1. Introduction

The study of photonic crystals is of increasing interest for its likeness with solid state physics and for coupling with non-linear optics. Improvement of nanocontrol engineering gives photonic band gaps, similar to electronics ones [1]. Former works on collective optical trapping and optical crystallization [2, 3, 4, 5] opened up a new possibility to create optically induced holograms. These optically maintained crystals could be a new type of third order non-linear material in colloidal suspensions [6, 7]. When some scattering particles are in an intense optical field, they optically induce mechanical forces on each others. Optical interaction forces, now called “binding forces” can lead to two dimensional crystals or chains [8] depending on the trap geometry and the particle size. The 1/r long range interaction dependency of the field scattered by a particle in the Rayleigh range [9] suggests possibly strong field enhancement in long self-organized chains. Some experimental work was done on longitudinal, one-dimensional interaction configuration [10, 11] and exhibited strong optical binding interactions. We discuss here the case of a transverse interaction for which it has been proven and experimentally observed [9, 12, 13] that dipolar interaction leads, for two spheres, to potential minima at distances roughly multiple of the wavelength. As the scattered field for a dipole is maximum along directions orthogonal to polarization, energy wells are deeper in those directions. For the simplicity of experimental setup and calculations, we study here the case of small spheres, trapped in a coherent, monophase fringe. In this configuration, dipoles tend to self-organize in the fringe, so as to give a chain perpendicular to the incident polarization. Calculations and results would be very similar for particles separations along the beam propagation axis. Here we consider a linear chain of dielectric or metallic spheres in a Rayleigh regime, trapped in an optical sinusoidal fringe lying in a plane, as shown on Fig. 1.

 

Fig. 1. Configuration of the trapped chain made of induced scattering dipoles.

Download Full Size | PDF

For small dielectric spheres, that is to say when ka < 1 (k being the wave vector and a the radius of the sphere), one can neglect radiation pressure, proportional to the squared volume of the sphere while gradient force is proportional to its volume. In this approximation, it can be checked numerically that potential minima are distributed every lambda with a good approximation even when the chain is finite (Fig. 2). This calculation has been performed neglecting scattering force and looking for the energy minimum of each dipole, starting from a λ periodicity. Then, we apply infinitesimal variations in a multi-iterative process.

 

Fig. 2. Deviation of position from lambda periodicity for a finite chain. The longer the chain, the weaker the edge effects.

Download Full Size | PDF

In the central part of the chain, the separation distance between particles is roughly constant and slightly larger than λ: dipoles in the middle of chain see an infinite-like chain. On the edges, the separation distance increases: edge effects are similar to periodicity defects in solid state physics, for layers close to the surface (the bulk appears after five or six layers). This result also looks like periodicity defects experimentally observed for an optically levitated chain of microspheres in the Mie regime[14].

2. Finite number of interacting induced coherent dipoles

The field E_s,j scattered by a dipole j in the transverse direction is [15]:

where E_j is the field seen by the j^th dipole, r the distance to this dipole, and $\alpha =\frac{m^2-1}{m^2+2}{a}^3$ (while ka < 1), m being the refractive index of the dielectric sphere and a its radius. Assuming these spheres self-organize into a “d-periodic” linear chain, the self-consistent equations to solve are then scalar:

∀n ∈〚1,N〛,

where the summation is performed over all the N dipoles of the chain except the n^th for which we calculate the field. Working with identical spheres, and the incident fields E _0,n being the same for every dipole we assume, when the chain is big, that the field seen by every dipole is the same (translation invariance). In this case, the system is easily solved and gives:

When the light wavelength is tuned on the chain step (i.e. kd = 2π), the real part of the summation at the denominator covers the range from zero to infinity when increasing the number N of interacting dipoles. The field E_n is then maximized when the real part of the denominator is zero. This enhancement also depends on the polarizability k ³ α as already numerically observed [16] for the two-dimensional case. For a given polarizability, the maximum binding field is then:

As kd = 2π and the summation $\sum _{p=1}^N\frac{1}{p^2}$ quickly converges to the well known result $\frac{{\pi }^2}{6}$. When performed numerically, the maximum enhancement condition is not exactly the one calculated with this simple model. The comparison between this model and the numerical calculation is shown on Figs. 3 and 4. The difference between the former approximate result and the numerical calculation is due to edge effects. We have supposed a translation invariance which is obviously false for a finite chain. In the real case, an n × n matrix must be inversed. As we are interested in translation-invarianced long structures, we set particles every lambda. A more realistic model should take into account periodicity defects, especially close to boundaries. However, looking for potential minima of each dipole requires much longer computational processes. Moreover, there may be not a single stable solution.

