Optical traps and anti-traps for glass nanoplates in hollow waveguides

M. C. Günendi; S. Xie; D. Novoa; P. StJ. Russell

doi:10.1364/OE.27.017708

1. Introduction

Radiation forces acting on a mechanically resonant membrane suspended inside an optical cavity have been extensively studied in the context of optomechanical cooling to the ground state [1]. In these experiments, the membrane is typically much thinner than half a wavelength so as to minimize heating by optical absorption, and the light propagates perpendicular to the membrane [2]. In recent years, interest has grown in optomechanical effects in waveguides [3,4]. Guidance of tightly confined optical modes significantly increases the optomechanical interaction length compared to free-space configurations. In these experiments, the transverse field profiles of the guided eigenmodes are typically only very weakly perturbed by the optomechanical interaction. An example of a system in which this is not the case is self-alignment by optomechanical back-action of a glass nanospike in a hollow-core photonic crystal fiber [5]. As the base of the nanospike is displaced from the fiber axis, the system jumps from a single-lobed to a two-lobed core mode at a certain critical point, exhibiting strong hysteresis when the base is returned to the axis. This means that one of the coupled modes – that of the hollow core – changes shape dramatically as a result of the interaction. In the field of optical tweezers, optomechanical back-action has also been explored as a means of reducing the power required to trap nanoparticles [6,7].

In this work we aim to explore the optomechanical forces acting on a thin plate of glass, supported by a base, after insertion into a single-ring hollow-core photonic crystal fiber (SR-PCF). We will show by finite element modelling that it can be stably trapped, both translationally and rotationally, in the hollow core and that under certain conditions dark-trapping is possible, which occurs when the tongue is anti-resonant with the hollow core mode. The analysis is the first rigorous attempt to investigate optomechanical back-action trapping of glass membranes, paving the way towards all-optical control of the spatial position and orientation of non-spherical nano-scale objects in hollow waveguides.

2. Planar system

To help understand the SR-PCF system we first consider the case of a hollow planar waveguide formed by two perfect mirrors placed at y = 0 and y = w, with a thin glass nanoplate of width h placed at position y = η (Fig. 1). The system is assumed infinite in x- and z-directions, and to support guided optical modes, infinitely wide in the x-direction, that propagate in the z-direction. It is straightforward to derive the dispersion relation for these modes (we use the formalism reported in [8]; for a brief derivation see Appendix). It takes the form:

\sin (p_{v} (1 - h_{n})) + (q \cos A \cos B - q^{- 1} \sin A \sin B) \tan p_{g} h_{n} = 0,

where

p_{v} = k w \sqrt{1 - n_{s m}^{2}}

,

p_{g} = k w \sqrt{n_{g}^{2} - n_{s m}^{2}}

, n_g is the glass index, n_sm the supermode index, k the vacuum wavevector,

q = p_{v} ξ_{v} / (p_{g} ξ_{g})

,

A = p_{v} (1 - η_{n} - h_{n} / 2)

,

B = p_{v} (η_{n} - h_{n} / 2)

, and the subscript “n” denotes normalization to w, e.g.,

h_{n} = h / w

. For TE modes (polarized in the x-direction),

ξ_{v} = ξ_{g} = 1

and for TM modes (magnetic field pointing along x),

ξ_{v} = ξ_{g} n_{g}^{2} = 1

. As the nanoplate is displaced across the waveguide, the transverse profiles and refractive indices n_SM of the supermodes of the “nanoplate plus waveguide” system, and thus the optical gradient forces will vary, giving rise to strong optomechanical back-action. Compared to previous configurations in which a membrane is placed within an optical cavity [2], in a hollow waveguide light propagates both along and perpendicular to the plane of the membrane. This means that the optomechanical gradient forces can be controlled by launching different optical modes into the system.

