May 1, 2014 / Vol. 39, No. 9 / OPTICS LETTERS
2823
Resonant optical gun A. V. Maslov* and M. I. Bakunov University of Nizhny Novgorod, Nizhny Novgorod, Russia *Corresponding author: [email protected] Received March 6, 2014; revised April 6, 2014; accepted April 6, 2014; posted April 7, 2014 (Doc. ID 206911); published April 30, 2014 We propose a concept of a structure—a resonant optical gun—to realize an efficient propulsion of dielectric microparticles by light forces. The structure is based on a waveguide in which a reversal of the electromagnetic momentum flow of the incident mode is realized by exciting a whispering gallery resonance in the microparticle. The propelling force can reach the value up to the theoretical maximum of twice the momentum flow of the initial wave. The force density oscillates along the particle periphery and has very large amplitude. © 2014 Optical Society of America OCIS codes: (140.7010) Laser trapping; (350.4855) Optical tweezers or optical manipulation. http://dx.doi.org/10.1364/OL.39.002823
Recently, it was experimentally demonstrated that the optical force created by a tapered-fiber mode on dielectric spheres with diameters 10–20 μm has strong resonances [1]. This behavior is characterized by high frequency (or size) selectivity and very large peak values. The large values can be explained by an efficient transformation of the momentum flow of the incident fiber mode to the propelling force [1,2]. The transformation is accompanied by the excitation of high-quality (Q) whispering gallery modes (WGMs) in the particles by the evanescent field of the mode. In contrast, small dielectric particles, in which WGMs do not exist, do not provide any resonant enhancement even with evanescent excitation [3]. Plane waves also give a much lower force enhancement due to the weak coupling to WGMs [4]. Here we extend work [1,2] and show that a further enhancement of force is possible. The proposed geometry of the structure—a resonant optical gun (ROG)—to realize the enhancement and its operation principle are illustrated in Fig. 1. In a ROG, a dielectric particle is placed inside a waveguide [with parallel plates as shown in Fig. 1(a) or cylindrical as shown in Fig. 1(b)] that can support surface waves. The theoretical analysis of Ref. [2] showed that the excitation of WGMs by a surface wave (guided by one interface) can create the propelling force approximately equal to the momentum flow of the incident mode. This agrees with the propulsion velocity measurements [1] from which the propelling force was estimated to reach the value comparable to the incident momentum flow. The excitation is accompanied by the creation of the scattered bulk radiation with anisotropic angular distribution. Although this bulk radiation carries significant power (up to 100% at resonance), its momentum flow along the incidence direction is rather small. To enhance the force, one can think of collecting the scattered radiation and redirecting it backward to provide a larger change of the incident momentum and, therefore, a larger propelling force. This can intuitively be accomplished by using a symmetric structure shown in Fig. 1(a) in which the excited WGM creates guided reflection. By the conservation of the momentum flow, the particle should experience a larger force as compared to the case when one interface is present. Instead of using the surface wave guided by one interface, one can launch symmetric or antisymmetric modes of the structure. 0146-9592/14/092823-04$15.00/0
The reflection that follows will create a large propelling force on the microparticle. This principle can be used to create forces on the particles not only in parallel plate waveguides [Fig. 1(a)] but in cylindrical structures as well [Fig. 1(b)]. The surface modes of the cylindrical structure may have various angular dependences. The ROG structure is similar to channel dropping filters in which the signal from one waveguide is transformed to the other waveguide by coupling them to a microring resonator [5]. The optical filter synthesis can be extended to find the forces [6]. The acceleration of nanoparticles was proposed in a V-groove metallic structure in which a particle with negative real part of polarizability will be pushed out by the strong field gradient [7]. In a ROG, the propulsion is achieved through the reversal of the incident momentum flow rather than through the field gradient. Recently, the exploration of various possible forces on the particles using waveguiding structure has become of significant interest. For example, pulling optical forces can be created using multimodal waveguide [8] or backward wave structures [9,10]. The propulsion of micrometer-sized particles inside a hollow core [11] and photonic crystal fibers [12] can also be realized. The excitation of WGMs of a microsphere integrated inside a microstructured optical fiber was recently demonstrated [13], but the optical forces were not studied. Similar structures can be used to realize the propulsion with surface modes illustrated in Fig. 1(b). In this Letter, we demonstrate the resonant enhancement of the propelling force using a rigorous analytical theory and simulations. We show that the evanescent nature of the waves plays a key role in the efficient excitation of WGMs that leads to the force enhancement.