The field on Fig. 3 reach a maximum for a given number of dipoles in the chain. The stronger the polarizability, the smaller this optimum number of dipoles. Two regimes can be distinguished here, the case of “short” chains and the case of “long” chains. For a small number of dipoles, that is to say, when the dipoles weakly scatter the light and when the real part of the summation at the denominator remains small compared to 1, the n^th dipole feels mostly the incident field, slightly enhanced by other dipoles. In this regime, the field increases as the logarithm of the number of interacting dipoles. As the scattered field of a dipole has a 1/r dependency, for a given number of dipoles, the scattered field becomes stronger than the incident field. When the number of dipoles is too big, the impedance of the resonator is too high to be efficiently coupled with the incident field. The field E_n is then phase-opposed to the incident field E _0,n. For a finite chain, all the dipoles might not be in the same regime. Indeed, spheres at the edges interact with roughly half less dipoles than the ones in the middle. The impedance of the chain can be defined as:$Z\left(\omega \right)=\frac{E_0}{E_n}$. Moreover, we can notice that the smaller the spheres, the higher the maximum. It means in particular that for an optimum number of optically trapped spheres in the Rayleigh regime, the trapping energy (proportional to α |E_n,res |²) increases when the polarizability decreases. Finally, in this case, the rate between excitation and response is purely imaginary.

For some applications in optical trapping, our purpose is to keep field enhancement even when the number of interacting dipoles is large. There are two possibilities to achieve this. The first one would consist in shrinking the coherence of the incident field in order to reduce the number of coherently interacting dipoles. The second possibility is to spatially modulate the phase of the incident field. We are going to study the relevance of these possibilities in the next paragraphs.

 

Fig. 3. Numerical calculation of field enhancement (blue curve) for the dipole in the middle of the chain, is compared to the presented simple approximation (red curve) of uniform field. The maximum enhancement does not happen exactly for the same number of dipoles. The difference between these two curves is due to edge effects. For small N_dip , the field increases logarithmically. Curves plotted for k ³ α = 0.86.

Download Full Size | PDF

 

Fig. 4. Field seen by dipoles as a function of their position in the chain (starting from the middle). The red line represents the numerical value obtained by the infinite chain model.

Download Full Size | PDF

3. Coherence length of the trapping light

In optical trapping experiments, potential minima are maxima of the squared modulus of the field. With an infinite number of dipoles and a perfectly coherent trapping laser, the trapping energy will always collapse for a given number of spheres. Tuning the coherence length of the incident light, we can reduce interactions to a finite number of dipoles. Indeed, in that case, for each dipole, correlations between incident field and scattered field by other incoherent induced dipoles, are zero when averaged over the characteristic response time of the mechanical dynamic of the system. It means that incoherent dipoles do not contribute, or contribute weakly, to binding effects. When trapping in water, Brownian motion acousto-modulates scattered fields. Relevant parameters are then mean free path and correlation time of speed.

3.1. Two dipoles case

In the simple case of two interacting identical dipoles, the equations to be solved are:

where d is the distance between the two dipoles and $f\left(kd\right)=k^3\alpha \left(\frac{1}{kd}-\frac{1}{{\left(kd\right)}^3}+\frac{i}{{\left(kd\right)}^2}\right)$. Which gives for Ẽ ₁ and Ẽ ₂:

The potential energy of a dipole is <W>_t= -4παε<|E(t)|²>_t. According to the Wiener-Khinchin theorem, <W>_t can be expressed in reciprocal domain, as:

Using the expression of Ẽ ₁, we get:

Here we can distinguish between two extreme regimes. When d is very small compared to the coherence length of the incident light, the pair of dipole sees a perfectly monochromatic incident light. In the former calculation, the characteristic widths of Ẽ ₁₀ and Ẽ ₂₀ are much smaller than variations of the impedance and than e^ikdf(kd). These terms can then be set as constants in the integral, which gives the results for the monochromatic case.

