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Communications in Physics, Vol. 27, No. 1 (2017), pp. 23-35 DOI:10.15625/0868-3166/27/1/9240 Invited Paper CHIRALITY OF LIGHT IN HYBRID MODES OF VACUUM-CLAD ULTRATHIN OPTICAL FIBERS FAM LE KIEN1,† , TH. BUSCH2 , VIET GIANG TRUONG3 and SÍLE NIC CHORMAIC3,4 1 Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan 2 Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan 3 Light-Matter Interactions Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan 4 School of Chemistry and Physics, University of KwaZulu-Natal, Durban, KwaZulu-Natal, 4001, South Africa † E-mail: kien pham@hotmail.com Received 21 February 2017 Accepted for publication 9 March 2017 Abstract. We investigate chirality of light in the quasicircularly polarized fundamental and higherorder hybrid modes of vacuum-clad ultrathin optical fibers. We show that, for a given fiber with the parameters in the range of experimental interest, the higher-order modes have smaller optical chirality per unit energy than the fundamental mode. The sign of the chirality per unit energy of a HE or EH mode is the same as or opposite to, respectively, the sign of the phase circulation direction. When the fiber radius is large enough, the field in a quasicircularly polarized hybrid mode can be superchiral outside the fiber. In particular, outside a fiber with a radius of 600 nm, the fields in the quasicircularly polarized HE11 , HE21 , and HE31 modes are superchiral. Keywords: chirality, higher orders, hybrid modes, ultrathin optical fibers. Classification numbers: 42.81.Qb, 42.25.Ja, 42.50.Tx . I. INTRODUCTION Guided fields of vacuum-clad ultrathin optical fibers [1–3] penetrate deeply into the surrounding medium and appear as an evanescent wave carrying a significant fraction of the power c 2017 Vietnam Academy of Science and Technology 24 CHIRALITY OF LIGHT IN HYBRID MODES OF VACUUM-CLAD ULTRATHIN OPTICAL FIBERS and having a complex polarization pattern [4–6]. Vacuum-clad ultrathin optical fibers have a wide range of potential practical applications [7, 8]. They have been used for trapping atoms [9–11], for probing atoms [12–19], molecules [20], quantum dots [21], and color centers in nanodiamonds [22, 23], and for mechanical manipulation of small particles [24–28]. An interesting property of guided fields of vacuum-clad ultrathin optical fibers is that they are, in general, chiral. The chirality of an object means that this object cannot be superimposed on its mirror image. The interaction between a chiral light field with a chiral emitter can lead to directional effects [29]. In particular, spontaneous emission and scattering from an atom with a circular dipole near a nanofiber can be asymmetric with respect to the opposite axial propagation directions [30–34]. These directional effects are optical chiral effects and the signature of spin-orbit coupling of light [35–41]. They are due to the existence of a nonzero longitudinal component of the nanofiber-guided field, which oscillates in phase quadrature with respect to the radial transverse component. Circularly polarized light is the simplest example of chiral light. The chirality of photons in circularly polarized light originates from their spin. It is well known that the spin of photons in circularly polarized light is maximum. This led to a common belief that circularly polarized light must have the maximum chirality. However, this belief is wrong. Although chirality and spin are related to each other, they are different characteristics of light. Tang and Cohen have introduced a chirality measure for the interaction between chiral light and chiral molecules with mixed electricmagnetic dipole polarizability. They have shown that the local values of this measure of chirality at the nodes of a circularly polarized standing wave can be larger than for circularly polarized light [42, 43]. They termed such light as being superchiral. A disadvantage of the superchiral light construction scheme of Refs. [42, 43] is that the superchirality appears at the nodes of a standing wave, where the field intensity and hence the molecular excitation rate are small. The purpose of this work is to study chirality of light in hybrid modes of vacuum-clad ultrathin optical fibers. We