(a)
y
L
R
εb
z
εs
(b) F
ϕ F
d
x
εp Fig. 1. (a) Geometry and operation principle of a ROG with parallel plate waveguide. (b) Cross section of possible 3D ROG realization. © 2014 Optical Society of America
OPTICS LETTERS / Vol. 39, No. 9 / May 1, 2014
0
eihx−x r~ Ay; y0 dh ; gb h 1 − r~ 2 −∞ ∞
(b)
1.5 h/k
s a
s0 a0 s1 a1
1 0.5 0 0
5
10 15 20 25 30 35 40 kL
Ns 5
(b)
0 1 0.8 0.6 0.4
s0 s1
0.2 (c)
0 1 0.8 0.6
s0 s1
0.4 0.2
(d)
0 1 0.8
s0
0.6
s1
0.4 0.2
1
0.5 y/L
2
15 10
(1)
p with gb εb ω2 ∕c2 − h2 , r~ r bp eigb L , Ay; y0 0 and By; y0 eigb y eigb y By; y0 B−y; −y0 , −ig y r~ e b ; r bp is the reflection coefficient from the wall. For high-Q structures, the analytical procedure can be more reliable than the finite-difference time domain method [16]. Figure 2 shows the dispersion properties and typical mode profiles for the parallel plate ROG structure. It (a)
(a)
Pr /P0
0
The scattering characteristics of s0 and s1 as functions of particle size kR at fixed kd 1.5 are shown in Fig. 3. We
+ Psc /P0
0
j≠i
s1
(e)
0 1 0.8
0 a0
-0.5
s0
-
i G x; y; x ; y 4π 1
Z
supports symmetric and antisymmetric with respect to H z y modes. Modes s0 and a0 tend to be confined to the interfaces. They are practically degenerate for kL > 5. The other modes propagate in the region jyj < p L∕2 with phase index n c∕vph h∕k < εb . The number of the propagating modes increases with kL. The cutoff width Lm for a symmetric propagating mode of order p m is determined by kLm 2mπ − 2 arctan εb ∕jεp j. We compare propulsion in two cases: excitation by evanescent mode s0 and propagating mode s1. The scattering of an incident mode with power P 0 results in the excitation of other modes. Let us define P j as the power carried by the mode of order j in the x direction. If i is the order of the incident mode, we define the reflected P r and transmitted P t powers in the same mode and scattered P sc powers using X Pt P P r P −i ; P P (2) sc j : i ;
Pt /P0
We show that the force density has a strongly oscillatory behavior and only the regions near the waveguiding surface give significant contributions to the total force. We adopt a 2D model shown in Fig. 1(a). Twodimensional models are common in studying the interaction of resonators with waveguides since they capture the main physical processes [8,14,15]. The particle that scatters the incident mode is characterized by its radius R and dielectric constant εs and separated from the walls by distance d. The dielectric constant of the background is εb and that of the metallic walls (with plasma-like properties) is εp . We choose εp −2 and transverse magnetic polarization topallow surface waveguiding. We also take εb 1 and εs 1.4. We first solve the scattering problem and then use the Lorentz formula and the balance of the momentum flow in the system to find the force. The scattered fields outside of the cylinder are represented as produced by some effective magnetic current on the surface. The current is expanded into the angular exponential functions with unknown coefficients. The fields inside are expanded into the cylindrical functions. Matching the fields at the boundary of the cylinder gives a set of equations for the expansion coefficients. The case of a particle near a single interface is described in Ref. [2]. To extend this approach to the structure with two interfaces we replace the Green’s function given by Eq. (18b) of Ref. [2] with that for a δ-source emitting in the region jyj < L∕2. Expanding the source into the Fourier integral reduces the problem to finding the field produced by a plane wave bouncing between the walls. Taking the inverse transform gives the required G1 x; y; x0 ; y0 as
Psc /P0
2824
s0 s1
0.6 0.4 0.2
-1 -1
0 1 2 Hz(arb. units)
Fig. 2. (a) Propagation wavenumber h (normalized to k ω∕c) as a function of the size parameter kL for the modes of the waveguide shown in Fig. 1(a) with εb 1 and εp −2. The modes are labeled according to their symmetry (s, symmetric; a, antisymmetric) and order. (b) Distribution of the magnetic field H z y for s0, a0, and s1 at kL 10.
