Optical orbital angular momentum of evanescent Bessel waves Zhenshan Yang∗ Shandong Provincial Key Laboratory of Optical Communication Science and Technology, School of Physical Science and Information Technology, Liaocheng University, Liaocheng, Shandong 252000, China ∗ [email protected]
Abstract: We show that the orbital angular momentum (OAM) of evanescent light is drastically different from that of traveling light. Specifically, the paraxial contribution (typically the most significant part in a traveling wave) to the OAM vanishes in an evanescent Bessel wave when averaged over the azimuthal angle. Moreover, the OAM energy for the evanescent per unit κ2 Bessel field is reduced by a factor of 1 + k2 from the standard result for the corresponding traveling field, where k and κ are the wave number and the evanescent decay rate, respectively. © 2015 Optical Society of America OCIS codes: (240.0240) Optics at surfaces; (260.2110) Electromagnetic optics; (020.1670) Coherent optical effects.
References and links 1. J. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London Ser. A 82, 560–567 (1909). 2. R. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936). 3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). 4. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3, 161–204 (2011). 5. Ed. by J. P. Torres and L. Torner, Twisted Photons: Applications of Light with Orbital Angular Momentum, (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011). 6. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997). 7. A. Lehmuskero, Y. Li, P. Johansson, and M. Kall, “Plasmonic particles set into fast orbital motion by an optical vortex beam,” Opt. Express 22, 4349–4356 (2014). 8. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabitscale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013). 9. M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commmun. 5, 3115 (2014). 10. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). 11. M. Agnew, J. Z. Salvail, J. Leach, and R. W. Boyd, “Generation of orbital angular momentum Bell states and their verification via accessible nonlinear witnesses,” Phys. Rev. Lett. 111, 030402 (2013). 12. V. Garces-Chavez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 031106 (2005). 13. L. C. Thomson, G. Whyte, M. Mazilu, and J. Courtial, “Simulated holographic three-dimensional intensity shaping of evanescent-wave fields,” J. Opt. Soc. Am. B 25, 849–853 (2008). 14. G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009). 15. D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4, 211–217 (2010).
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16. V. E. Lembessis, M. Babiker, and D. L. Andrews, “Surface optical vortices,” Phys. Rev. A 79, 011806 (2009). 17. V. E. Lembessis, S. Al-Awfi, M. Babiker, and D. L. Andrews, “Surface plasmon optical vortices and their influence on atoms,” J. Opt. 13, 064002 (2011). 18. S. Ruschin and A. Leizer, “Evanescent Bessel beams,” J. Opt. Soc. Am. A 15, 1139–1143 (1998). 19. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006). 20. S. N. Kurilkina, V. N. Belyi, and N. S. Kazak, “Features of evanescent Bessel light beams formed in structures containing a dielectric layer,” Opt. Commun. 238, 3860–3868 (2010). 21. L. W. Davis, “Theory of electromagnetics beams,” Phys. Rev. A 19, 1177–1179 (1979). 22. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014). 23. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
1.
Introduction
It has long been recognized, dating back to Poynting [1] and Beth [2], that circularly polarized light carries spin angular mometum (SAM) associated with the spin of photons. Much later, in 1992 Allen et al. [3] made the breakthrough observation that Laguerre-Gaussian (LG) modes, and more generally light beams with an azimuthal phase dependence of exp (imφ ) [m = 0, ±1, ...], possess orbital angular momentum (OAM) in the overall propagating direction (z-direction). Following Allen’s discovery, the OAM of light has rapidly developed into an exciting research area [4,5]. Besides bringing new insight into our fundamental understanding of light, the optical OAM has found potential applications in optical tweezers and spanners [6,7], optical communications [8,9], and quantum information processing [10,11]. Since Allen’s poineering work, most studies in this area have focused on traveling beams, and much less attention has been paid to the OAM of evanescent waves. Evanescent optical fields can be employed to actively manipulate small particles near a surface [12,13] or to provide optomechanical coupling between light and mechanical objects via gradient forces [14,15], and the added dimension of OAM is expected to introduce new aspects to these applications. While evanescent light with azimuthal phase dependence has been investigated in the context of surface optical vortex [16,17], the focus was mainly on its phase and intensity properties. A quantitative study on the evanescent optical OAM is still lacking. In this paper, we present a detailed analysis of the optical OAM in evanescent Bessel fields, which have transverse (x-y plane) spatial profiles identical to those of traveling Bessel beams, but decay exponentially in the z-direction [18-20]. We derive an analytic formula for the OAM density of the evanescent Bessel wave, and show that the paraxial part of the OAM vanishes when integrated over the azimuthal angle φ , which is in stark contrast to the traveling-wave case. Even more remarkably, for an evanescent mth order Bessel field, the OAM per unit en 2 −1 m κ2 from the standard result ωm for the ergy is 1 + κk2 ω , reduced by a factor of 1 + k2 corresponding traveling beam (here κ is the amplitude decay rate of the evanescent light, and ω , k are the angular frequency and the wave number, respectively). 2.
Linear momentum of evanecsent fields
It is well known that an evanescent field does not carry linear momentum in the z-direction (along which the field decays). However, if the azimuthal component of the linear momentum exists, then the field possesses angular momentum in the z-direction. From now on, unless explicitly stated otherwise, “angular momentum” refers to its z-component. In this section, we derive a general formula for the linear momentum of evanescent light, which will not only suggest some remarkable difference between the angular momenta of the evenescent and the traveling beams, but also serve as a basis for further study on the OAM of evanescent Bessel #233857 - $15.00 USD © 2015 OSA
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12701
waves in the next section. We choose to work in the Lorentz gauge and consider a monochromatic field polarized in the x-y plane [21] A (r,t) = A (r) e−iω t = A (r) (α ex + β ey ) e−iω t , |α |2 + |β |2 = 1,
(1)
where A is the vector potential, ex and ey are the unit vectors in the x and y directions, respectively, and ω is the optical angular frequency. A (r) satisfies the Helmholtz equation ∇2 A (r) + k2 A (r) = 0, k =
ω , c
(2)
with c being the speed of light in vacuum. We emphasize that it is consistent to assume that the vector potential is polarized as in Eq. (1) even if the light field is non-paraxial, which can be seen as follows. The Lorentz gauge ∇ · A (r) − icω2 ϕ (r) = 0 gives the scalar potential as
ϕ (r) =
c2 ∇ · A (r) . iω
With A (r) = A (r) (α ex + β ey ) and A (r) being a solution of Eq. (2), it is easy to show that both the vector and the scalar potentials obey the wave equations (i.e., the Maxwell’s equations in the Lorentz gauge) ∇2 A (r) + k2 A (r) = 0, ∇2 φ (r) + k2 φ (r) = 0. Thus the assumption on the polarization of the vector potential in Eq. (1) is entirely consistent. For an evanescent wave, we take A (r) = u (x, y) e−κ z ,
(3)
where the detailed form of u (x, y) is intentionally left unspecified, so that we can first achieve a relatively general formula for the linear momentum. According to Maxwell’s equations, the magnetic field B and the electric field E are given by B (r) = ∇ × A (r) , E (r) = i Substituting Eq. (3) into Eq. (4) yields B = e−κ z β κ u,
c2 ∇ × B (r) . ω
−ακ u, β ∂∂ ux − α ∂∂ uy
(4)
(5)
and E=
i ω −κ z ∂ 2 u 2 β ∂ x∂ y − α ∂∂ yu2 − ακ 2 u, e k2
α ∂∂x∂uy − β ∂∂ xu2 − β κ 2 u, −ακ ∂∂ ux − β κ ∂∂ uy 2
2
The linear momentum density of the field is determined by −iω t Ee + E∗ eiω t Be−iω t + B∗ eiω t ε0 × = [E × B∗ + c.c.] , p = ε0 2 2 4
. (6)
(7)
where ε0 is the electric permittivity, stands for the time-average over one optical oscillation period, and “c.c.” represents the complex conjugate. By inserting Eqs. (5) and (6) into Eq. (7)
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Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12702
and after a fairly lengthy calculation, we obtain ⎧ (−) (↔) ⎪ κ 2 |α |2 − |β |2 u∇⊥ u∗ − c.c + 2κ 2 Re (αβ ∗ ) u∇⊥ u∗ − c.c ⎪ ⎪ iε0 ω e−2κ z ⎨ 2 ∂u 2 ∂u ∂ u∗ ∂ u∗ | | | | α ∇ − c.c + β ∇ − c.c p = + ⊥ ∂y ∂ x ⊥∂ x ⎪ 4k2
∂ y ⎪ ⎪ ∂ u∗ ∂u ∂ u∗ ∗ ∗ ⎩ + αβ ∂ x ∇⊥ ∂ y + α β ∂ y ∇⊥ ∂∂ ux − c.c ⎡ ⎤ 2 ∂ ∗ ∂ u − c.c + |β |2 κ ∂ u∗ ∂ u − c.c −2 κ z | | u α κ iε0 ω e ∂y
∂ y ∗ ∂ x ∂ x∗ ⎦ , ez ⎣ + ∂ ∂u ∗ 4k2 u − c.c + ∂ u ∂ u − c.c +Re (αβ ) κ ∂x
with ∇⊥ = ex
∂y
∂y
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (8)
∂x
∂ ∂ ∂ ∂ ∂ ∂ (−) (↔) + ey , ∇⊥ = ex − ey , ∇⊥ = ex + ey . ∂x ∂y ∂x ∂y ∂y ∂x
Note that we have not made the paraxial approximation, and thus the non-paraxial terms (i.e., products containing high-order derivatives or more than one first-order derivatives) are retained in Eq. (8). This is necessary because for the evanescent field, the variation in the transverse plane is usually comparable to or even larger than that in the z-direction. As expected, the integral of pz (z component of the linear momentum) over x and y vanishes due to the totalderivative feature of each term in pz . However, our main interest is in the transverse (x-y plane) part p⊥ , particularly the azimuthal component pφ , which is related to the angular momentum density by jz = ρ pφ , with ρ as the radius of the polar coordinates. To conclude this section, we make a comparison between the linear momentum density p in Eq. (8) and that of the traveling field Atra (r,t) = u (x, y) eikz (α ex + β ey ) e−iω t [the subscript “tra” means “traveling”] ⎫ ⎧ ∗ − α ∗ β ) k2 ∇ |u|2 × e kz2 (u∇⊥ u∗ − c.c) + (αβ ⎪ z ⊥ z ⎪ ⎪ ⎪ iε0 ω ⎨ + |α |2 ∂ u ∇ ∂ u∗ − c.c + |β |2 ∂ u ∇ ∂ u∗ − c.c ⎬ ⊥ ∂y ptra = ∂ x ⊥∂ x
∂ y ⎪ 4k2 ⎪ ⎪ ⎪ ∂ u∗ ∂u ∂ u∗ ∗ ∗ ⎭ ⎩ ∇⊥ + αβ ∇⊥ ∂ u − c.c + α β ⎡
+
ε0 ω ⎢ ⎢ ez ⎢ 4k2 ⎣
∂y
∂x
∂x
∂y
2 2 2kz3 |u|2 + 2 |α |2 kz ∂∂ uy + 2 |β |2 kz ∂∂ ux 2 +2Re (αβ ∗ ) kz u∗ ∂∂x∂uy + c.c. − |α |2 kz ∂∂y u∗ ∂∂ uy + c.c. − |β |2 kz ∂∂x u∗ ∂∂ ux + c.c.