calculate, analytically and numerically, the chirality measure introduced by Tang and Cohen [42, 43]. The paper is organized as follows. In Sec. II we briefly review the basic properties of hybrid modes of optical fibers. In Sec. III we calculate the chirality of guided light and our conclusions are given in Sec. IV. II. HYBRID MODES OF OPTICAL FIBERS In this section, we review the basic properties of hybrid modes of optical fibers. Consider the model of a step-index fiber that is a dielectric cylinder of radius a and refractive index n1 , surrounded by an infinite background medium of refractive index n2 , where n2 < n1 . We use Cartesian coordinates {x, y, z}, where z is the coordinate along the fiber axis. We also use cylindrical coordinates {r, ϕ, z}, where r and ϕ are the polar coordinates in the fiber transverse plane xy. For a guided light field of frequency ω (free-space wavelength λ = 2πc/ω and free-space wave number k = ω/c), the propagation constant β is determined by the fiber eigenvalue equation [44]. This eigenvalue equation leads to hybrid HE and EH modes, and to TE and TM modes [44]. We do not consider TE and TM modes in this paper because the integral chirality of light in these modes and the corresponding chirality density are zero. FAM LE KIEN et al. 25 Hybrid HE and EH modes are labeled as HElm and EHlm , where l = 1, 2, . . . and m = 1, 2, . . . are the azimuthal and radial mode orders, respectively. The azimuthal mode order l determines the helical phasefront and the associated phase gradient in the fiber transverse plane. The radial mode order m implies that the HElm or EHlm mode is the m-th solution to the corresponding eigenvalue equation. We examine the mode functions of hybrid modes [44]. We present the electric and magnetic components of the field in the form E = (E e−iωt + c.c.)/2 and H = (H e−iωt + c.c.)/2, where E and H are the envelopes. The spatial functions E and H obey the Helmholtz equation and are called the mode functions. For a guided mode with a propagation constant β and an azimuthal mode order l, we can write E = eeiβ z+ilϕ and H = heiβ z+ilϕ . Here, e and h are the mode profile functions of the electric and magnetic components of the field, and β and l can have positive and negative values. The explicit expressions for the mode profile functions e and h can be found in Refs. [44, 45]. For convenience, we use the notations β > 0 for the propagation constant and l > 0 for the azimuthal mode order. We introduce the index f = +1 or −1 (or simply f = + or −) for the positive or negative propagation direction, which indicates that the propagation phase factor is eiβ z or e−iβ z . We also introduce the index p = +1 or −1 (or simply p = + or −) for the counterclockwise or clockwise phase circulation direction, which indicates that the azimuthal phase factor is eilϕ or e−ilϕ . We label quasicircularly polarized hybrid modes by the mode index µ = ( f l p). In cylindrical coordinates, the mode profile functions e( f l p) (r) and h( f l p) (r) of the electric and magnetic components of quasicircularly polarized hybrid modes are given by e( f l p) = r̂er + pϕ̂eϕ + f ẑez , h( f l p) = f pr̂hr + f ϕ̂hϕ + pẑhz . (1) Here, the notations r̂ = x̂ cos ϕ + ŷ sin ϕ, ϕ̂ = −x̂ sin ϕ + ŷ cos ϕ, and ẑ stand for the unit basis vectors of the cylindrical coordinate system {r, ϕ, z}, with x̂ and ŷ being the unit basis vectors of the Cartesian coordinate system for the fiber transverse plane xy. The notations er , eϕ , and ez and the notations hr , hϕ , and hz stand for the cylindrical-coordinate components of the electric and magnetic mode functions e and h for the corresponding hybrid mode with the positive propagation direction f = + and the counterclockwise phase circulation direction p = +. Note that e and h depend on the type (HE or EH) of the mode and on the mode orders l and m. An important property of the mode functions is that the longitudinal components ez and hz are nonvanishing and are in quadrature (π/2 out of phase) with the radial components er and hr , respectively. In addition, the azimuthal components eϕ and hϕ are also nonvanishing and are in quadrature with the radial components er and hr , respectively. Note that the full mode functions for quasicircularly polarized hybrid modes