0 0
5
10
15
20
25
30
35
40
kR
Fig. 3. Dependence of the scattering characteristics of ROG on size kR when the incident mode is s0 or s1. (a) Number of guided symmetric modes. (b) Transmitted power P t ∕P 0 , and (c) reflected power P r ∕P 0 , p (d) and (e) scattered powers εs 1.4, and kd 1.5. P sc ∕P 0 for εb 1, εp −2,
confirmed that the energy balance holds. Due to symmetry, the scattering can produce only symmetric modes. The number of modes grows with kR since L 2R d. The scattering of s0 is characterized by the appearance of periodic narrow dips in the transmission spectrum with corresponding peaks in the reflection and scattering spectra. These peaks are due to the excitation of WGMs. In addition, there are some transmission dips at sizes that lie just below the cutoffs. In contrast to s0, s1 mode does not excite WGMs and its scattering is characterized by a rather monotonic decrease of transmission with increasing kR. There are a few pronounced and narrow dips (for kR ≈ 3.34 and kR ≈ 3.83). The dips can perhaps be explained qualitatively either as the formation of a bound resonant state by the modes below their cutoffs or as an interference effect related to the reduced number of modes. Since the particle fills a rather large portion of the waveguide, the fractions of the mode propagating outside and through the particle are comparable. The field that propagates through the particle acquires a p p phase comparable to 2kR εb − εs ∼ π and interferes destructively with the directly transmitted field, giving rise to a strong reflection. The scattered fields allow us to calculate the propelling force using two independent approaches. In the first approach, we use the Lorentz formula with the fields inside and on the boundary of the cylinder; see Eqs. (28) and (29) in Ref. [2]. In the second approach, we use the change of the electromagnetic momentum flow ΔM x : X − cΔM x ni P 0 − nj P j − P j ;
(3)
j≠i
where nj is the phase index of mode j. Figure 4 shows the propelling force cF x ∕P 0 that corresponds to the cases in Fig. 3. The excitation using s0 gives cF x ∕P 0 reaching about 2.7. This value corresponds to an almost complete reflection of the incident momentum p flow ≈2nP 0 ∕c ≈ 2 2P 0 ∕c. The efficient coupling to WGMs and large force at kR ≳ 30 correlate with phase matching. The phase index for WGMs (the ratio of c and the phase velocity at the cylinder boundary) is c∕vph ≈ 1.23 for kR ≈ 35 and increases with kR [2]. This value is rather close to the phase index ≈1.41 of the exciting s0. The exact matching is not needed due to strong interaction between the WGMs and the guided mode. For smaller kR, however, the phase index of the WGMs becomes too small. This explains the lower reflection and the force. To provide an efficient coupling to WGMs at smaller kR (with smaller Q factors), one needs to take
kr
May 1, 2014 / Vol. 39, No. 9 / OPTICS LETTERS 35 33 31 29 27
4 2 0 -2 -4 π/4
0
an exciting mode with a lower phase index and to vary the gap kd. We also note that the incidence of s1 results in the appearance of a small negative force. The value of the negative force reaches cF x ∕P 0 ≈ −0.016 (at kR 2.61) and is barely visible on Fig. 4. The negative force can be explained by forward scattering into s0 that has a larger wavenumber and creates a larger momentum flow for the same energy. Negative forces due to scattering into modes with larger wavenumbers were also investigated in Ref. [8]. To understand the spatial properties of the force we can write time-averaged F x as an integral of the angular density of its electric and magnetic components: Z 2π dφF ex φ F m Fx x φ; 0 ZR Fm φ drrF m (4) x x r; φ; 0
where the magnetic component F m x φ is obtained from the radial integration. The distribution F m x r; φ for the resonant excitation at kR 34.96 (WGM with orbital number n 43; see [2]) is shown in Fig. 5. Due to the symmetric excitation and geometry, all fields and forces are symmetric with respect to φ → −φ so we plot only φ ∈ 0∶π. The force density is localized near the periphery of the particle and oscillates rapidly due to the creation of the standing wave by the WGMs excited by the symmetric s0. The angular densities of forces are shown in Fig. 6. The electric and magnetic components show fast oscillations. The amplitude for the magnetic component exceeds significantly that for the electric one. At resonance, we can represent the rapidly oscillating propelling force using slowly varying envelopes: F x φ F 0 φ − F s φ sin2nφ ξφ;
(ϕ)/P0
1
200
-200
Fm x e Fx
0
-400
0
0 0
5
10
15
20
25
30
35
kR
Fig. 4. Propelling force created by modes s0 and s1.