On the contrary, when the distance between dipoles is much bigger than the coherence length of the source, impedance oscillates quickly in the envelope (Fig. 5) of Ẽ _0j, centered on ω ₀. For large distances, f(kd) can be approximated by $\frac{1}{k_{0}d}$ at first order. We will suppose moreover that Ẽ ₁₀ = Ẽ ₂₀ = Ẽ ₀. That is to say that the two dipoles are coherently excited. In this approximation, the calculation of the three parts of <W>_t gives:

 

Fig. 5. Inverse of squared impedance $\left(\frac{1}{{\mid 1-e^{2\mathit{ikd}}{f}^2\left(kd\right)\mid }^2}\right)$ of a pair of interacting dipoles for different values of distance: d=λ ₀ (blue), 2λ ₀ (green), 3λ ₀ (red), 4λ ₀ (yellow), 5λ ₀ (cyan). Curves plotted as a function of λ ₀/λ.

Download Full Size | PDF

All these terms have a simple physical meaning. The first one in Eq. (9): <|Ẽ ₀|²>_ω, is a constant, and is the energy well where dipole 1 sits, due to the single incident field. The second one of the same equation is the auto-coherence of the twice scattered field Ẽ₀₁. It is the product between the incident field and the field scattered by dipole 1, which has traveled the distance d twice, after a reflection on the second dipole. This double scattering explains the $\frac{1}{{\left(2k_{0}d\right)}^2}$ dependency. The integral will be zero for a large incident spectrum. In Eq. (10), we performed the calculation of the energy well due to scattering by the second dipole only. In Eq. (11), we have a mutual coherence term between incident fields on dipoles 1 and 2. The exponential term in the integral being the retardation (Fig. 6). Here we have the $\frac{1}{kd}$ dependency of the interaction potential energy. When the retardation is so big that the scattered field 2 and the incident field 1 are incoherent, the average over time is zero, as well as the second term of Eq. (9).

As a conclusion, for two dipoles, the interaction energy has a $\frac{1}{kd}$ dependency while d is smaller than the coherence length of the source. The relevant parameter is then the first harmonic of the autocorrelation function of the incident field. When d is bigger than the coherence length of the binding light, the interaction between the two dipoles is weaker and more precisely, averaged interaction over time has a $\frac{1}{{\left(kd\right)}^2}$ dependency. The line width can be interpreted as the inverse of the duration of a pulsed beam. If the incident light is a quasi-monochromatic impulse, the distance between dipoles is bigger than the coherence length if excitation and first scattering do not overlap. Each sphere will feel a succession of impulses due to multiple reflections on both dipoles. The first scattering is the biggest one and will not interact with the incident impulse. The interaction energy will then be merely proportional to the squared scattered field, that is to say to $\frac{1}{{\left(kd\right)}^2}$.

 

Fig. 6. Interaction between two dipoles.

Download Full Size | PDF

This phenomenon can be studied in the infinite number of dipoles case. In the first paragraph, we obtained that the field enhancement happened for a precise number of dipole in the chain. Then, we expect similar effect tuning the coherence length of the incident light on the critical chain length. This way, the number of coherently interacting dipoles will be reduced.

3.2. Infinite chain of dipoles

Relating to the first paragraph, when the number of particles in the chain becomes too large, the field collapses. This phenomenon appears because the summation at the denominator diverges. The critical number of dipole is approximately given by $ln\left(N\right)\simeq \frac{1}{k^3\alpha }$. Then, it seems impossible to keep a strong field enhancement for chains with more than this typical number of dipoles. In the case of an infinite chain, as the incident field is the same for all the dipoles, it can be supposed that the felt field is the same for all of them. With this hypothesis of translation invariance, the infinite-chain case can be solved exactly:

In this extreme regime, the resonance appears for a chain step different from λ (see Fig. 7). For a self-organized chain of dipoles, the step will self-tune so as to keep field enhancement. There are here two possibilities, two stable crystalline states, similar to allotropic phases in solid state physics. The difference between these two phases will be a slightly different step. On the curve presented on Fig. 7, the relative step difference is typically 0.2%. According to the first paragraph, the binding energy is maximum for a finite number of interacting dipoles. A solution is to widen the spectrum of the source in order to reduce the number of coherently interacting dipoles. Multiplying the presented curve (Fig. 7) by a Gaussian spectrum, the spectrum of the propagating field in the chain has two peaks. In time domain, the field beats. In this configuration as in the one presented in the first part, the enhancement of the field increases when k ³ α decreases. Meanwhile, the infinite chain approximation is correct for a larger number of dipoles when k ³ α is small.