are given by E ( f l p) = e( f l p) ei f β z+iplϕ , H ( f l p) = h( f l p) ei f β z+iplϕ . (2) We are interested in ultrathin fibers, whose diameters are on the order of a micron. Such fibers can support not only the fundamental HE11 mode but also several higher-order modes in the optical region. We note that the excitation of higher-order modes has been studied [46, 47]. The production of ultrathin fibers for higher-order mode propagation with high transmission has been 26 CHIRALITY OF LIGHT IN HYBRID MODES OF VACUUM-CLAD ULTRATHIN OPTICAL FIBERS demonstrated [48–50]. Experimental studies on the interaction between higher-order modes and atoms [19] or particles [27, 28] have been reported. III. CHIRALITY According to [42], the cycle-averaged optical chirality density of a monochromatic field can be characterized by the quantity ρ chir = cε0 n2 cµ0 Re[E ∗ · (∇ × E )] + Re[H ∗ · (∇ × H )]. 4ω 4ω (3) Here, n(r) = n1 for r < a and n2 for r > a. The quantity ρ chir has been studied in Refs. [42, 43, 51–54]. When we use the Maxwell equations ∇ × E = iω µ0 H and ∇ × H = −iωε0 n2 E for the fields in the frequency domain, we obtain [53] n2 Im[E · H ∗ ]. (4) 2c For TE and TM guided modes, the chirality density (4) is zero. The optical chirality density defined by Eq. (4) can be measured from the asymmetry in the rates of excitation between a small chiral molecule and its mirror image [42, 43]. According to Ref. [55], there is no single measure of chirality and, for any measure, there exists a chiral body for which the measure is zero. The chirality of a light beam is closely related to the optical helicity. Indeed, the cycleaveraged optical helicity density of a monochromatic field is given by [51, 56–60] ρ chir = ε0 n2 1 Re(A · H ∗ ) − Re(C · E ∗ ), (5) 4c 4 where A and C are the positive frequency components of the magnetic and electric vector potentials, respectively. With the help of the relations A = E /iω and n2 C = µ0 cH /iω between the vector potentials and the fields in the frequency domain, we obtain [51, 56–60] ρ hlcy = ρ hlcy = 1 Im(E · H ∗ ). 2cω (6) It is clear that ρ chir = n2 ωρ hlcy . Thus, in the frequency domain, the chirality density is proportional to the helicity density. The proportionality factor is n2 ω. For quasicircularly polarized hybrid modes, we find from Eq. (4) the following expression for the chirality density: n2 Im(er h∗r + eϕ h∗ϕ + ez h∗z ). 2c For guided light, the optical chirality per unit length is ρ chir = f p J chir = Z ρ chir dr, (7) (8) where dr = 02π dϕ 0∞ r dr. It is clear from Eqs. (7) and (8) that the sign of chirality per unit length changes when the propagation direction f or the phase circulation direction p is reversed. R R R FAM LE KIEN et al. 27 We calculate analytically the chirality per unit length (8). For this purpose, we use the explicit expressions for the mode functions e and h [44, 45]. When we insert these expressions into Eq. (7) and calculate the integral (8), we obtain the explicit expression J chir = πa2 ε0 n21 β 2 2 2 {n1 k (1 − s)(1 − s1 )[Jl−1 (ha) − Jl−2 (ha)Jl (ha)] 4h2 k 2 − n21 k2 (1 + s)(1 + s1 )[Jl+1 (ha) − Jl+2 (ha)Jl (ha)] f p|A|2 − 2h2 s[Jl2 (ha) − Jl+1 (ha)Jl−1 (ha)]} πa2 ε0 n22 β Jl2 (ha) 2 2 2 + f p|A|2 {n k (1 − s)(1 − s2 )[Kl−2 (qa)Kl (qa) − Kl−1 (qa)] 4q2 k Kl2 (qa) 2 2 − n22 k2 (1 + s)(1 + s2 )[Kl+2 (qa)Kl (qa) − Kl+1 (qa)] − 2q2 s[Kl+1 (qa)Kl−1 (qa) − Kl2 (qa)]}. (9) Here, the parameters h = (n21 k2 − β 2 )1/2 and q = (β 2 − n22 k2 )1/2 characterize the scales of the spatial variations of the field inside and outside the fiber, respectively. The parameters s, s1 , and s2 are given as  s=l 1 1 + h2 a2 q2 a2 s1 = β2 s, k2 n21 s2 = β2 s. k2 n22  K 0 (qa) Jl0 (ha) + l haJl (ha) qaKl (qa) −1 (10) The coefficient A is a constant that can be determined from the propagating power of the field. The optical chirality per unit length can be normalized to the energy per unit length, which is given by U= ε0 4 Z n2 |E |2 dr + µ0 4 Z |H |2 dr. (11) The first and second terms on the right-hand side of expression (11) correspond to the electric and magnetic parts of the energy of the field, respectively. For guided modes, these parts are equal to each other. We have U = Uin +Uout , where Uin and Uout are the energies per unit length inside and outside the