40
(5)
where F 0 φ and F s φ are the amplitudes and ξφ is the phase, and n is the orbital number of the excited WGM.
e,m
2
π
3π/2
Fig. 5. Density of the normalized magnetic component of 2 force cF m x r; φ∕k P 0 (unitless) at resonance kR 34.96.
cFx
cFx/P0
s0 s1
π/2 ϕ (rad)
400 3
2825
π/4
π/2
3π/2
π
ϕ (rad)
Fig. 6. Angular dependence of the electric F ex φ and magnetic Fm x φ force components at resonance kR 34.96.
OPTICS LETTERS / Vol. 39, No. 9 / May 1, 2014 1
π/2
0.5
π/4
0
0
F0 /15 Fs /400 ξ
-0.5 -1 0
π/4
ξ (rad)
cF0,s/P0
2826
-π/4 π/2 ϕ (rad)
3π/2
π
-π/2
balance of the electromagnetic momentum. Although we considered a structure with surface waves supported by metallic interfaces, similar effects can be obtained in dielectric structures. For example, a cylindrical dielectric shell can guide waves that can be evanescent inside and provide efficient propulsion of the particles.
Fig. 7. Angular dependence of the envelopes F 0 φ, F s φ, and ξφ for the propelling force at kR 34.96.
AVM is grateful to V. N. Astratov for discussions. This work was supported in part by the Ministry of Education and Science of the Russian Federation through Agreement No. 11.G34.31.0011.
This representation works very well due to a small bandwidth of F 0;s φ as compared to the frequency of the fast oscillations. The amplitudes F 0;s φ and the phase ξφ are shown in Fig. 7. The contribution to the propelling force comes only from F 0 . In particular, the small bump on F 0 φ near φ π∕2 determines the force. The angular position φ π∕2 corresponds to the particle region in the proximity of the waveguide surface. The transverse force on the particle in the center of ROG is zero. A deviation from the center would give rise to a transverse force that does not create an equilibrium in the center. The particle may hit the wall and bounce. The propulsion may not be a steady process but a series of events where the velocity changes rapidly, as with tapered fibers [1]. The stability can be provided by a repulsive transverse force similar to that observed in Ref. [17] and whose origin remains under discussion. To conclude, we proposed and showed through simulations that a waveguiding structure that simultaneously couples its modes into and out of a particle with WGMs can provide its efficient propulsion. The enhancement by a factor of 2 as compared to a particle coupled to one waveguide or interface is due to directing the scattered radiation into a guided mode rather then to bulk radiation. The force density is strongly peaked near the surface of the particle, consistent with the WGM excitation. The density of the force can reach extremely high values inside the particle. However, the spatial integration of this force gives a value that is limited by the
References 1. Y. Li, O. V. Svitelskiy, A. V. Maslov, D. Carnegie, E. Rafailov, and V. N. Astratov, Light Sci. Appl. 2, e64 (2013). 2. A. V. Maslov, V. N. Astratov, and M. I. Bakunov, Phys. Rev. A 87, 053848 (2013). 3. B. S. Schmidt, A. H. J. Yang, D. Erickson, and M. Lipson, Opt. Express 15, 14322 (2007). 4. A. Ashkin and J. M. Dziedzic, Phys. Rev. Lett. 38, 1351 (1977). 5. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997). 6. P. T. Rakich, M. A. Popović, and Z. Wang, Opt. Express 17, 18116 (2009). 7. A. S. Shalin and S. V. Sukhov, Plasmonics 8, 625 (2013). 8. V. Intaraprasonk and S. Fan, Opt. Lett. 38, 3264 (2013). 9. A. Salandrino and D. N. Christodoulides, Opt. Lett 36, 3103 (2011). 10. A. V. Maslov, Phys. Rev. Lett. 112, 113903 (2014). 11. M. J. Renn, R. Pastel, and H. J. Lewandowski, Phys. Rev. Lett. 82, 1574 (1999). 12. O. A. Schmidt, M. K. Garbos, T. G. Euser, and P. S. J. Russell, Opt. Lett. 37, 91 (2012). 13. K. Kosma, G. Zito, K. Schuster, and S. Pissadakis, Opt. Lett. 38, 1301 (2013). 14. J. J. Xiao, J. Ng, Z. F. Lin, and C. T. Chan, Appl. Phys. Lett. 94, 011102 (2009). 15. Z. Guo and H. Quan, J. Heat Transfer 129, 44 (2007). 16. A. V. Boriskin, S. V. Boriskina, A. Rolland, R. Sauleau, and A. I. Nosich, J. Opt. Soc. Am. A 25, 1169 (2008). 17. R. Khakimov, A. Shevchenko, A. Havukainen, K. Lindfors, and M. Kaivola, Opt. Commun. 285, 4571 (2012).