⎤
⎥ ⎥ ⎥, ⎦
(9)
which can be achieved by taking similar steps that led to Eq. (8). Comparing Eqs. (8) and (9) we immediately find that the paraxial terms in p⊥ and ptra⊥ are evidently different, which indicates a corresponding difference between the angular momenta of the two fields. Indeed, as we will see in the next section, for an evanescent Bessel wave the paraxial part of the OAM vanishes when intergrated over the azimuthal angle. This is in stark contrast to the traveling-wave case, where the paraxial contribution to the OAM is typically the most significant. It is worth noting that some other interesting properties of the linear and angular momenta of evanescent waves were recently reported in [22]. 3.
OAM of Evanescent Bessel Waves
Equation (8), obtained without specifying the transverse spatial profile u (x, y) in Eq. (3), gives the linear momentum density p of an evanescent field, which in turn determines the angular momentum density (in the z-direction) via jz = ρ pφ . In this section, we specifically investigate
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Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12703
the angular momentum of the evanescent Bessel wave, with u (x, y) in Eq. (3) as (converted to the polar-coordinate system) (10) u (ρ , φ ) = Jm (μρ ) eimφ , μ = k2 + κ 2 , m = 0, ±1, ... We mainly focus on the OAM, although the SAM is also briefly discussed at the end of the section. For convenience, we will frequently adopt the simplified notation a (ρ ) = Jm (μρ ) in the analysis. For the study of OAM, we only consider linear polarization (so that SAM does not come into play), which is characterized by real αβ ∗ in Eq. (1). Without loss of generality, we take both α and β to be real. Substituting Eq. (10) into Eq. (8) and using the polar-coordinate differential operators in Eq. (25), we derive the azimuthal component of the linear momentum density for the evanescent Bessel wave (see the Appendix for more details) −κ 2 a2 (ρ ) cos 2 (φ − φ0 ) mε0 ω e−2κ z 2 2 , (11) pφ = a (ρ ) cos2 (φ − φ0 ) − ρ1 a (ρ ) a (ρ ) + a2 (ρ ) sin2 (φ − φ0 ) + m 2k2 ρ ρ2 where φ0 is the azimuthal angle of the polarization cos φ0 = α , sin φ0 = β .
(12)
The first line in the square bracket on the right-hand side of Eq. (11) comes from the paraxial terms in Eq. (8), and the second line from the non-paraxial ones. The OAM density (in the z-direction) is achieved by multiplying pφ with the radius ρ −κ 2 a2 (ρ ) cos 2 (φ − φ0 ) mε0 ω e−2κ z 2 2 . (13) jz = a (ρ ) cos2 (φ − φ0 ) − ρ1 a (ρ ) a (ρ ) + a2 (ρ ) sin2 (φ − φ0 ) + m 2k2 ρ2 For comparison purpose, we also calculate the OAM density from Eq. (9) for the traveling Bessel wave, with u (x, y) and α , β the same as in Eq. (8), kz2 a2 (ρ ) mε0 ω (tra) 2 2 . (14) jz = a (ρ ) cos2 (φ − φ0 ) − ρ1 a (ρ ) a (ρ ) + a2 (ρ ) sin2 (φ − φ0 ) + m 2k2 ρ2 The most remarkable feature of the evanescent Bessel wave OAM, according to Eq. (13), is that the paraxial part vanishes when integrated over the azimuthal angle. This is drastically different from the traveling wave OAM in Eq. (14), where the paraxial contribution is typically the most significant. A result of essential importance in the optical OAM theory is that the OAM per unit energy in a linearly-polarized traveling beam with azimuthal phase dependence exp (imφ ) [but otherwise rotationally symmetric in the transverse plane] obeys the simple relation [23] (tra)
Jφ
E (tra)
=
m , ω
(15)
where (tra)
Jz
Etra #233857 - $15.00 USD © 2015 OSA
= =
∞
−∞ ∞ −∞
(tra)
dxdy jz
(x, y, z) =
dxdy wtra (x, y, z) =
2π
0 2π 0
dφ
dφ
∞
0 ∞
0
(tra)
d ρ ρ jz
(ρ , φ , z) ,
d ρ ρ wtra (ρ , φ , z) ,
(16) (17)
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12704
and wtra is the energy density of the light field ∗ (r) eiω t 2 ∗ (r) eiω t 2 Btra (r) e−iω t + Btra 1 ε0 Etra (r) e−iω t + Etra + wtra = 2 2 2μ0 2 =
1 ε0 |Btra |2 . |Etra |2 + 4 4μ0
(18)
Equation (15) is a well-known formula, and we can confirm it specifically for the traveling Bessel beam [following a procedure similar to the derivation of Eq. (19) below] but will not show the details here. However, for an evanescent wave this relation is violated. Particularly, for mth order Bessel wave, the OAM-energy ratio is reduced by a factor of the2 evanescent κ 1 + k2 , i.e., −1 κ2 m Jz = 1+ 2 , (19) E k ω where k and κ [see Eqs. (2) and (3)] are the wave number and the evanescent decay rate, respectively, and Jz and E are defined as in Eqs. (16) - (18) without the sub- or super-scripts “tra”. √ To prove Eq. (19), we recall the explicit form of a (ρ ), i.e., Jm (μρ ) with μ = k2 + κ 2 . We only consider m = 0 because the validity of Eq. (19) is obvious for m = 0. First we insert Eq. (13) into the evanescent-wave counterpart of Eq. (16) and integrate over the azimuthal angle to obtain m2 2 π mε0 ω e−2κ z ∞ 1 2 dρ ρ a (ρ ) + a (ρ ) , (20) Jz = k2 2 ρ 0 where it has been taken into account that ∞ 0
d ρ a (ρ ) a (ρ ) =
1 2
∞ 0
dρ
d 2 a (ρ ) = 0, dρ
since Jm =0 (μρ ) → 0 for both ρ → 0 and ρ → ∞. We then calculate the energy of the eavenescent Bessel beam in the Appendix and present the result here ! πε0 ω 2 e−2κ z ∞ κ2 κ2 1 m2 2 2 2 2 1 − 1 + d ρ κ ρ a ( ρ ) + ρ a ( ρ ) + a ( ρ ) . E = k2 k2 2 k2 ρ 0 (21) We need to point out that for an ideal Bessel function, the integrals in both Eqs. (20) and (21) diverge. Thus in the definition of Jz and E , we have implicitly assumed a cutoff at a sufficiently large radius ρcut [i.e., a (ρ > ρcut ) = 0] such that the essential features of the Bessel distribution are preserved while the integrals are kept finite. To go further, we note the asymptotic behavior of the Bessel function, i.e., as ρ → ∞ " 2 π mπ , cos μρ − − Jm (μρ ) → π μρ 4 2 and thus
∂ Jm (μρ ) →− ∂ρ
"
2μ π mπ . sin μρ − − πρ 4 2
It follows that in the same limit
ρ a2 (ρ ) →
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2 π mπ , cos2 μρ − − πμ 4 2
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12705
ρ a2 (ρ ) → m2 2 a (ρ ) → ρ
2μ 2 π mπ , sin μρ − − π 4 2 2m2 2m2 π mπ 2 − ≤ cos μρ − . π μρ 2 4 2 π μρ 2
(22) #
m2 2 2m2 m2 2 ρ a (ρ ) ≤ π μρ 2 for ρ → ∞ and ρ a (ρ ) → 0 for ρ → 0 [due to Jm (s → 0) → 2 integral of mρ a2 (ρ ) over ρ is finite. On the other hand, the other two terms in
Since
− 2s
$|m|
],
Eq. (22) the become periodic (rather than vanish) for ρ → ∞ and the integrals of them actually diverge. As mentioned above, we cut off the integrals at some sufficiently large radius. The cutoff radius ρcut should be many periods after the Bessel function has settled into the asymptotic cosine 2 function, such that (i) compared to the other terms, the integrals of mρ a2 (ρ ) in Jz and E are