Fig. 1 Sketch of the planar system. A glass nanoplate of a thickness h is placed inside a hollow planar waveguide of width w. The optical modes propagate in the z-direction and the nanoplate is considered to be attached to imaginary springs, which in the absence of any optical forces will return the nanoplate (position y = η) to the base position y = η_B.

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The system has two main regimes of operation, corresponding to whether the nanoplate is resonant or off-resonant. In the first case, the nanoplate supports an open resonance (a leaky mode) that propagates along the nanoplate with refractive index:

n_{R} = \sqrt{n_{g}^{2} - {(m π / (k w h_{n}))}^{2}}, n_{R} < 1

where m is an integer representing the mode order. If the supermode index equals this value, only very weak (if any) optical forces will act on the nanoplate since light will pass through without being reflected – a feature characteristic of a resonance. As the nanoplate is gradually detuned from resonance, which in practice can be realized by tuning laser frequency or the width of hollow waveguide, m becomes increasingly non-integral and Eq. (2) no longer holds [9]. Under these conditions the supermodes of the entire structure are substantially altered by strong reflections at the nanoplate, which as a result experiences higher and higher gradient forces.

Once the field distribution f (y_n) across the structure is known, the optical pressure acting on the nanoplate can be calculated by taking the difference between the Maxwell stress tensors on its two sides [3]:

σ_{n e t} = σ (η_{n} + h_{n} / 2) - σ (η_{n} - h_{n} / 2),

where

σ (y_{n})

is derived in the Appendix. The net optical pressure σ_net can be conveniently expressed in terms of a dimensionless parameter [10]:

{\hat{σ}}_{n e t} = \frac{σ_{n e t} c}{n_{sm} S_{a v}},

where

S_{a v} = \int S_{z} d y / w

, S_z is the z component of the time-averaged Poynting vector, c is the vacuum speed of light.

{\hat{σ}}_{net}

corresponds to the efficiency of momentum transfer to the membrane.

We now present results for a system in which kw = 100 and n_g = 1.45. First we place the nanoplate at η_n = 0.5 and choose h_n so that the two-lobed second order resonance of the nanoplate (m = 2 in Eq. (2)) has the same effective index as the second order TE supermode of the whole structure (see Table 1). Then we calculate the optical pressure and supermode refractive index as a function of nanoplate position η_n. As expected from the above discussion, the optical pressure is zero and the refractive index does not vary. We then follow the same procedure for off-resonant nanoplate modes (m = 1.50, 1.90, 1.95 and 1.99). The results are plotted in Fig. 2. It is clear that the optical forces become stronger as the nanoplate is detuned away from resonance, and that at the same time, the supermode index varies more and more strongly with nanoplate position. For a 100 nm thick anti-resonant (m = 1.5) nanoplate of silica glass (density 2000 kg/m³), in a system arranged horizontally, the gravitational pressure is exactly balanced by the optical pressure for S_av = 10 mW/µm² ( ${\hat{σ}}_{net} = 6 \times 10^{- 5}$ ).

Table 1. Membrane mode order and thickness at which the membrane mode and the supermode have the same refractive index. As m = 2 is approached, the index changes very little.

View Table

Fig. 2 (a) Dimensionless optical pressure for five different values of nanoplate mode order m, plotted against nanoplate position. On resonance (m = 2) the forces are zero. As m deviates more and more from 2, the anti-resonance grows in strength, producing stronger and stronger optical forces. (b) The corresponding modal refractive indices. For m = 2 (dashed line) the nanoplate is fully transparent, i.e., it acts like an anti-reflection coating. As a result the refractive index does not vary with the position of the nanoplate.