fiber and are given as [44, 45] 2 πa Uin = |A| 2 ε n2 0 1  1 2 [β 2 (1 − s)2 + n21 k2 (1 − s1 )2 ][Jl−1 (ha) − Jl−2 (ha)Jl (ha)] 2h2 4 1 2 + 2 [β 2 (1 + s)2 + n21 k2 (1 + s1 )2 ][Jl+1 (ha) − Jl+2 (ha)Jl (ha)] 2h    β 2 s2 2 + 1 + 2 2 [Jl (ha) − Jl−1 (ha)Jl+1 (ha)] n1 k (12) 28 CHIRALITY OF LIGHT IN HYBRID MODES OF VACUUM-CLAD ULTRATHIN OPTICAL FIBERS and 2 πa Uout = |A| 2 ε n2 0 2 4  Jl2 (ha) 1 2 [β 2 (1 − s)2 + n22 k2 (1 − s2 )2 ][Kl−2 (qa)Kl (qa) − Kl−1 (qa)] 2 Kl (qa) 2q2 j chir 1 2 + 2 [β 2 (1 + s)2 + n22 k2 (1 + s2 )2 ][Kl+2 (qa)Kl (qa) − Kl+1 (qa)] 2q    β 2 s2 2 + 1 + 2 2 [Kl−1 (qa)Kl+1 (qa) − Kl (qa)] . n2 k HE11 HE21 HE12 HE31 HE41 HE22 HE51 (13) EH11 EH21 EH31 a (nm) Fig. 1. (Color online) Chirality per unit energy jchir as a function of the fiber radius a for quasicircularly polarized hybrid modes with the positive propagation direction f = + and the positive phase circulation direction p = +. The refractive indices of the fiber and the surrounding medium are n1 = 1.4537 and n2 = 1, respectively. The wavelength of light is λ = 780 nm. The vertical dotted lines indicate the positions of the cutoffs for higher-order modes. The horizontal dotted line separates the positive and negative sides of the vertical axis. We introduce the notation jchir = J chir /U for the chirality per unit energy of guided light. We show jchir as a function of the fiber radius a in Fig. 1. We see from the figure that, for the positive propagation direction f = + and the positive phase circulation direction p = +, the chirality per unit energy jchir is positive for HE and negative for EH modes. We note that the chirality per unit energy of a circularly polarized light field in a dielectric medium of refractive index n is equal to n. We observe that the absolute values of jchir obtained for the guided modes do not exceed the refractive index n1 of the fiber. FAM LE KIEN et al. 29 HE11 HE21 HE31 ηchir n1 n2 Radial distance r (nm) Fig. 2. Chirality factor η chir as a function of the radial distance r for the counterclockwise quasicircularly polarized HE11 (red line), HE21 (green line), and HE31 (blue line) modes. The fiber radius is a = 600 nm. Other parameters are as for Fig. 1. The vertical dotted line indicates the position of the fiber surface. The horizontal dotted lines indicate the refractive index values n1 = 1.4537 and n2 = 1. According to [42,43], the asymmetry in the excitation rates between a small chiral molecule and its mirror image is proportional to the factor η chir = ρ chir , 2ue (14) where ε0 n2 2 ε0 n2 |E | = (|er |2 + |eϕ |2 + |ez |2 ) (15) 4 4 is the local electric energy density of the field. Note that the value of the factor η chir for a counterclockwise or clockwise circularly polarized field in a medium with refractive index n is equal to n or −n. According to [42, 43], a light field for which the absolute value of the chirality factor η chir is larger than the refractive index n in a spatial region is called a superchiral field. We plot in Fig. 2 the factor η chir as a function of the radial distance r for the quasicircularly polarized HE11 , HE21 , and HE31 modes. We observe from the figure that the values of η chir inside and outside the fiber are very different from each other. We see that η chir is discontinuous at the position of the fiber surface. This discontinuity results from the boundary condition for the normal (radial) component er of the electric field. Since the difference between the refractive indices n1 of the silica core and n2 of the vacuum cladding is large, the discontinuity of η chir at the fiber surface is dramatic. We observe from the figure that η chir is positive. Comparison between the colored curves of Fig. 2 shows that, outside the fiber, η chir is larger for the HE11 mode (red curve) than for ue = 30 CHIRALITY OF LIGHT IN HYBRID MODES OF VACUUM-CLAD ULTRATHIN OPTICAL FIBERS the higher-order HE21 (green curve) and HE31 (blue curve) modes. An interesting feature is that, outside the fiber, the values of η chir for the HE11 , HE21 , and HE31 modes are larger than unity. Although the enhancement of η chir is significant, it is limited by the magnitude of the fiber radius and the difference between the refractive indices of the fiber and the surrounding medium. Thus, for an ultrathin