negligible, (ii) the integrals of ρ a2 (ρ ) and ρ a #2 (ρ ) are dominated by contributions from $ # $ the asymptotic region, and (iii) the integrals of sin2 μρ − π4 − m2π and cos2 μρ − π4 − m2π in the asymptotic region are essentially equivalent. With such arrangement, we simplify Eqs. (20) and (21) into Jz → and E
π mε0 ω e−2κ z 2k2
d ρ ρ a2 (ρ ) →
π μ 2 mε0 ω −2κ z e 2k2
d ρ ρ a2 (ρ )
! 2 πε0 ω 2 e−2κ z κ2 μ2 κ2 2 2 1 − 1 + d ρ κ ρ a ( ρ ) + ρ a ( ρ ) k2 k2 2 k2 κ 2 π μ 2 ε0 ω 2 e−2κ z → 1+ 2 d ρ ρ a2 (ρ ) , k 2k2
(23)
→
(24)
where we have used μ 2 = k2 + κ 2 in the last step. Dividing Eq. (23) by Eq. (24) immediately yields Eq. (19). the OAM per unit energy of the evenescent Bessel field is reduced by a factor of Thus, 2 1 + κk2 compared to that of the traveling Bessel beam. For κ = 0, which is the “transitional point” between the evanescent and the traveling waves, Eqs. (19) and (15) are identical. As κ gets larger, the OAM-energy ratio becomes smaller, suggesting that an evanescent wave more tightly localized to the surface possesses less OAM. This qualitative difference between the OAM of the evanescent and the traveling fields is one of the main points of this paper. Since the cutoff of the integrals plays a significant role in the above argument, it is helpful to know more precisely how large the cutoff radius ρcut should be. To this end, we numerically 2 plot mρ a2 (ρ ), μ 2 ρ a2 (ρ ), ρ a2 (ρ ) as functions of ρ in Fig. 1, and the integrals of them versus the upper-bound ρ√up of the integration interval in Fig. 2. The parameters are set as κ = k (or equivalently μ = 2k) and m = 3. From Fig. 1 we see that for ρ > 4λ (λ is the wavelength), m2 2 2 2 2 ρ a (ρ ) is essentially zero and μ ρ a (ρ ), ρ a (ρ ) behave asymptotically. Figure 2 shows that as the upper-bound ρup exceeds 40λ , % ρup
% ρup 0
dρ
m2 2 ρ a (ρ )
is less than 1% of
% ρup 0
d ρ μ 2 ρ a2 (ρ )
and 0 d ρ ρ a2 (ρ ), and the difference between the later two integrals is also less than 1%. So in this case we can safely set the cutoff radius at ρcut = 40λ . Of course, generally the appropriate ρcut depends on the specific values of κ , m, and k. We can draw a similar conclusion for the SAM in a circularly-polarized evanescent 0th-order Bessel field, i.e., with u = J0 (μρ ), |α | = |β | = √12 , and αβ ∗ being imaginary in Eqs. (1) and (3). Based on Eq. (8), we achieve after a fairly lengthy calculation Jz = #233857 - $15.00 USD © 2015 OSA
i (αβ ∗ − α ∗ β ) πε0 ω e−2κ z k2
dρ
1 2 ρ a (ρ ) , 2
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12706
Integrand
8
4
0
0
2
4
6
U[O] μ 2 ρ a2 (ρ ) [black solid], ρ a2 (ρ ) [green dashed] as functions √ of ρ . Here a (ρ ) = Jm (μρ ), with μ = 2k and m = 3. Fig. 1.
m2 2 ρ a (ρ ) [red dotted],
150
Integral
100
50
0
0
25
50
Uup [ O ] % ρup
% ρup
%ρ
d ρ μ 2 ρ a2 (ρ ) [black solid], 0 up d ρ ρ a2 (ρ ) √ [green dashed] versus ρup . Here a (ρ ) = Jm (μρ ), with μ = 2k and m = 3.
Fig. 2.
0
and E =
dρ
m2 2 ρ a (ρ )
πε0 ω 2 e−2κ z k2
[red dotted],
dρ
1+
0
κ2 k2
κ 2 ρ a2 (ρ ) +
κ2 1 1 − 2 ρ a2 (ρ ) . 2 k
Following essentially the same steps that led to Eq. (19), we have −1 Jz κ2 σ = 1+ 2 , E k ω where σ = i (αβ ∗ − α ∗ β ) = ±1, dependingon the handness of the polarization. This SAM2 energy ratio is also a reduction by a factor of 1 + κk2 from that for the corresponding traveling #233857 - $15.00 USD © 2015 OSA
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12707
Bessel wave
(tra)
σ Jz = . (tra) ω E Finally, we remark that evanescent Bessel beams may be generated via total internal reflection or surface plasmon resonance excitation, as proposed in [18–20]. In principle, our prediction in this paper can be tested by measuring the torque acting on a mechanical object in the evanescent field and comparing the result with that in the corresponding traveling field. 4.
Summary
In summary, there are crucial differences between the OAM of evanescent and traveling light. Paritcularly, we studied the evanescent Bessel field and found that the traditionally dominant paraxial part of the OAM vanishes when averaged over the azimuthal angle. The standard OAM-energy relation, i.e., the OAM per unit energy equals ωm in a linearly-polarized traveling wave with azimuthal phase dependence exp (imφ ), is violated in the evanescent wave. For 2 −1 m an mth-order evanescent Bessel field, the OAM-energy ratio is instead given by 1 + κk2 ω, which indicates that the OAM decreases as the field is more firmly localized to the surface. Our study gives insight into the distinctive features of the OAM in evanescent light and may provide theoretical support for potential experiments. 5.
Appendix
In the appendix, we provide some mathematical details that have been intentionally skipped in the main sections to smooth the discussions there. By use of the following differential operators in the polar-coordinate system ∂ 1 ∂ ∂ 1 ∂ (−) − eφ sin 2φ , = eρ cos 2φ − sin 2φ + cos 2φ ∇⊥ ∂ρ ρ ∂φ ∂ρ ρ ∂φ ∂ 1 ∂ ∂ 1 ∂ (↔) + eφ cos 2φ , = eρ sin 2φ + cos 2φ − sin 2φ ∇⊥ ∂ρ ρ ∂φ ∂ρ ρ ∂φ ' & '& ' & ∂ ∂ cos φ − ρ1 sin φ ∂ 1 ∂ ∂ρ ∂x = , (25) + eφ , ∇⊥ = eρ ∂ 1 ∂ sin φ ∂ρ ρ ∂φ ρ cos φ ∂y ∂φ we will derive Eqs. (11) and (21), i.e., the azimuthal linear momentum density pφ and the energy E (per unit length in the z-direction) for the linearly-polarized evanescent Bessel wave defined in Eqs. (1) and (3), with u = a (ρ ) eimφ = Jm (μρ ) eimφ , and α , β being real. I. In this item, we elaborate the steps to get Eq. (11) from Eq. (8), or more precisely, from the transverse part of Eq. (8) ⎧ ⎫ (−) ∗ 2 |α |2 − |β |2 2 Re (αβ ∗ ) u∇(↔) u∗ − c.c ⎪ ⎪ u∇ κ u − c.c + 2 κ ⎪ ⎪ ⊥ ⊥ ⎪ ⎪ ⎬ iε0 ω e−2κ z ⎨ 2 ∂u 2 ∂u ∂ u∗ ∂ u∗ | | | | α ∇ − c.c + β ∇ − c.c p⊥ = + . ⊥ ∂y ∂ x ⊥∂ x ⎪ ⎪ 4k2
∂ y ⎪ ⎪ ∗ ∗ ⎪ ⎪ ⎭ ⎩ + αβ ∗ ∂∂ux ∇⊥ ∂∂ uy + α ∗ β ∂∂uy ∇⊥ ∂∂ ux − c.c (26) p⊥ includes the azimuthal and the radial components, but we will only consider the azimuthal one, i.e., pφ . We first look at the paraxial terms (each containing at most one first-order derivative and no high-order derivative) (−) (↔) |α |2 − |β |2 κ 2 u∇⊥ u∗ − c.c. + 2κ 2 Re (αβ ∗ ) u∇⊥ u∗ − c.c φ
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φ
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12708
= =
$ m # m 2i α 2 − β 2 κ 2 a2 (ρ ) cos 2φ + 4iαβ κ 2 a2 (ρ ) sin 2φ ρ ρ $ ) (# m 2iκ 2 a2 (ρ ) α 2 − β 2 cos 2φ + 2αβ sin 2φ . ρ
(27)
To further simplify this, we note that the azimuthal angle of the polarization φ0 is determined by α , β via Eq. (12), i.e., (28) cos φ0 = α , sin φ0 = β , from which it follows
α2 − β 2 2αβ
=
cos2 φ0 − sin2 φ0 = cos 2φ0 ,
=
2 sin φ0 cos φ0 = sin 2φ0 .