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Four representative supermode field distributions for the resonant (m = 2) and anti-resonant (m = 1.5) cases are shown in Fig. 3, calculated following the analytical approach reported in [8]. In each case the membrane is designed so that its resonance or anti-resonance is phase-matched to the supermode at η_n = 0.5. In the resonant case the intensities on the two membrane sides are always equal, so that there is no net pressure on the membrane. In the anti-resonant case the pressure changes strongly with position. For instance, at η_n = 0.64 the force is strongly negative (point A in Fig. 2, upper plot in Fig. 3b) whereas at η_n = 0.70 it is strongly positive (point B in Fig. 2, lower plot in Fig. 3b). Between these two positions the field intensity differential between opposite membrane sides changes sign.

Fig. 3 Sketch Supermode field intensity profiles in the resonant and anti-resonant cases. The insets are magnified versions of the fields in the vicinity of the nanoplate. (a) Intensity profiles for two positions of an m = 2 resonant nanoplate (TE polarization). The profile of the hollow waveguide mode is almost unaffected by the nanoplate, which acts like an anti-reflection coating. (b) Intensity profiles for two positions of an m = 1.5 anti-resonant nanoplate (TE polarization). The supermode field profiles are strongly affected by the nanoplate, which acts like a strong reflector. The upper plot refers to point A in Fig. 2, and the lower plot to point B.

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2.1. Effects of detuning

Detuning can be explored by changing kw while keeping everything else constant. In practice this could be achieved either by tuning the laser frequency or adjusting the width of the hollow waveguide. In the calculations, the nanoplate is first centered at η/w = 0.52 and kw = 100 at zero detuning (δkw = 0). Positive pressures denote instability, and negative pressures will result in nanoplate trapping (shaded area). Figure 4(b) shows the optical pressure as a function of nanoplate position in the m = 2 resonant case, revealing that positive (blue) detuning creates an anti-trap with negative optical stiffness, while negative (red) detuning results in stable trapping. Indeed, the forces on the nanoplate behave in a manner similar to those experienced by cold atoms as the trapping laser frequency is tuned in the vicinity of an atomic resonance [11]. For values of m ≈2, stable trapping is usually possible by detuning the laser appropriately. Note that, once the membrane is trapped, the system will not switch its state unless the membrane position is strongly perturbed.

Fig. 4 (a) Effects of laser detuning δkw on the optical pressure. Each value of m is exactly matched in the nanoplate at zero detuning, by adjusting h_n (values shown). In every case, η_n = 0.52. (b) Optical pressure in the resonant case plotted versus nanoplate position in the vicinity of η_n = 0.52 for positive (blue) and negative (red) detuning. It is interesting that positive detuning from a zero creates a positive pressure and thus an anti-trap, whereas negative detuning creates a trap (shaded region in (a)). There is a close analogy with red and blue detuning of an atomic resonance.

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2.2. Mechanically constrained membrane

We now consider the case when the nanoplate is mechanically constrained by connecting it to a base with imaginary springs (Fig. 1). As the base position is moved from η_n = 0 to η_n = 1, the analysis above allows calculation of the nanoplate positions at which mechanical and optical forces cancel. A plot of these positions versus base position for a spring constant of 0.98 (η_n – η_b) pN/µm² is shown in Fig. 5. The dotted line represents the situation in the absence of light, when η_n = η_b. Although the forces are balanced along the dashed curves, the system is unstable. As the base position increases, the system jumps to a new state at certain critical values of η_B, and when it is cycled back, the switching positions are quite different. The system thus displays strong bistability, not unlike that seen in experiments on nanospike trapping in HC-PCF [5].

Fig. 5 Bistability in the position of a sprung anti-resonant nanoplate (m = 1.5, kw = 100) as the base position is varied (see Fig. 1). With the light turned off, the base and nanoplate positions coincide (dotted line). If the mechanical spring constant is set to zero, the nanoplate will be stably trapped at the position marked by the small white circle at the center. Although the optical and mechanical forces are balanced at the yellow circles, these positions are anti-traps. Note that the nanoplate touches the mirrors at the edges of the gray-shaded regions.

