent, J and dJ/dt vanish for 0 b γ ≤ 1. When it has infinitesimal extent, however, both J and dJ/dt survive, except γ = 1. For 0 b γ ≤ 1, the torque of dJ/dt thus leads directly to the Faraday rotation; the Faraday rotation is not induced for a linearly polarized incident plane wave of infinite extent, while it occurs for that of finite or infinitesimal extent. When a plane wave has small extent, therefore, the relative Faraday rotation angle ΘF/θF shows an abrupt decrease near γ = 1. It indicates that it is possible to estimate δ experimentally for a realistic light from the drastic variation in ΘF/θF near γ = 1. When the experiment was done well, therefore, the transverse extent of a realistic light is first revealed and the paradox would be resolved and settled. Finally, we hope that the experiment proposed here will be carried out in the near future. Appendix A It is desirable to choose a representation that most conveniently expresses the relationship between the Faraday effect and the ε and μ tensors. Here we consider the incident plane wave to be a superposition of equal amplitude right and left circularly polarized waves. The complex ε and μ tensors for an optically active medium with threefold or higher rotational symmetry about the z axis has the following form [20] 0
1 ε0 −iε1 0 ½ε = @+ iε1 ε0 0 A 0 0 εz and
: Infinitesimal extent : Intermediate extent : Infinite extent δ
0
( (
μ −iκ ½μ = @+iκ μ 0 0
1
ðA:1Þ
Θ /θ F F
where the symbols ε0, ε1, μ and κ are adopted to conform to accepted conventions. When the linearly polarized incident wave propagating along the z axis entered such a medium, it splits to the right and left circularly polarized waves in a medium. Of course it never splits in a medium without the off-diagonal elements of the ε and μ tensors. The E and H fields for two monochromatic plane waves propagating along the z axis can be expressed as
0.5
0
1 0 0 A μz
0.98
γ (=b/a)
1
iðωt−krzÞ
E = ðE0 = 2Þ½ðe
iðωt−krzÞ
Fig. 4. The relative Faraday rotation angle ΘF/θF calculated as a function of γ (= b/a) for various values of δ, where ΘF = θF{1 − exp[− 2a(1 − γ)/δ)]} and θF is the intrinsic Faraday rotation angle of a medium. The symbols ○ and ● show that the dashed line takes a value of 0 at γ = 1.
H = ðH0 = 2Þ½ðe
iðωt−k‘zÞ
+e
iðωt−k‘zÞ
−e
iðωt−krzÞ
Þe1 −ðe
iðωt−krzÞ
Þe1 +ðe
iðωt−k‘zÞ
−e
Þe2 ;
iðωt−k‘zÞ
+e
Þe2 ;
ðA:2Þ ðA:3Þ
where E0 and H0 are the amplitude of electric and magnetic fields which are related to each other by the relations H0/E0 = (ε‘/μ ‘)1/2 = (εr/μ r)1/2
O. Yamashita / Optics Communications 284 (2011) 4248–4253
and the subscripts “r” and “‘” refer to the right and left circularly polarized waves. The right and left circularly polarized waves describe a circle in the x-y plane that is traced out clockwise and counterclockwise, respectively, when the plane is viewed from the negative z direction, as shown in Fig. 3. In a medium with the rotational symmetry around the optical axis, the electric D and magnetic B induction fields are expressed using the ε and μ tensors of Eq. (A.1) as D = Σj = x ; y εx j Ej e1 + εyj Ej e2 ;
ðA:4Þ
B = Σj = x ; y μx j Hj e1 + μ yj Hj e2 :
ðA:5Þ
The products of ε and μ tensors for the right and left circularly polarized waves are obtained as the eigenvalues of the [ε]·[μ] tensor, as follows ½ε · ½μ =
εr μr 0 ; 0 ε‘ μ ‘
ðA:6Þ
where εr = (ε0 + ε1), ε‘ = (ε0 − ε1), μ r = (μ + κ) and μ ‘ = (μ − κ), and εzμz is dropped because it is presumed that there is no active signal field component in the z direction [20]. The vector potential A is obtained by the time integration of E after substituting Eq. (A.2) into Eq. (9) as
½
A = iE0 = ð2ωÞ
h i iðωt−krzÞ iðωt−k‘zÞ iðωt−krzÞ iðωt−k‘zÞ e1 − e e2 ; e +e −e
ðA:7Þ
where kr = ω(εrμ r) 1/2 and k‘ = ω(ε‘ μ ‘) 1/2. Substituting Eqs. (A.3) and (A.7) into the second term in Eq. (18), ∂Sz/∂t is obtained for a cylindrical medium with radius a and length L along the z axis as ∂Sz E H 2 = −∫τ dv∇z ðA ·H Þ = 0 0 sin½ðkr −k‘ ÞLπb ; ω ∂t
ðA:8Þ
4253
where the radius b of beam is assumed to be equal to or less than a. Here the inner product of vectors was calculated using the conjugate complex to obtain their real parts.
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