(29)
Inserting Eq. (29) into Eq. (27) yields (−) (−) (↔) |α |2 − |β |2 κ 2 u∇⊥ u∗ − u∗ ∇⊥ u + 2κ 2 Re (αβ ∗ ) u∇⊥ u∗ − c.c φ
=
2iκ
2m 2
ρ
a (ρ ) [cos 2φ0 cos 2φ + sin 2φ0 sin 2φ ] = 2iκ
φ
2m 2
ρ
a (ρ ) cos 2 (φ − φ0 ) . (30)
We then turn to the non-paraxial terms (each containing a high-order derivative or more than one first-order derivatives) m ∂u ∂ u∗ ∂ u∗ 2 ∂u | | |α |2 ∇⊥ − c.c. + β ∇⊥ − c.c. = 2i 2 a (ρ ) a (ρ ) ∂y ∂y ∂ x ∂ x ρ φ φ ) ) ( ( m3 m −2i a2 (ρ ) α 2 sin2 φ + β 2 cos2 φ − 2i 3 a2 (ρ ) α 2 cos2 φ + β 2 sin2 φ ρ ρ and
∂ u∗ ∂ u∗ ∂u ∂u ∇⊥ + α ∗β ∇⊥ − c.c. αβ ∗ ∂x ∂y ∂y ∂x φ =
m m3 2i a2 (ρ ) [2αβ sin φ cos φ ] − 2i 3 a2 (ρ ) [2αβ cos φ sin φ ] . ρ ρ
By combining these two equations and applying Eq. (28), we have ∂u ∂ u∗ ∂u ∂ u∗ | α |2 ∇⊥ − c.c. + |β |2 ∇⊥ − c.c. ∂y ∂y ∂x ∂x φ φ ∗ ∗ ∂ u ∂ u u ∂ u ∂ ∇⊥ + α ∗β ∇⊥ − c.c. + αβ ∗ ∂x ∂y ∂y ∂x φ 3 m 2 m m = −2i a (ρ ) [α sin φ − β cos φ ]2 + 3 a2 (ρ ) [α cos φ + β sin φ ]2 − 2 a (ρ ) a (ρ ) ρ ρ ρ 3 m 2 m 2 m 2 2 = −2i a (ρ ) sin (φ − φ0 ) + 3 a (ρ ) cos (φ − φ0 ) − 2 a (ρ ) a (ρ ) . (31) ρ ρ ρ Finally, we substitute Eqs. (30) and (31) into Eq. (26) to achieve Eq. (11), i.e., −κ 2 a2 (ρ ) cos 2 (φ − φ0 ) mε0 ω e−2κ z 2 2 . pφ = a (ρ ) cos2 (φ − φ0 ) − ρ1 a (ρ ) a (ρ ) + a2 (ρ ) sin2 (φ − φ0 ) + m 2k2 ρ ρ2 #233857 - $15.00 USD © 2015 OSA
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12709
II. We now present a detailed derivation of Eq. (21), where the energy density of the monochromatic electromagnetic field is [see Eq. (18)] w=
ε0 1 |B (r)|2 . |E (r)|2 + 4 4μ0
(32)
For clarity, we will treat the magnetic (∝ |B (r)|2 ) and the electric (∝ |E (r)|2 ) parts of the energy separately and combine the results to obtain Eq. (21). With the magnetic field B given in Eq. (5), it is straightforward to show 2 |Bx |2 + By = e−2κ z κ 2 |u|2 = e−2κ z κ 2 a2 (ρ ) ,
(33)
and |Bz |
2
=
∂u ∂ u 2 β ∂ x − α ∂ y
−2κ z
e
−2κ z
=
e
=
e−2κ z
m2 a (ρ ) [β cos φ − α sin φ ] + 2 a2 (ρ ) [β sin φ + α cos φ ]2 ρ ! 2 m a2 (ρ ) sin2 (φ − φ0 ) + 2 a2 (ρ ) cos2 (φ − φ0 ) , ρ 2
!
2
(34)
where Eq. (28) has been taken into account. Thus we have the magnetic field energy as ∞
2π ∞ 2 1 1 |B|2 ≡ dφ d ρ ρ |Bx |2 + By + |Bz |2 4μ0 4μ0 0 −∞ 0 1 1 m2 2 πε0 ω 2 e−2κ z ∞ 2 2 2 d ρ ρ κ a (ρ ) + a (ρ ) + a (ρ ) . 2k2 2 2 ρ2 0
dxdy
=
(35)
To calculate the electric field energy, we use the second equation in (4) to write |E|2 =
ω2 c4 2 |∇ = (∇ × B) · (∇ × B∗ ) . × B| ω2 k4
(36)
Setting b = ∇ × B, a = B∗ in the formula b · (∇ × a) = ∇ · (a × b) + a · (∇ × b) , we get (∇ × B) · (∇ × B∗ ) = ∇ · [B∗ × (∇ × B)] + B∗ · (∇ × [∇ × B]) ( ) = ∇ · [B∗ × (∇ × B)] + B∗ · ∇ (∇ · B) − ∇2 B = ∇ · [B∗ × (∇ × B)] + k2 |B|2 , where ∇ · B = 0 and ∇2 B + k2 B = 0 (from Maxwell’s equations) have been applied. Further noting the symmetry between B and B∗ in the above equation, one has (∇ × B) · (∇ × B∗ ) =
1 {∇ · [B∗ × (∇ × B)] + c.c.} + k2 |B|2 . 2
With ∇ · [B∗ × (∇ × B)] = ∇⊥ · [B∗ × (∇ × B)] +
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(37)
∂ ∗ [B × (∇ × B)]z , ∂z
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12710
and
= =
∂ ∗ [B × (∇ × B)]z + c.c. ∂z ∂ By ∂ ∂ Bx ∂ Bz ∂ Bz B∗x − B∗x − B∗y − B∗y + c.c. ∂z ∂z ∂x ∂y ∂z 2 # $ # $ # $ ∂ 2 |Bx |2 + By − |Bz |2 ∂ Bx B∗z ∂ By B∗z ∂ ∂ (B∗x Bz ) ∂ B∗y Bz + + + , − ∂ z2 ∂z ∂x ∂y ∂x ∂y
Eq. (37) becomes 1 ∂2 2 2 2 |B |B | | (∇ × B) · (∇ × B∗ ) = k2 |B|2 + + B − x y z 2 ∂ z2 # ∗ $ # $ # $ ∗ ∂ Bx B∗z ∂ By B∗z 1 ∂ ∂ (Bx Bz ) ∂ By Bz + + + − 2 ∂z ∂x ∂y ∂x ∂y 1 + {∇⊥ · [B∗ × (∇ × B)] + c.c} . 2
(38)
With Eqs. (33) - (36), (38), and noting that the integrals (over x and y) of the second and third lines in Eq. (38) vanish, we achieve the electric field energy ∞
∞ ε0 2 ε0 ω 2 |E| = dxdy (∇ × B) · (∇ × B∗ ) 4 4k4 −∞ −∞ ∞ ε0 ω 2 2 1 ε0 ω 2 ∂ 2 2 2 2 |B| + |Bx | + By − |Bz | = dxdy 4k2 2 4k4 ∂ z2 −∞ 1 2 1 m2 2 πε0 ω 2 e−2κ z ∞ 2 2 d ρ ρ κ a (ρ ) + a (ρ ) + a (ρ ) = 2k2 2 2 ρ2 0 πε0 ω 2 e−2κ z κ 2 ∞ 1 2 1 m2 2 2 2 d ρ ρ κ a (ρ ) − a (ρ ) − a (ρ ) . + k2 k2 0 2 2 ρ2
dxdy
(39)
The total electromagnetic energy E is the sum of the magnetic part in Eq. (35) and the electric part in Eq. (39) ! πε0 ω 2 e−2κ z ∞ κ2 κ2 1 m2 2 2 2 2 1− 2 1 + 2 κ ρ a (ρ ) + dρ ρ a (ρ ) + a (ρ ) , E = k2 k 2 k ρ 0 i.e., Eq. (21). Acknowledgments We gratefully acknowledge financial supports from the National Natural Science Foundation of China (No. 11375081), the Shandong Provincial Natural Science Foundation (No. ZR2013AL007), and the Start-up Fund of Liaocheng University.