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3. Single-ring hollow-core PCF

We are now in a position to discuss the case of a silica nanoplate inserted into the core of a SR-PCF, which consist of a number of thin-walled capillaries attached to the inner wall of a thick-walled capillary fiber and arranged in an equally-spaced ring around the hollow core [12]. They guide by anti-resonant reflection [13] and despite their very simple structure can provide very low transmission loss [14]. Since analytical solutions do not exist for this (much more complicated) system, we employ finite element modelling. The system is illustrated in Fig. 6. The hollow core has diameter D and the capillaries have diameter d and wall thickness t. A glass nanoplate with thickness h and width L is inserted into the SR-PCF and oriented along the y axis (θ = 0). The parameters, chosen so that the nanoplate is anti-resonant (m = 1.5) and phase-matched to the supermode when placed at core center, were kD = 162, h/D = 0.0093, L/D = 0.492, and d/D = 0.5. In addition, the capillary wall thickness, t/D = 0.01, was chosen to ensure that the LP₁₁-like core mode does not couple to capillary wall resonances. The supermode intensity profile and index were obtained using two-dimensional boundary mode analysis in finite element modelling, and the optical forces were calculated by integrating the Maxwell stress tensor over the membrane surface.

Fig. 6 Schematic of the SR-PCF plus nanoplate system. Left: sketch of the whole structure. The diagonal distance between the capillaries is D, and the inner diameter and wall thickness of the capillaries are d and t. Right: Detail showing the nanoplate dimensions and orientation.

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The upper (lower) panel in Fig. 7(a) plots against η_x/D (η_y/D) the optical force in the x-direction (y-direction) acting on a vertically aligned nanoplate. The optical forces vary dramatically with nanoplate position, consistent with the behavior of the planar system. Figure 7(b) plots the S_z distributions for the three positions marked in Fig. 7(a), showing that as the nanoplate moves away from core center, the supermode field distributions are strongly perturbed. Note that at position A, the nanoplate is stably trapped at the anti-node of an LP₁₁-like supermode. Given a SR-PCF with D = 30 μm, the estimated trapping stiffness at position A using the aforementioned parameters is ~10 pN∙μm⁻¹m⁻¹W⁻¹. Note that stable trapping at the center of the fiber core is also feasible along the y-direction (lower panel in Fig. 7(a)), despite the much lower mechanical compliance of the membrane along this axis. The trapping stiffness is, however, orders of magnitude smaller than that obtained in the perpendicular direction since the LP₁₁-like core mode has a nodal line aligned with the y-axis. In practice, the nanoplate can be mechanically clamped at its base before being introduced into the hollow core, thus eliminating the need to consider gravitational forces.

Fig. 7 (a) The optical force acting on a vertically aligned nanoplate (per unit fiber length per Watt of launched power) plotted against its horizontal (upper) and vertical (lower) position. Within the gray shaded regions, complex surface states appear that strongly distort the forces (not studied in this paper). (b) Normalized distributions of the z component of Poynting vector for the three different nanoplate positions marked in (a). The scale-bar marks a length of 10λ.

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We further analyze the rotational stability of the anti-resonant nanoplate by calculating the net optical torque per unit length per Watt acting on it:

τ_{o p t} = \oint r \times σ_{n e t} d s,

where the integral is over the closed surface of the nanoplate and r is a vector pointing from the nanoplate center (marked by the white circle in Fig. 6) to points on its surface. Figure 8(a) plots the calculated optical torque as a function of the position of the nanoplate center, (η_x, η_y). As expected the torque vanishes at core center.

Fig. 8 (a) Spatial map of the optical torque acting on the nanoplate near core center. (b) Upper: Optical torque plotted against nanoplate orientation θ for the three nanoplate positions marked in (a). Lower: The curve for position A plotted on an expanded vertical scale. The vertical gray lines mark the positions of the capillary centers. The white circles label the rotationally stable trapping positions, and the red, unstable positions.