fiber with a radius of 600 nm, superchirality of light can be obtained for the quasicircularly polarized HE11 , HE21 , and HE31 modes. Such superchirality appears as a consequence of the complex polarization structure of the guided field. The total internal reflection of light from the fiber surface is the physical mechanism that leads to this superchirality. This mechanism is different from the interference between two counterpropagating circularly polarized plane waves with opposite handedness, equal frequency, and slightly different intensity, which was used by Tang and Cohen to produce a standing-wave superchiral light field [42,43]. An advantage of the waveguide scheme is that superchirality of the HE11 , HE21 , and HE31 modes can occur everywhere in the evanescent region of the field (outside the fiber). In contrast, superchirality of the field constructed in the standing-wave scheme is limited to the nodes, where the intensity is small. One can increase the enhancement of the chirality factor in the waveguide scheme by increasing the fiber radius and by using a material with a higher refractive index. HE12 n1 ηchir n2 Radial distance r (nm) Fig. 3. Chirality factor η chir as a function of the radial distance r for the counterclockwise quasicircularly polarized higher-order HE12 mode. The parameters used are the same as for Fig. 2. The vertical dotted line indicates the position of the fiber surface. The horizontal dotted lines indicate the refractive index values n1 = 1.4537 and n2 = 1. In Fig. 3, we plot η chir as a function of r for the quasicircularly polarized HE12 mode. We see from the figure that η chir is positive. Outside the fiber, this factor is smaller than unity. In Fig. 4, we plot η chir as a function of r for the quasicircularly polarized EH11 mode. We see from the figure that η chir is positive in a limited region containing the fiber center, but is negative everywhere else. This behavior is consistent with the fact that the chirality per unit energy FAM LE KIEN et al. 31 ηchir EH11 -n2 -n1 Radial distance r (nm) Fig. 4. Chirality factor η chir as a function of the radial distance r for the counterclockwise quasicircularly polarized higher-order EH11 mode. The parameters used are the same as for Fig. 2. The vertical dotted line indicates the position of the fiber surface. The horizontal dotted lines indicate the values −n1 = −1.4537 and −n2 = −1. jchir is negative for EH modes (see Fig. 1). Outside the fiber, η chir is negative, and its absolute value is smaller than unity. Thus, according to Figs. 3 and 4, for a fiber with a radius of 600 nm, the fields in the quasicircularly polarized HE12 and EH11 modes are not superchiral outside the fiber. However, these field are superchiral in some regions inside the fiber. Furthermore, additional calculations show that, when the fiber radius is large enough, the fields in the HE12 and EH11 modes can become superchiral outside the fiber (see also Fig. 5). We see from Figs. 2–4 that, in the limit of large radial distances, the chirality factor η chir slowly varies and tends to a finite value η∞chir , which depends on the mode type and the azimuthal and radial mode orders. We confirm this by performing a simple analysis below. In the limit r → ∞, we have er eϕ ez r β Jl (ha) π −qr = iA e , q Kl (qa) 2qr r β s Jl (ha) π −qr = A e , q Kl (qa) 2qr r Jl (ha) π −qr = A e , Kl (qa) 2qr (16) 32 CHIRALITY OF LIGHT IN HYBRID MODES OF VACUUM-CLAD ULTRATHIN OPTICAL FIBERS and hr hϕ hz r ωε0 n22 s2 Jl (ha) π −qr = −A e , q Kl (qa) 2qr r ωε0 n22 Jl (ha) π −qr = iA e , q Kl (qa) 2qr r β s Jl (ha) π −qr = iA e . ω µ0 Kl (qa) 2qr (17) Then, Eqs. (7) and (15) yield the asymptotic expressions ρ chir = − f p|A|2 ε0 n22 3 Jl2 (ha) π −2qr β s 2 e kq2 Kl (qa) 2qr (18) and ue = |A|2 2 ε0 n22 2 2 2 Jl (ha) π −2qr [q + β (1 + s )] e , 4q2 Kl2 (qa) 2qr (19) which lead to η∞chir = −2 f p β 3s . k[q2 + β 2 (1 + s2 )] HE11 HE21 (20) HE31 ηchir ∞ n2 HE12 EH11 -n2 Fiber radius a (nm) Fig. 5. Dependence of the limiting value η∞chir of the chirality factor η chir on the fiber radius a for the counterclockwise quasicircularly polarized HE11 , HE21 , HE31 , HE12 , and EH11 modes. All other parameters are as for Fig. 1. The horizontal dotted lines indicate the values n2 = 1 and −n2 = −1.