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Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12711
Abstract: We show that the orbital angular momentum (OAM) of evanescent light is drastically different from that of traveling light. Specifically, the paraxial contribution (typically the most significant part in a traveling wave) to the OAM vanishes in an evanescent Bessel wave when averaged over the azimuthal angle. Moreover, the OAM energy for the evanescent per unit κ2 Bessel field is reduced by a factor of 1 + k2 from the standard result for the corresponding traveling field, where k and κ are the wave number and the evanescent decay rate, respectively. © 2015 Optical Society of America OCIS codes: (240.0240) Optics at surfaces; (260.2110) Electromagnetic optics; (020.1670) Coherent optical effects.
References and links 1. J. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London Ser. A 82, 560–567 (1909). 2. R. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936). 3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). 4. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3, 161–204 (2011). 5. Ed. by J. P. Torres and L. Torner, Twisted Photons: Applications of Light with Orbital Angular Momentum, (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011). 6. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997). 7. A. Lehmuskero, Y. Li, P. Johansson, and M. Kall, “Plasmonic particles set into fast orbital motion by an optical vortex beam,” Opt. Express 22, 4349–4356 (2014). 8. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabitscale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013). 9. M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commmun. 5, 3115 (2014). 10. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). 11. M. Agnew, J. Z. Salvail, J. Leach, and R. W. Boyd, “Generation of orbital angular momentum Bell states and their verification via accessible nonlinear witnesses,” Phys. Rev. Lett. 111, 030402 (2013). 12. V. Garces-Chavez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 031106 (2005). 13. L. C. Thomson, G. Whyte, M. Mazilu, and J. Courtial, “Simulated holographic three-dimensional intensity shaping of evanescent-wave fields,” J. Opt. Soc. Am. B 25, 849–853 (2008). 14. G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009). 15. D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4, 211–217 (2010).
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16. V. E. Lembessis, M. Babiker, and D. L. Andrews, “Surface optical vortices,” Phys. Rev. A 79, 011806 (2009). 17. V. E. Lembessis, S. Al-Awfi, M. Babiker, and D. L. Andrews, “Surface plasmon optical vortices and their influence on atoms,” J. Opt. 13, 064002 (2011). 18. S. Ruschin and A. Leizer, “Evanescent Bessel beams,” J. Opt. Soc. Am. A 15, 1139–1143 (1998). 19. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006). 20. S. N. Kurilkina, V. N. Belyi, and N. S. Kazak, “Features of evanescent Bessel light beams formed in structures containing a dielectric layer,” Opt. Commun. 238, 3860–3868 (2010). 21. L. W. Davis, “Theory of electromagnetics beams,” Phys. Rev. A 19, 1177–1179 (1979). 22. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014). 23. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
1.
Introduction
It has long been recognized, dating back to Poynting [1] and Beth [2], that circularly polarized light carries spin angular mometum (SAM) associated with the spin of photons. Much later, in 1992 Allen et al. [3] made the breakthrough observation that Laguerre-Gaussian (LG) modes, and more generally light beams with an azimuthal phase dependence of exp (imφ ) [m = 0, ±1, ...], possess orbital angular momentum (OAM) in the overall propagating direction (z-direction). Following Allen’s discovery, the OAM of light has rapidly developed into an exciting research area [4,5]. Besides bringing new insight into our fundamental understanding of light, the optical OAM has found potential applications in optical tweezers and spanners [6,7], optical communications [8,9], and quantum information processing [10,11]. Since Allen’s poineering work, most studies in this area have focused on traveling beams, and much less attention has been paid to the OAM of evanescent waves. Evanescent optical fields can be employed to actively manipulate small particles near a surface [12,13] or to provide optomechanical coupling between light and mechanical objects via gradient forces [14,15], and the added dimension of OAM is expected to introduce new aspects to these applications. While evanescent light with azimuthal phase dependence has been investigated in the context of surface optical vortex [16,17], the focus was mainly on its phase and intensity properties. A quantitative study on the evanescent optical OAM is still lacking. In this paper, we present a detailed analysis of the optical OAM in evanescent Bessel fields, which have transverse (x-y plane) spatial profiles identical to those of traveling Bessel beams, but decay exponentially in the z-direction [18-20]. We derive an analytic formula for the OAM density of the evanescent Bessel wave, and show that the paraxial part of the OAM vanishes when integrated over the azimuthal angle φ , which is in stark contrast to the traveling-wave case. Even more remarkably, for an evanescent mth order Bessel field, the OAM per unit en 2 −1 m κ2 from the standard result ωm for the ergy is 1 + κk2 ω , reduced by a factor of 1 + k2 corresponding traveling beam (here κ is the amplitude decay rate of the evanescent light, and ω , k are the angular frequency and the wave number, respectively). 2.
Linear momentum of evanecsent fields
It is well known that an evanescent field does not carry linear momentum in the z-direction (along which the field decays). However, if the azimuthal component of the linear momentum exists, then the field possesses angular momentum in the z-direction. From now on, unless explicitly stated otherwise, “angular momentum” refers to its z-component. In this section, we derive a general formula for the linear momentum of evanescent light, which will not only suggest some remarkable difference between the angular momenta of the evenescent and the traveling beams, but also serve as a basis for further study on the OAM of evanescent Bessel #233857 - $15.00 USD © 2015 OSA
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12701
waves in the next section. We choose to work in the Lorentz gauge and consider a monochromatic field polarized in the x-y plane [21] A (r,t) = A (r) e−iω t = A (r) (α ex + β ey ) e−iω t , |α |2 + |β |2 = 1,
(1)
where A is the vector potential, ex and ey are the unit vectors in the x and y directions, respectively, and ω is the optical angular frequency. A (r) satisfies the Helmholtz equation ∇2 A (r) + k2 A (r) = 0, k =
ω , c
(2)
with c being the speed of light in vacuum. We emphasize that it is consistent to assume that the vector potential is polarized as in Eq. (1) even if the light field is non-paraxial, which can be seen as follows. The Lorentz gauge ∇ · A (r) − icω2 ϕ (r) = 0 gives the scalar potential as
ϕ (r) =
c2 ∇ · A (r) . iω
With A (r) = A (r) (α ex + β ey ) and A (r) being a solution of Eq. (2), it is easy to show that both the vector and the scalar potentials obey the wave equations (i.e., the Maxwell’s equations in the Lorentz gauge) ∇2 A (r) + k2 A (r) = 0, ∇2 φ (r) + k2 φ (r) = 0. Thus the assumption on the polarization of the vector potential in Eq. (1) is entirely consistent. For an evanescent wave, we take A (r) = u (x, y) e−κ z ,
(3)
where the detailed form of u (x, y) is intentionally left unspecified, so that we can first achieve a relatively general formula for the linear momentum. According to Maxwell’s equations, the magnetic field B and the electric field E are given by B (r) = ∇ × A (r) , E (r) = i Substituting Eq. (3) into Eq. (4) yields B = e−κ z β κ u,
c2 ∇ × B (r) . ω
−ακ u, β ∂∂ ux − α ∂∂ uy
(4)
(5)
and E=
i ω −κ z ∂ 2 u 2 β ∂ x∂ y − α ∂∂ yu2 − ακ 2 u, e k2
α ∂∂x∂uy − β ∂∂ xu2 − β κ 2 u, −ακ ∂∂ ux − β κ ∂∂ uy 2
2
The linear momentum density of the field is determined by −iω t Ee + E∗ eiω t Be−iω t + B∗ eiω t ε0 × = [E × B∗ + c.c.] , p = ε0 2 2 4
. (6)
(7)
where ε0 is the electric permittivity, stands for the time-average over one optical oscillation period, and “c.c.” represents the complex conjugate. By inserting Eqs. (5) and (6) into Eq. (7)
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and after a fairly lengthy calculation, we obtain ⎧ (−) (↔) ⎪ κ 2 |α |2 − |β |2 u∇⊥ u∗ − c.c + 2κ 2 Re (αβ ∗ ) u∇⊥ u∗ − c.c ⎪ ⎪ iε0 ω e−2κ z ⎨ 2 ∂u 2 ∂u ∂ u∗ ∂ u∗ | | | | α ∇ − c.c + β ∇ − c.c p = + ⊥ ∂y ∂ x ⊥∂ x ⎪ 4k2
∂ y ⎪ ⎪ ∂ u∗ ∂u ∂ u∗ ∗ ∗ ⎩ + αβ ∂ x ∇⊥ ∂ y + α β ∂ y ∇⊥ ∂∂ ux − c.c ⎡ ⎤ 2 ∂ ∗ ∂ u − c.c + |β |2 κ ∂ u∗ ∂ u − c.c −2 κ z | | u α κ iε0 ω e ∂y
∂ y ∗ ∂ x ∂ x∗ ⎦ , ez ⎣ + ∂ ∂u ∗ 4k2 u − c.c + ∂ u ∂ u − c.c +Re (αβ ) κ ∂x
with ∇⊥ = ex
∂y
∂y
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (8)
∂x
∂ ∂ ∂ ∂ ∂ ∂ (−) (↔) + ey , ∇⊥ = ex − ey , ∇⊥ = ex + ey . ∂x ∂y ∂x ∂y ∂y ∂x
Note that we have not made the paraxial approximation, and thus the non-paraxial terms (i.e., products containing high-order derivatives or more than one first-order derivatives) are retained in Eq. (8). This is necessary because for the evanescent field, the variation in the transverse plane is usually comparable to or even larger than that in the z-direction. As expected, the integral of pz (z component of the linear momentum) over x and y vanishes due to the totalderivative feature of each term in pz . However, our main interest is in the transverse (x-y plane) part p⊥ , particularly the azimuthal component pφ , which is related to the angular momentum density by jz = ρ pφ , with ρ as the radius of the polar coordinates. To conclude this section, we make a comparison between the linear momentum density p in Eq. (8) and that of the traveling field Atra (r,t) = u (x, y) eikz (α ex + β ey ) e−iω t [the subscript “tra” means “traveling”] ⎫ ⎧ ∗ − α ∗ β ) k2 ∇ |u|2 × e kz2 (u∇⊥ u∗ − c.c) + (αβ ⎪ z ⊥ z ⎪ ⎪ ⎪ iε0 ω ⎨ + |α |2 ∂ u ∇ ∂ u∗ − c.c + |β |2 ∂ u ∇ ∂ u∗ − c.c ⎬ ⊥ ∂y ptra = ∂ x ⊥∂ x
∂ y ⎪ 4k2 ⎪ ⎪ ⎪ ∂ u∗ ∂u ∂ u∗ ∗ ∗ ⎭ ⎩ ∇⊥ + αβ ∇⊥ ∂ u − c.c + α β ⎡
+
ε0 ω ⎢ ⎢ ez ⎢ 4k2 ⎣
∂y
∂x
∂x
∂y
2 2 2kz3 |u|2 + 2 |α |2 kz ∂∂ uy + 2 |β |2 kz ∂∂ ux 2 +2Re (αβ ∗ ) kz u∗ ∂∂x∂uy + c.c. − |α |2 kz ∂∂y u∗ ∂∂ uy + c.c. − |β |2 kz ∂∂x u∗ ∂∂ ux + c.c.