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Figure 8(b) plots the dependence of the optical torque on nanoplate rotation θ at three different positions (marked as A-C) in the hollow core. At core center (A) the optical torque is almost independent of nanoplate orientation. As the nanoplate is moved from core center, however, the optical torque changes more strongly with θ. Note that it switches from counter-clockwise to clockwise as θ passes π/2. The lower plot in Fig. 8(b) zooms into the curve at position A. The symmetry of the SR-PCF structure (the solid-gray lines mark the positions of the capillary centers) dictates that the optical torque should vary periodically with θ. The nanoplate is stably trapped (in the dark) at angles marked by the white circles where the torque is zero.

4. Conclusions

An optically anti-resonant nano-scale dielectric object, placed inside a hollow waveguide, can strongly distort the transverse mode profile, causing large position-dependent variations in the optical gradient forces and torque. When an anti-resonant nanoplate is placed at the field node of a two-lobed hollow waveguide mode, it is passively trapped “in the dark” by optomechanical back-action. When on the other hand an open nanoplate resonance has the same axial index as the single-lobed (fundamental) supermode, there are no trapping forces because the nanoplate acts as an anti-reflection coating. If the laser frequency is blue-detuned from this condition, the nanoplate is pushed away from the intensity maximum, i.e., an anti-trap forms, while for red detuning a trap forms. This effect has a close analogy with the (anti-)trapping of cold atoms. The results suggest that the spatial position and angular orientation of nano-scale objects can be optically controlled in hollow waveguides by launching beams with appropriate transverse field profiles, synthesized using spatial light modulators. It may also be possible, by temporally modulating the power and spatial profile of the launched laser light, to control the rotational speed of non-circular objects. Such a glass nanoplate in a hollow-core photonic crystal fiber might be used as a novel highly-controllable cover-slip or solution stirrer for applications in photochemistry and bio-photonics.

Appendix analysis of planar structure

The dispersion relation in Eq. (1) is derived by solving the Helmholtz equation:

\frac{d^{2} f}{d y^{2}} + (k^{2} n^{2} (y) - β^{2}) f = 0

where f is the field component in the x-direction (E_x for TE modes and H_x for TM modes), subject to the boundary conditions:

f_{1} = f_{2} and ξ_{1} \frac{d f_{1}}{d y} = ξ_{2} \frac{d f_{2}}{d y},

between medium 1 (high index) and medium 2 (low index), where

ξ_{j} = 1

for TE modes (polarized in the x-direction) and

ξ_{j} = 1 / n_{j}^{2}

for TM modes. The optical pressure at position y_n, calculated using the Maxwell stress tensor, is given by:

σ (y_{n}) = \frac{S_{av}}{2 c n_{sm}} [(n_{sm}^{2} ξ^{2} - 1) f^{2} (y_{n}) - ξ^{2} {(\frac{f^{'} (y_{n})}{k^{2} w^{2}})}^{2}]

where f (y_n) is normalized so that:

\int_{0}^{1} ξ (y_{n}) f^{2} (y_{n}) d y_{n} = 1

Note that in Eqs. (8) and (9), as before,

ξ = 1

for TE modes and

ξ = 1 / n^{2} (y_{n})

for TM modes.

Acknowledgments

We thank R. E. Noskov and A. A. Sukhorukov for helpful discussions during the initial stages of the work.

References

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m	h_n	n_sm = n_R
1.50	0.0448	0.997918
1.90	0.0567	0.997765
1.95	0.0582	0.997765
1.99	0.0594	0.997765
2.00	0.0597	0.997765

Optical traps and anti-traps for glass nanoplates in hollow waveguides

Abstract

1. Introduction

2. Planar system

2.1. Effects of detuning

2.2. Mechanically constrained membrane

3. Single-ring hollow-core PCF

4. Conclusions

Appendix analysis of planar structure

Acknowledgments

References

Cited By

Figures (8)

Tables (1)

Equations (9)

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