2823
Resonant optical gun A. V. Maslov* and M. I. Bakunov University of Nizhny Novgorod, Nizhny Novgorod, Russia *Corresponding author: [email protected] Received March 6, 2014; revised April 6, 2014; accepted April 6, 2014; posted April 7, 2014 (Doc. ID 206911); published April 30, 2014 We propose a concept of a structure—a resonant optical gun—to realize an efficient propulsion of dielectric microparticles by light forces. The structure is based on a waveguide in which a reversal of the electromagnetic momentum flow of the incident mode is realized by exciting a whispering gallery resonance in the microparticle. The propelling force can reach the value up to the theoretical maximum of twice the momentum flow of the initial wave. The force density oscillates along the particle periphery and has very large amplitude. © 2014 Optical Society of America OCIS codes: (140.7010) Laser trapping; (350.4855) Optical tweezers or optical manipulation. http://dx.doi.org/10.1364/OL.39.002823
Recently, it was experimentally demonstrated that the optical force created by a tapered-fiber mode on dielectric spheres with diameters 10–20 μm has strong resonances [1]. This behavior is characterized by high frequency (or size) selectivity and very large peak values. The large values can be explained by an efficient transformation of the momentum flow of the incident fiber mode to the propelling force [1,2]. The transformation is accompanied by the excitation of high-quality (Q) whispering gallery modes (WGMs) in the particles by the evanescent field of the mode. In contrast, small dielectric particles, in which WGMs do not exist, do not provide any resonant enhancement even with evanescent excitation [3]. Plane waves also give a much lower force enhancement due to the weak coupling to WGMs [4]. Here we extend work [1,2] and show that a further enhancement of force is possible. The proposed geometry of the structure—a resonant optical gun (ROG)—to realize the enhancement and its operation principle are illustrated in Fig. 1. In a ROG, a dielectric particle is placed inside a waveguide [with parallel plates as shown in Fig. 1(a) or cylindrical as shown in Fig. 1(b)] that can support surface waves. The theoretical analysis of Ref. [2] showed that the excitation of WGMs by a surface wave (guided by one interface) can create the propelling force approximately equal to the momentum flow of the incident mode. This agrees with the propulsion velocity measurements [1] from which the propelling force was estimated to reach the value comparable to the incident momentum flow. The excitation is accompanied by the creation of the scattered bulk radiation with anisotropic angular distribution. Although this bulk radiation carries significant power (up to 100% at resonance), its momentum flow along the incidence direction is rather small. To enhance the force, one can think of collecting the scattered radiation and redirecting it backward to provide a larger change of the incident momentum and, therefore, a larger propelling force. This can intuitively be accomplished by using a symmetric structure shown in Fig. 1(a) in which the excited WGM creates guided reflection. By the conservation of the momentum flow, the particle should experience a larger force as compared to the case when one interface is present. Instead of using the surface wave guided by one interface, one can launch symmetric or antisymmetric modes of the structure. 0146-9592/14/092823-04$15.00/0
The reflection that follows will create a large propelling force on the microparticle. This principle can be used to create forces on the particles not only in parallel plate waveguides [Fig. 1(a)] but in cylindrical structures as well [Fig. 1(b)]. The surface modes of the cylindrical structure may have various angular dependences. The ROG structure is similar to channel dropping filters in which the signal from one waveguide is transformed to the other waveguide by coupling them to a microring resonator [5]. The optical filter synthesis can be extended to find the forces [6]. The acceleration of nanoparticles was proposed in a V-groove metallic structure in which a particle with negative real part of polarizability will be pushed out by the strong field gradient [7]. In a ROG, the propulsion is achieved through the reversal of the incident momentum flow rather than through the field gradient. Recently, the exploration of various possible forces on the particles using waveguiding structure has become of significant interest. For example, pulling optical forces can be created using multimodal waveguide [8] or backward wave structures [9,10]. The propulsion of micrometer-sized particles inside a hollow core [11] and photonic crystal fibers [12] can also be realized. The excitation of WGMs of a microsphere integrated inside a microstructured optical fiber was recently demonstrated [13], but the optical forces were not studied. Similar structures can be used to realize the propulsion with surface modes illustrated in Fig. 1(b). In this Letter, we demonstrate the resonant enhancement of the propelling force using a rigorous analytical theory and simulations. We show that the evanescent nature of the waves plays a key role in the efficient excitation of WGMs that leads to the force enhancement.