⎤
⎥ ⎥ ⎥, ⎦
(9)
which can be achieved by taking similar steps that led to Eq. (8). Comparing Eqs. (8) and (9) we immediately find that the paraxial terms in p⊥ and ptra⊥ are evidently different, which indicates a corresponding difference between the angular momenta of the two fields. Indeed, as we will see in the next section, for an evanescent Bessel wave the paraxial part of the OAM vanishes when intergrated over the azimuthal angle. This is in stark contrast to the traveling-wave case, where the paraxial contribution to the OAM is typically the most significant. It is worth noting that some other interesting properties of the linear and angular momenta of evanescent waves were recently reported in [22]. 3.
OAM of Evanescent Bessel Waves
Equation (8), obtained without specifying the transverse spatial profile u (x, y) in Eq. (3), gives the linear momentum density p of an evanescent field, which in turn determines the angular momentum density (in the z-direction) via jz = ρ pφ . In this section, we specifically investigate
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Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12703
the angular momentum of the evanescent Bessel wave, with u (x, y) in Eq. (3) as (converted to the polar-coordinate system) (10) u (ρ , φ ) = Jm (μρ ) eimφ , μ = k2 + κ 2 , m = 0, ±1, ... We mainly focus on the OAM, although the SAM is also briefly discussed at the end of the section. For convenience, we will frequently adopt the simplified notation a (ρ ) = Jm (μρ ) in the analysis. For the study of OAM, we only consider linear polarization (so that SAM does not come into play), which is characterized by real αβ ∗ in Eq. (1). Without loss of generality, we take both α and β to be real. Substituting Eq. (10) into Eq. (8) and using the polar-coordinate differential operators in Eq. (25), we derive the azimuthal component of the linear momentum density for the evanescent Bessel wave (see the Appendix for more details) −κ 2 a2 (ρ ) cos 2 (φ − φ0 ) mε0 ω e−2κ z 2 2 , (11) pφ = a (ρ ) cos2 (φ − φ0 ) − ρ1 a (ρ ) a (ρ ) + a2 (ρ ) sin2 (φ − φ0 ) + m 2k2 ρ ρ2 where φ0 is the azimuthal angle of the polarization cos φ0 = α , sin φ0 = β .
(12)
The first line in the square bracket on the right-hand side of Eq. (11) comes from the paraxial terms in Eq. (8), and the second line from the non-paraxial ones. The OAM density (in the z-direction) is achieved by multiplying pφ with the radius ρ −κ 2 a2 (ρ ) cos 2 (φ − φ0 ) mε0 ω e−2κ z 2 2 . (13) jz = a (ρ ) cos2 (φ − φ0 ) − ρ1 a (ρ ) a (ρ ) + a2 (ρ ) sin2 (φ − φ0 ) + m 2k2 ρ2 For comparison purpose, we also calculate the OAM density from Eq. (9) for the traveling Bessel wave, with u (x, y) and α , β the same as in Eq. (8), kz2 a2 (ρ ) mε0 ω (tra) 2 2 . (14) jz = a (ρ ) cos2 (φ − φ0 ) − ρ1 a (ρ ) a (ρ ) + a2 (ρ ) sin2 (φ − φ0 ) + m 2k2 ρ2 The most remarkable feature of the evanescent Bessel wave OAM, according to Eq. (13), is that the paraxial part vanishes when integrated over the azimuthal angle. This is drastically different from the traveling wave OAM in Eq. (14), where the paraxial contribution is typically the most significant. A result of essential importance in the optical OAM theory is that the OAM per unit energy in a linearly-polarized traveling beam with azimuthal phase dependence exp (imφ ) [but otherwise rotationally symmetric in the transverse plane] obeys the simple relation [23] (tra)
Jφ
E (tra)
=
m , ω
(15)
where (tra)
Jz
Etra #233857 - $15.00 USD © 2015 OSA
= =
∞
−∞ ∞ −∞
(tra)
dxdy jz
(x, y, z) =
dxdy wtra (x, y, z) =
2π
0 2π 0
dφ
dφ
∞
0 ∞
0
(tra)
d ρ ρ jz
(ρ , φ , z) ,
d ρ ρ wtra (ρ , φ , z) ,
(16) (17)
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12704
and wtra is the energy density of the light field ∗ (r) eiω t 2 ∗ (r) eiω t 2 Btra (r) e−iω t + Btra 1 ε0 Etra (r) e−iω t + Etra + wtra = 2 2 2μ0 2 =
1 ε0 |Btra |2 . |Etra |2 + 4 4μ0
(18)
Equation (15) is a well-known formula, and we can confirm it specifically for the traveling Bessel beam [following a procedure similar to the derivation of Eq. (19) below] but will not show the details here. However, for an evanescent wave this relation is violated. Particularly, for mth order Bessel wave, the OAM-energy ratio is reduced by a factor of the2 evanescent κ 1 + k2 , i.e., −1 κ2 m Jz = 1+ 2 , (19) E k ω where k and κ [see Eqs. (2) and (3)] are the wave number and the evanescent decay rate, respectively, and Jz and E are defined as in Eqs. (16) - (18) without the sub- or super-scripts “tra”. √ To prove Eq. (19), we recall the explicit form of a (ρ ), i.e., Jm (μρ ) with μ = k2 + κ 2 . We only consider m = 0 because the validity of Eq. (19) is obvious for m = 0. First we insert Eq. (13) into the evanescent-wave counterpart of Eq. (16) and integrate over the azimuthal angle to obtain m2 2 π mε0 ω e−2κ z ∞ 1 2 dρ ρ a (ρ ) + a (ρ ) , (20) Jz = k2 2 ρ 0 where it has been taken into account that ∞ 0
d ρ a (ρ ) a (ρ ) =
1 2
∞ 0
dρ
d 2 a (ρ ) = 0, dρ
since Jm =0 (μρ ) → 0 for both ρ → 0 and ρ → ∞. We then calculate the energy of the eavenescent Bessel beam in the Appendix and present the result here ! πε0 ω 2 e−2κ z ∞ κ2 κ2 1 m2 2 2 2 2 1 − 1 + d ρ κ ρ a ( ρ ) + ρ a ( ρ ) + a ( ρ ) . E = k2 k2 2 k2 ρ 0 (21) We need to point out that for an ideal Bessel function, the integrals in both Eqs. (20) and (21) diverge. Thus in the definition of Jz and E , we have implicitly assumed a cutoff at a sufficiently large radius ρcut [i.e., a (ρ > ρcut ) = 0] such that the essential features of the Bessel distribution are preserved while the integrals are kept finite. To go further, we note the asymptotic behavior of the Bessel function, i.e., as ρ → ∞ " 2 π mπ , cos μρ − − Jm (μρ ) → π μρ 4 2 and thus
∂ Jm (μρ ) →− ∂ρ
"
2μ π mπ . sin μρ − − πρ 4 2
It follows that in the same limit
ρ a2 (ρ ) →
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2 π mπ , cos2 μρ − − πμ 4 2
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12705
ρ a2 (ρ ) → m2 2 a (ρ ) → ρ
2μ 2 π mπ , sin μρ − − π 4 2 2m2 2m2 π mπ 2 − ≤ cos μρ − . π μρ 2 4 2 π μρ 2
(22) #
m2 2 2m2 m2 2 ρ a (ρ ) ≤ π μρ 2 for ρ → ∞ and ρ a (ρ ) → 0 for ρ → 0 [due to Jm (s → 0) → 2 integral of mρ a2 (ρ ) over ρ is finite. On the other hand, the other two terms in
Since
− 2s
$|m|
],
Eq. (22) the become periodic (rather than vanish) for ρ → ∞ and the integrals of them actually diverge. As mentioned above, we cut off the integrals at some sufficiently large radius. The cutoff radius ρcut should be many periods after the Bessel function has settled into the asymptotic cosine 2 function, such that (i) compared to the other terms, the integrals of mρ a2 (ρ ) in Jz and E are