In the case of a self-organized chain of dipoles, the collapsing of the field for the central frequency and the two local maxima might lead to mechanical instabilities. We reach here a limit of our numerical model. A more realistic model should take into account the building process of the chain. For an adiabatically built chain, built particle by particle, mainly driven by edge effects, we should obtain a single step. However, for a given initial linear density of particles equal to 1/λ over an infinite distance, the chain should be a melting of the two possible allotropic crystalline states.

 

Fig. 7. ${\left(\frac{E_n}{E_0}\right)}^2$ for an infinite chain of dipoles as a function of λ ₀/λ. Curve plotted for $k_0^3$ α= 0.4.

Download Full Size | PDF

Finally, we can notice that the effect of coherence observed for two dipoles cannot be exactly extrapolated to an infinite chain because of collective effects.

4. Space modulation of the phase of the trapping light

A second possibility to keep an enhancement effect when the chain is big is to spatially modulate the phase of the incident field. Here, we consider a slightly different experimental configuration where the chain of dipoles is still trapped in a plane fringe with a sinusoidal energy distribution, but the phase of the incident field is modulated. Such an experimental configuration can be obtained by giving a small relative angle between the two (nearly) counter-propagating trapping beams. This way, in a bright fringe, the phase varies linearly. Then, we replace E ₀ in the former equations by E ₀ e ^2iπn/N. Now, the “N” is the phase modulation period of the incident field and not the number of dipoles (like in the first paragraph) as the chain we consider is infinite. We assume the same periodic dependency for the amplitude of the field seen by the dipoles. In Eq. (1), E_n is then replaced by E_ne ^2iπn/N. With this hypothesis, the field seen by the n^th dipole is:

Space modulation of the phase of the incident light offers a free parameter. Changing N, it is possible to keep a field enhancement for a given grating step kd. In this summation, only a few terms have a simple mathematical expression [17]. However, some numerical results are presented on Figs. 8 and 9.

 

Fig. 8. Color coded intensity as a function of the grating step $\frac{kd}{2\pi }$ and the phase modulation period N. Curve plotted for k ³ α = 0.86. One of the tails is enlarged in the upper right corner. Field enhancement is the strongest in this part of the curve.

Download Full Size | PDF

When kd = 2π and N tends to infinity, the field falls down to zero as already observed in the previous paragraphs. When the step of the phase modulation is short enough, a second maximum appears, two crystalline states are then stable. In this experimental configuration, the chain will have very likely two allotropic crystalline phases. We assume the chain could be formed with a single phase or a binary melting phase. The final separation distance between particles will depend on the building process of the chain, on the relative stability of the two allotropic states and on external fluctuations as already discussed in the previous paragraph. We can also notice that maximum field enhancement is only one half than the one for a finite chain. By using a more complicated structure for phase modulation, it should be possible to improve this ratio and to get closer to the finite chain case.

5. Conclusion

Relying on previous works of optical binding and on the study of the field enhancement structure in chains of particles in the Rayleigh range, we performed numerical calculations to predict the stability of long optically bound linear chains. The slowly decaying law of the scattered field first suggested a logarithmical increase of the binding energy with the number of particles. This increasing law is only correct for small chains. For too big a periodic chain, the incoming beam cannot be efficiently coupled with the chain and the field experienced by each dipole collapses. The critical number of particles was shown to depend on their cross sections. We proposed two solutions to keep a strong enhancement and to reduce the number of interacting dipoles even for infinite chains. The first consisted in reducing the coherence length of the trapping light but the intuition inspired by the two dipoles case could not apply to infinite chains due to collective effects. The second possibility we developed was a spatially phase-modulated trapping fringe. This solution seems to be much more promising as we obtained a field enhancement. The study of a more complicated phase structure of the trapping fringe may give even stronger enhancements.

 

Fig. 9. Cross section of the curve on Fig. 8 for N=100 (blue curve), N=150 (green curve) and N=200 (red curve).

Download Full Size | PDF

Our calculations were performed in the Rayleigh range. For particles in the Mie range, multiple-scattering calculations should take into account multimodal coupling of spherical harmonics[18]. However, when microspheres are far away from each other (compared to their radius), the far field scattered by each sphere is dominated by the $\frac{e^{\mathit{ikd}}}{kd}$ law like in the dipolar case. Similar results concerning infinite periodic chains are then expected.