(a)
y
L
R
εb
z
εs
(b) F
ϕ F
d
x
εp Fig. 1. (a) Geometry and operation principle of a ROG with parallel plate waveguide. (b) Cross section of possible 3D ROG realization. © 2014 Optical Society of America
OPTICS LETTERS / Vol. 39, No. 9 / May 1, 2014
0
eihx−x r~ Ay; y0 dh ; gb h 1 − r~ 2 −∞ ∞
(b)
1.5 h/k
s a
s0 a0 s1 a1
1 0.5 0 0
5
10 15 20 25 30 35 40 kL
Ns 5
(b)
0 1 0.8 0.6 0.4
s0 s1
0.2 (c)
0 1 0.8 0.6
s0 s1
0.4 0.2
(d)
0 1 0.8
s0
0.6
s1
0.4 0.2
1
0.5 y/L
2
15 10
(1)
p with gb εb ω2 ∕c2 − h2 , r~ r bp eigb L , Ay; y0 0 and By; y0 eigb y eigb y By; y0 B−y; −y0 , −ig y r~ e b ; r bp is the reflection coefficient from the wall. For high-Q structures, the analytical procedure can be more reliable than the finite-difference time domain method [16]. Figure 2 shows the dispersion properties and typical mode profiles for the parallel plate ROG structure. It (a)
(a)
Pr /P0
0
The scattering characteristics of s0 and s1 as functions of particle size kR at fixed kd 1.5 are shown in Fig. 3. We
+ Psc /P0
0
j≠i
s1
(e)
0 1 0.8
0 a0
-0.5
s0
-
i G x; y; x ; y 4π 1
Z
supports symmetric and antisymmetric with respect to H z y modes. Modes s0 and a0 tend to be confined to the interfaces. They are practically degenerate for kL > 5. The other modes propagate in the region jyj < p L∕2 with phase index n c∕vph h∕k < εb . The number of the propagating modes increases with kL. The cutoff width Lm for a symmetric propagating mode of order p m is determined by kLm 2mπ − 2 arctan εb ∕jεp j. We compare propulsion in two cases: excitation by evanescent mode s0 and propagating mode s1. The scattering of an incident mode with power P 0 results in the excitation of other modes. Let us define P j as the power carried by the mode of order j in the x direction. If i is the order of the incident mode, we define the reflected P r and transmitted P t powers in the same mode and scattered P sc powers using X Pt P P r P −i ; P P (2) sc j : i ;
Pt /P0
We show that the force density has a strongly oscillatory behavior and only the regions near the waveguiding surface give significant contributions to the total force. We adopt a 2D model shown in Fig. 1(a). Twodimensional models are common in studying the interaction of resonators with waveguides since they capture the main physical processes [8,14,15]. The particle that scatters the incident mode is characterized by its radius R and dielectric constant εs and separated from the walls by distance d. The dielectric constant of the background is εb and that of the metallic walls (with plasma-like properties) is εp . We choose εp −2 and transverse magnetic polarization topallow surface waveguiding. We also take εb 1 and εs 1.4. We first solve the scattering problem and then use the Lorentz formula and the balance of the momentum flow in the system to find the force. The scattered fields outside of the cylinder are represented as produced by some effective magnetic current on the surface. The current is expanded into the angular exponential functions with unknown coefficients. The fields inside are expanded into the cylindrical functions. Matching the fields at the boundary of the cylinder gives a set of equations for the expansion coefficients. The case of a particle near a single interface is described in Ref. [2]. To extend this approach to the structure with two interfaces we replace the Green’s function given by Eq. (18b) of Ref. [2] with that for a δ-source emitting in the region jyj < L∕2. Expanding the source into the Fourier integral reduces the problem to finding the field produced by a plane wave bouncing between the walls. Taking the inverse transform gives the required G1 x; y; x0 ; y0 as
Psc /P0
2824
s0 s1
0.6 0.4 0.2
-1 -1
0 1 2 Hz(arb. units)
Fig. 2. (a) Propagation wavenumber h (normalized to k ω∕c) as a function of the size parameter kL for the modes of the waveguide shown in Fig. 1(a) with εb 1 and εp −2. The modes are labeled according to their symmetry (s, symmetric; a, antisymmetric) and order. (b) Distribution of the magnetic field H z y for s0, a0, and s1 at kL 10.