negligible, (ii) the integrals of ρ a2 (ρ ) and ρ a #2 (ρ ) are dominated by contributions from $ # $ the asymptotic region, and (iii) the integrals of sin2 μρ − π4 − m2π and cos2 μρ − π4 − m2π in the asymptotic region are essentially equivalent. With such arrangement, we simplify Eqs. (20) and (21) into Jz → and E
π mε0 ω e−2κ z 2k2
d ρ ρ a2 (ρ ) →
π μ 2 mε0 ω −2κ z e 2k2
d ρ ρ a2 (ρ )
! 2 πε0 ω 2 e−2κ z κ2 μ2 κ2 2 2 1 − 1 + d ρ κ ρ a ( ρ ) + ρ a ( ρ ) k2 k2 2 k2 κ 2 π μ 2 ε0 ω 2 e−2κ z → 1+ 2 d ρ ρ a2 (ρ ) , k 2k2
(23)
→
(24)
where we have used μ 2 = k2 + κ 2 in the last step. Dividing Eq. (23) by Eq. (24) immediately yields Eq. (19). the OAM per unit energy of the evenescent Bessel field is reduced by a factor of Thus, 2 1 + κk2 compared to that of the traveling Bessel beam. For κ = 0, which is the “transitional point” between the evanescent and the traveling waves, Eqs. (19) and (15) are identical. As κ gets larger, the OAM-energy ratio becomes smaller, suggesting that an evanescent wave more tightly localized to the surface possesses less OAM. This qualitative difference between the OAM of the evanescent and the traveling fields is one of the main points of this paper. Since the cutoff of the integrals plays a significant role in the above argument, it is helpful to know more precisely how large the cutoff radius ρcut should be. To this end, we numerically 2 plot mρ a2 (ρ ), μ 2 ρ a2 (ρ ), ρ a2 (ρ ) as functions of ρ in Fig. 1, and the integrals of them versus the upper-bound ρ√up of the integration interval in Fig. 2. The parameters are set as κ = k (or equivalently μ = 2k) and m = 3. From Fig. 1 we see that for ρ > 4λ (λ is the wavelength), m2 2 2 2 2 ρ a (ρ ) is essentially zero and μ ρ a (ρ ), ρ a (ρ ) behave asymptotically. Figure 2 shows that as the upper-bound ρup exceeds 40λ , % ρup
% ρup 0
dρ
m2 2 ρ a (ρ )
is less than 1% of
% ρup 0
d ρ μ 2 ρ a2 (ρ )
and 0 d ρ ρ a2 (ρ ), and the difference between the later two integrals is also less than 1%. So in this case we can safely set the cutoff radius at ρcut = 40λ . Of course, generally the appropriate ρcut depends on the specific values of κ , m, and k. We can draw a similar conclusion for the SAM in a circularly-polarized evanescent 0th-order Bessel field, i.e., with u = J0 (μρ ), |α | = |β | = √12 , and αβ ∗ being imaginary in Eqs. (1) and (3). Based on Eq. (8), we achieve after a fairly lengthy calculation Jz = #233857 - $15.00 USD © 2015 OSA
i (αβ ∗ − α ∗ β ) πε0 ω e−2κ z k2
dρ
1 2 ρ a (ρ ) , 2
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12706
Integrand
8
4
0
0
2
4
6
U[O] μ 2 ρ a2 (ρ ) [black solid], ρ a2 (ρ ) [green dashed] as functions √ of ρ . Here a (ρ ) = Jm (μρ ), with μ = 2k and m = 3. Fig. 1.
m2 2 ρ a (ρ ) [red dotted],
150
Integral
100
50
0
0
25
50
Uup [ O ] % ρup
% ρup
%ρ
d ρ μ 2 ρ a2 (ρ ) [black solid], 0 up d ρ ρ a2 (ρ ) √ [green dashed] versus ρup . Here a (ρ ) = Jm (μρ ), with μ = 2k and m = 3.
Fig. 2.
0
and E =
dρ
m2 2 ρ a (ρ )
πε0 ω 2 e−2κ z k2
[red dotted],
dρ
1+
0
κ2 k2
κ 2 ρ a2 (ρ ) +
κ2 1 1 − 2 ρ a2 (ρ ) . 2 k
Following essentially the same steps that led to Eq. (19), we have −1 Jz κ2 σ = 1+ 2 , E k ω where σ = i (αβ ∗ − α ∗ β ) = ±1, dependingon the handness of the polarization. This SAM2 energy ratio is also a reduction by a factor of 1 + κk2 from that for the corresponding traveling #233857 - $15.00 USD © 2015 OSA
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12707
Bessel wave
(tra)
σ Jz = . (tra) ω E Finally, we remark that evanescent Bessel beams may be generated via total internal reflection or surface plasmon resonance excitation, as proposed in [18–20]. In principle, our prediction in this paper can be tested by measuring the torque acting on a mechanical object in the evanescent field and comparing the result with that in the corresponding traveling field. 4.
Summary
In summary, there are crucial differences between the OAM of evanescent and traveling light. Paritcularly, we studied the evanescent Bessel field and found that the traditionally dominant paraxial part of the OAM vanishes when averaged over the azimuthal angle. The standard OAM-energy relation, i.e., the OAM per unit energy equals ωm in a linearly-polarized traveling wave with azimuthal phase dependence exp (imφ ), is violated in the evanescent wave. For 2 −1 m an mth-order evanescent Bessel field, the OAM-energy ratio is instead given by 1 + κk2 ω, which indicates that the OAM decreases as the field is more firmly localized to the surface. Our study gives insight into the distinctive features of the OAM in evanescent light and may provide theoretical support for potential experiments. 5.
Appendix
In the appendix, we provide some mathematical details that have been intentionally skipped in the main sections to smooth the discussions there. By use of the following differential operators in the polar-coordinate system ∂ 1 ∂ ∂ 1 ∂ (−) − eφ sin 2φ , = eρ cos 2φ − sin 2φ + cos 2φ ∇⊥ ∂ρ ρ ∂φ ∂ρ ρ ∂φ ∂ 1 ∂ ∂ 1 ∂ (↔) + eφ cos 2φ , = eρ sin 2φ + cos 2φ − sin 2φ ∇⊥ ∂ρ ρ ∂φ ∂ρ ρ ∂φ ' & '& ' & ∂ ∂ cos φ − ρ1 sin φ ∂ 1 ∂ ∂ρ ∂x = , (25) + eφ , ∇⊥ = eρ ∂ 1 ∂ sin φ ∂ρ ρ ∂φ ρ cos φ ∂y ∂φ we will derive Eqs. (11) and (21), i.e., the azimuthal linear momentum density pφ and the energy E (per unit length in the z-direction) for the linearly-polarized evanescent Bessel wave defined in Eqs. (1) and (3), with u = a (ρ ) eimφ = Jm (μρ ) eimφ , and α , β being real. I. In this item, we elaborate the steps to get Eq. (11) from Eq. (8), or more precisely, from the transverse part of Eq. (8) ⎧ ⎫ (−) ∗ 2 |α |2 − |β |2 2 Re (αβ ∗ ) u∇(↔) u∗ − c.c ⎪ ⎪ u∇ κ u − c.c + 2 κ ⎪ ⎪ ⊥ ⊥ ⎪ ⎪ ⎬ iε0 ω e−2κ z ⎨ 2 ∂u 2 ∂u ∂ u∗ ∂ u∗ | | | | α ∇ − c.c + β ∇ − c.c p⊥ = + . ⊥ ∂y ∂ x ⊥∂ x ⎪ ⎪ 4k2
∂ y ⎪ ⎪ ∗ ∗ ⎪ ⎪ ⎭ ⎩ + αβ ∗ ∂∂ux ∇⊥ ∂∂ uy + α ∗ β ∂∂uy ∇⊥ ∂∂ ux − c.c (26) p⊥ includes the azimuthal and the radial components, but we will only consider the azimuthal one, i.e., pφ . We first look at the paraxial terms (each containing at most one first-order derivative and no high-order derivative) (−) (↔) |α |2 − |β |2 κ 2 u∇⊥ u∗ − c.c. + 2κ 2 Re (αβ ∗ ) u∇⊥ u∗ − c.c φ
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φ
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12708
= =
$ m # m 2i α 2 − β 2 κ 2 a2 (ρ ) cos 2φ + 4iαβ κ 2 a2 (ρ ) sin 2φ ρ ρ $ ) (# m 2iκ 2 a2 (ρ ) α 2 − β 2 cos 2φ + 2αβ sin 2φ . ρ
(27)
To further simplify this, we note that the azimuthal angle of the polarization φ0 is determined by α , β via Eq. (12), i.e., (28) cos φ0 = α , sin φ0 = β , from which it follows
α2 − β 2 2αβ
=
cos2 φ0 − sin2 φ0 = cos 2φ0 ,
=
2 sin φ0 cos φ0 = sin 2φ0 .
(29)
Inserting Eq. (29) into Eq. (27) yields (−) (−) (↔) |α |2 − |β |2 κ 2 u∇⊥ u∗ − u∗ ∇⊥ u + 2κ 2 Re (αβ ∗ ) u∇⊥ u∗ − c.c φ
=
2iκ
2m 2
ρ
a (ρ ) [cos 2φ0 cos 2φ + sin 2φ0 sin 2φ ] = 2iκ
φ
2m 2
ρ
a (ρ ) cos 2 (φ − φ0 ) . (30)
We then turn to the non-paraxial terms (each containing a high-order derivative or more than one first-order derivatives) m ∂u ∂ u∗ ∂ u∗ 2 ∂u | | |α |2 ∇⊥ − c.c. + β ∇⊥ − c.c. = 2i 2 a (ρ ) a (ρ ) ∂y ∂y ∂ x ∂ x ρ φ φ ) ) ( ( m3 m −2i a2 (ρ ) α 2 sin2 φ + β 2 cos2 φ − 2i 3 a2 (ρ ) α 2 cos2 φ + β 2 sin2 φ ρ ρ and
∂ u∗ ∂ u∗ ∂u ∂u ∇⊥ + α ∗β ∇⊥ − c.c. αβ ∗ ∂x ∂y ∂y ∂x φ =
m m3 2i a2 (ρ ) [2αβ sin φ cos φ ] − 2i 3 a2 (ρ ) [2αβ cos φ sin φ ] . ρ ρ
By combining these two equations and applying Eq. (28), we have ∂u ∂ u∗ ∂u ∂ u∗ | α |2 ∇⊥ − c.c. + |β |2 ∇⊥ − c.c. ∂y ∂y ∂x ∂x φ φ ∗ ∗ ∂ u ∂ u u ∂ u ∂ ∇⊥ + α ∗β ∇⊥ − c.c. + αβ ∗ ∂x ∂y ∂y ∂x φ 3 m 2 m m = −2i a (ρ ) [α sin φ − β cos φ ]2 + 3 a2 (ρ ) [α cos φ + β sin φ ]2 − 2 a (ρ ) a (ρ ) ρ ρ ρ 3 m 2 m 2 m 2 2 = −2i a (ρ ) sin (φ − φ0 ) + 3 a (ρ ) cos (φ − φ0 ) − 2 a (ρ ) a (ρ ) . (31) ρ ρ ρ Finally, we substitute Eqs. (30) and (31) into Eq. (26) to achieve Eq. (11), i.e., −κ 2 a2 (ρ ) cos 2 (φ − φ0 ) mε0 ω e−2κ z 2 2 . pφ = a (ρ ) cos2 (φ − φ0 ) − ρ1 a (ρ ) a (ρ ) + a2 (ρ ) sin2 (φ − φ0 ) + m 2k2 ρ ρ2 #233857 - $15.00 USD © 2015 OSA
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12709
II. We now present a detailed derivation of Eq. (21), where the energy density of the monochromatic electromagnetic field is [see Eq. (18)] w=
ε0 1 |B (r)|2 . |E (r)|2 + 4 4μ0
(32)
For clarity, we will treat the magnetic (∝ |B (r)|2 ) and the electric (∝ |E (r)|2 ) parts of the energy separately and combine the results to obtain Eq. (21). With the magnetic field B given in Eq. (5), it is straightforward to show 2 |Bx |2 + By = e−2κ z κ 2 |u|2 = e−2κ z κ 2 a2 (ρ ) ,
(33)
and |Bz |
2
=
∂u ∂ u 2 β ∂ x − α ∂ y
−2κ z
e
−2κ z
=
e
=
e−2κ z
m2 a (ρ ) [β cos φ − α sin φ ] + 2 a2 (ρ ) [β sin φ + α cos φ ]2 ρ ! 2 m a2 (ρ ) sin2 (φ − φ0 ) + 2 a2 (ρ ) cos2 (φ − φ0 ) , ρ 2
!
2
(34)
where Eq. (28) has been taken into account. Thus we have the magnetic field energy as ∞
2π ∞ 2 1 1 |B|2 ≡ dφ d ρ ρ |Bx |2 + By + |Bz |2 4μ0 4μ0 0 −∞ 0 1 1 m2 2 πε0 ω 2 e−2κ z ∞ 2 2 2 d ρ ρ κ a (ρ ) + a (ρ ) + a (ρ ) . 2k2 2 2 ρ2 0
dxdy
=
(35)
To calculate the electric field energy, we use the second equation in (4) to write |E|2 =
ω2 c4 2 |∇ = (∇ × B) · (∇ × B∗ ) . × B| ω2 k4
(36)
Setting b = ∇ × B, a = B∗ in the formula b · (∇ × a) = ∇ · (a × b) + a · (∇ × b) , we get (∇ × B) · (∇ × B∗ ) = ∇ · [B∗ × (∇ × B)] + B∗ · (∇ × [∇ × B]) ( ) = ∇ · [B∗ × (∇ × B)] + B∗ · ∇ (∇ · B) − ∇2 B = ∇ · [B∗ × (∇ × B)] + k2 |B|2 , where ∇ · B = 0 and ∇2 B + k2 B = 0 (from Maxwell’s equations) have been applied. Further noting the symmetry between B and B∗ in the above equation, one has (∇ × B) · (∇ × B∗ ) =
1 {∇ · [B∗ × (∇ × B)] + c.c.} + k2 |B|2 . 2
With ∇ · [B∗ × (∇ × B)] = ∇⊥ · [B∗ × (∇ × B)] +
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(37)
∂ ∗ [B × (∇ × B)]z , ∂z
Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12710
and
= =
∂ ∗ [B × (∇ × B)]z + c.c. ∂z ∂ By ∂ ∂ Bx ∂ Bz ∂ Bz B∗x − B∗x − B∗y − B∗y + c.c. ∂z ∂z ∂x ∂y ∂z 2 # $ # $ # $ ∂ 2 |Bx |2 + By − |Bz |2 ∂ Bx B∗z ∂ By B∗z ∂ ∂ (B∗x Bz ) ∂ B∗y Bz + + + , − ∂ z2 ∂z ∂x ∂y ∂x ∂y
Eq. (37) becomes 1 ∂2 2 2 2 |B |B | | (∇ × B) · (∇ × B∗ ) = k2 |B|2 + + B − x y z 2 ∂ z2 # ∗ $ # $ # $ ∗ ∂ Bx B∗z ∂ By B∗z 1 ∂ ∂ (Bx Bz ) ∂ By Bz + + + − 2 ∂z ∂x ∂y ∂x ∂y 1 + {∇⊥ · [B∗ × (∇ × B)] + c.c} . 2
(38)
With Eqs. (33) - (36), (38), and noting that the integrals (over x and y) of the second and third lines in Eq. (38) vanish, we achieve the electric field energy ∞
∞ ε0 2 ε0 ω 2 |E| = dxdy (∇ × B) · (∇ × B∗ ) 4 4k4 −∞ −∞ ∞ ε0 ω 2 2 1 ε0 ω 2 ∂ 2 2 2 2 |B| + |Bx | + By − |Bz | = dxdy 4k2 2 4k4 ∂ z2 −∞ 1 2 1 m2 2 πε0 ω 2 e−2κ z ∞ 2 2 d ρ ρ κ a (ρ ) + a (ρ ) + a (ρ ) = 2k2 2 2 ρ2 0 πε0 ω 2 e−2κ z κ 2 ∞ 1 2 1 m2 2 2 2 d ρ ρ κ a (ρ ) − a (ρ ) − a (ρ ) . + k2 k2 0 2 2 ρ2
dxdy
(39)
The total electromagnetic energy E is the sum of the magnetic part in Eq. (35) and the electric part in Eq. (39) ! πε0 ω 2 e−2κ z ∞ κ2 κ2 1 m2 2 2 2 2 1− 2 1 + 2 κ ρ a (ρ ) + dρ ρ a (ρ ) + a (ρ ) , E = k2 k 2 k ρ 0 i.e., Eq. (21). Acknowledgments We gratefully acknowledge financial supports from the National Natural Science Foundation of China (No. 11375081), the Shandong Provincial Natural Science Foundation (No. ZR2013AL007), and the Start-up Fund of Liaocheng University.
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Received 4 Feb 2015; revised 2 Apr 2015; accepted 6 Apr 2015; published 6 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.012700 | OPTICS EXPRESS 12711