0 0
5
10
15
20
25
30
35
40
kR
Fig. 3. Dependence of the scattering characteristics of ROG on size kR when the incident mode is s0 or s1. (a) Number of guided symmetric modes. (b) Transmitted power P t ∕P 0 , and (c) reflected power P r ∕P 0 , p (d) and (e) scattered powers εs 1.4, and kd 1.5. P sc ∕P 0 for εb 1, εp −2,
confirmed that the energy balance holds. Due to symmetry, the scattering can produce only symmetric modes. The number of modes grows with kR since L 2R d. The scattering of s0 is characterized by the appearance of periodic narrow dips in the transmission spectrum with corresponding peaks in the reflection and scattering spectra. These peaks are due to the excitation of WGMs. In addition, there are some transmission dips at sizes that lie just below the cutoffs. In contrast to s0, s1 mode does not excite WGMs and its scattering is characterized by a rather monotonic decrease of transmission with increasing kR. There are a few pronounced and narrow dips (for kR ≈ 3.34 and kR ≈ 3.83). The dips can perhaps be explained qualitatively either as the formation of a bound resonant state by the modes below their cutoffs or as an interference effect related to the reduced number of modes. Since the particle fills a rather large portion of the waveguide, the fractions of the mode propagating outside and through the particle are comparable. The field that propagates through the particle acquires a p p phase comparable to 2kR εb − εs ∼ π and interferes destructively with the directly transmitted field, giving rise to a strong reflection. The scattered fields allow us to calculate the propelling force using two independent approaches. In the first approach, we use the Lorentz formula with the fields inside and on the boundary of the cylinder; see Eqs. (28) and (29) in Ref. [2]. In the second approach, we use the change of the electromagnetic momentum flow ΔM x : X − cΔM x ni P 0 − nj P j − P j ;
(3)
j≠i
where nj is the phase index of mode j. Figure 4 shows the propelling force cF x ∕P 0 that corresponds to the cases in Fig. 3. The excitation using s0 gives cF x ∕P 0 reaching about 2.7. This value corresponds to an almost complete reflection of the incident momentum p flow ≈2nP 0 ∕c ≈ 2 2P 0 ∕c. The efficient coupling to WGMs and large force at kR ≳ 30 correlate with phase matching. The phase index for WGMs (the ratio of c and the phase velocity at the cylinder boundary) is c∕vph ≈ 1.23 for kR ≈ 35 and increases with kR [2]. This value is rather close to the phase index ≈1.41 of the exciting s0. The exact matching is not needed due to strong interaction between the WGMs and the guided mode. For smaller kR, however, the phase index of the WGMs becomes too small. This explains the lower reflection and the force. To provide an efficient coupling to WGMs at smaller kR (with smaller Q factors), one needs to take
kr
May 1, 2014 / Vol. 39, No. 9 / OPTICS LETTERS 35 33 31 29 27
4 2 0 -2 -4 π/4
0
an exciting mode with a lower phase index and to vary the gap kd. We also note that the incidence of s1 results in the appearance of a small negative force. The value of the negative force reaches cF x ∕P 0 ≈ −0.016 (at kR 2.61) and is barely visible on Fig. 4. The negative force can be explained by forward scattering into s0 that has a larger wavenumber and creates a larger momentum flow for the same energy. Negative forces due to scattering into modes with larger wavenumbers were also investigated in Ref. [8]. To understand the spatial properties of the force we can write time-averaged F x as an integral of the angular density of its electric and magnetic components: Z 2π dφF ex φ F m Fx x φ; 0 ZR Fm φ drrF m (4) x x r; φ; 0
where the magnetic component F m x φ is obtained from the radial integration. The distribution F m x r; φ for the resonant excitation at kR 34.96 (WGM with orbital number n 43; see [2]) is shown in Fig. 5. Due to the symmetric excitation and geometry, all fields and forces are symmetric with respect to φ → −φ so we plot only φ ∈ 0∶π. The force density is localized near the periphery of the particle and oscillates rapidly due to the creation of the standing wave by the WGMs excited by the symmetric s0. The angular densities of forces are shown in Fig. 6. The electric and magnetic components show fast oscillations. The amplitude for the magnetic component exceeds significantly that for the electric one. At resonance, we can represent the rapidly oscillating propelling force using slowly varying envelopes: F x φ F 0 φ − F s φ sin2nφ ξφ;
(ϕ)/P0
1
200
-200
Fm x e Fx
0
-400
0
0 0
5
10
15
20
25
30
35
kR
Fig. 4. Propelling force created by modes s0 and s1.
40
(5)
where F 0 φ and F s φ are the amplitudes and ξφ is the phase, and n is the orbital number of the excited WGM.
e,m
2
π
3π/2
Fig. 5. Density of the normalized magnetic component of 2 force cF m x r; φ∕k P 0 (unitless) at resonance kR 34.96.
cFx
cFx/P0
s0 s1
π/2 ϕ (rad)
400 3
2825
π/4
π/2
3π/2
π
ϕ (rad)
Fig. 6. Angular dependence of the electric F ex φ and magnetic Fm x φ force components at resonance kR 34.96.
OPTICS LETTERS / Vol. 39, No. 9 / May 1, 2014 1
π/2
0.5
π/4
0
0
F0 /15 Fs /400 ξ
-0.5 -1 0
π/4
ξ (rad)
cF0,s/P0
2826
-π/4 π/2 ϕ (rad)
3π/2
π
-π/2
balance of the electromagnetic momentum. Although we considered a structure with surface waves supported by metallic interfaces, similar effects can be obtained in dielectric structures. For example, a cylindrical dielectric shell can guide waves that can be evanescent inside and provide efficient propulsion of the particles.
Fig. 7. Angular dependence of the envelopes F 0 φ, F s φ, and ξφ for the propelling force at kR 34.96.
AVM is grateful to V. N. Astratov for discussions. This work was supported in part by the Ministry of Education and Science of the Russian Federation through Agreement No. 11.G34.31.0011.
This representation works very well due to a small bandwidth of F 0;s φ as compared to the frequency of the fast oscillations. The amplitudes F 0;s φ and the phase ξφ are shown in Fig. 7. The contribution to the propelling force comes only from F 0 . In particular, the small bump on F 0 φ near φ π∕2 determines the force. The angular position φ π∕2 corresponds to the particle region in the proximity of the waveguide surface. The transverse force on the particle in the center of ROG is zero. A deviation from the center would give rise to a transverse force that does not create an equilibrium in the center. The particle may hit the wall and bounce. The propulsion may not be a steady process but a series of events where the velocity changes rapidly, as with tapered fibers [1]. The stability can be provided by a repulsive transverse force similar to that observed in Ref. [17] and whose origin remains under discussion. To conclude, we proposed and showed through simulations that a waveguiding structure that simultaneously couples its modes into and out of a particle with WGMs can provide its efficient propulsion. The enhancement by a factor of 2 as compared to a particle coupled to one waveguide or interface is due to directing the scattered radiation into a guided mode rather then to bulk radiation. The force density is strongly peaked near the surface of the particle, consistent with the WGM excitation. The density of the force can reach extremely high values inside the particle. However, the spatial integration of this force gives a value that is limited by the
References 1. Y. Li, O. V. Svitelskiy, A. V. Maslov, D. Carnegie, E. Rafailov, and V. N. Astratov, Light Sci. Appl. 2, e64 (2013). 2. A. V. Maslov, V. N. Astratov, and M. I. Bakunov, Phys. Rev. A 87, 053848 (2013). 3. B. S. Schmidt, A. H. J. Yang, D. Erickson, and M. Lipson, Opt. Express 15, 14322 (2007). 4. A. Ashkin and J. M. Dziedzic, Phys. Rev. Lett. 38, 1351 (1977). 5. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997). 6. P. T. Rakich, M. A. Popović, and Z. Wang, Opt. Express 17, 18116 (2009). 7. A. S. Shalin and S. V. Sukhov, Plasmonics 8, 625 (2013). 8. V. Intaraprasonk and S. Fan, Opt. Lett. 38, 3264 (2013). 9. A. Salandrino and D. N. Christodoulides, Opt. Lett 36, 3103 (2011). 10. A. V. Maslov, Phys. Rev. Lett. 112, 113903 (2014). 11. M. J. Renn, R. Pastel, and H. J. Lewandowski, Phys. Rev. Lett. 82, 1574 (1999). 12. O. A. Schmidt, M. K. Garbos, T. G. Euser, and P. S. J. Russell, Opt. Lett. 37, 91 (2012). 13. K. Kosma, G. Zito, K. Schuster, and S. Pissadakis, Opt. Lett. 38, 1301 (2013). 14. J. J. Xiao, J. Ng, Z. F. Lin, and C. T. Chan, Appl. Phys. Lett. 94, 011102 (2009). 15. Z. Guo and H. Quan, J. Heat Transfer 129, 44 (2007). 16. A. V. Boriskin, S. V. Boriskina, A. Rolland, R. Sauleau, and A. I. Nosich, J. Opt. Soc. Am. A 25, 1169 (2008). 17. R. Khakimov, A. Shevchenko, A. Havukainen, K. Lindfors, and M. Kaivola, Opt. Commun. 285, 4571 (2012).
