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Abstract
We experimentally demonstrate that the next-nearest-neighbor-coupling (NNN-coupling) in an array of waveguides can naturally be negative. To do so, dielectric zig-zag shaped waveguide arrays are fabricated with direct laser writing (DLW). By changing the angle of the zig-zag shape it is possible to tune between positive and negative ratios of nearest and next-nearest-neighbor coupling, which also allows to reduce the impact of the NNN-coupling to zero at the correct respective angle. We confirm the existence of negative NNN-couplings experimentally and show the improved accuracy of a refined tight-binding model. Our results are also of importance for other discrete systems in which the tight-binding model is often used.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The tight-binding model is an approximation that is able to reduce the complexity of a system to the point where the dynamics can be described by discrete coupled mode equations. This strong simplification allows one to experiment with interesting theoretical models at different physical platforms, for example cold gases [1], electrical circuits [2], evanescently coupled dielectric [3], plasmonic [4] or magnonic [5] waveguides and many more. In waveguides, we can use the tight-binding approximation because the light is mostly confined inside the core of one waveguide. The waveguides nearby are coupled to each other due to the overlap of their evanescent electric fields. Often in these structures, only the couplings to the nearest-neighboring waveguides are considered and those further away are neglected. However, this is valid only in the case of large distances between waveguides, due to the exponential decay of the confined electric field. We take a closer look at the coupling to the next-nearest-neighbors (NNN) in an array of closely spaced waveguides. Thereby we discover that counterintuitively, the NNN-coupling can naturally be negative. Negative couplings to the nearest-neighbor (NN) have been reportedly implemented by detuning the potential [6–9], dynamic modulation of the position [10,11], or usage of the complex valued field amplitudes of a higher mode [12]. None of these mechanisms is implemented in our system. It is composed of unmodulated identical single-mode waveguides, but still shows negative coupling. Other previous experimental investigations of the NNN-coupling are either more constraint in their geometry [13,14] or have only measured the amplitude of the NNN-coupling [15,16] and assumed it to be naturally positive. The sign of the NNN-coupling, however, is of great importance, since in some systems the NNN-coupling is the defining parameter for the creation of topologically non-trivial phases or states [1,17–19].
2. Model
We discuss a periodic array of waveguides arranged in a zig-zag shape as shown in Fig. 1(a). By changing the angle $\alpha$, the distance $d_2$ to the NNN can be varied, allowing us to change the NNN-coupling $c_2$ without changing the distance $d_1$ to the NN, thereby keeping the NN-coupling $c_1$ almost constant. Coupling terms to the waveguides further away like to the 3rd or 4th neighbor ($c_3$ and $c_4$), are only relevant for large $\alpha$. Calculations based on non-orthogonal coupled mode equations for this arrangement, with the parameters of the sample used in our experiment, reveal that the NNN-coupling $c_2$ can be negative, as can be seen in Fig. 1(b). $c_2$ increases with the angle $\alpha$ of the array, as the distance to the NNN decreases. For a straight chain ($\alpha = {0}^{\circ}$) $c_2$ has a negative value, that turns to zero at an angle of approximately $ {40}^{\circ}$. At $\alpha = {60}^{\circ}$, the distances $d_1$ and $d_2$ are equal, therefore, we do not consider the cases where $\alpha > {60}^{\circ}$ as the role between NN and NNN interchanges. The coupling $c_3$ and $c_4$ to more distant waveguides are close to zero for angles $\alpha < {40}^{\circ}$ and have only a small impact. The specific angle at which $c_2$ turns to zero is not fixed to 40°, but depends on the parameters of the sample. However, we want to emphasise that there always has to be an angle at which the NNN-coupling is zero due to the fact that it has to be a real number which changes steadily with $\alpha$ and is negative for $\alpha = {0}^{\circ}$ and positive at $\alpha = {60}^{\circ}$. The ability to tune the NNN-coupling to zero could be useful also in other experiments if the influence of NNN-coupling is disturbing or unwanted [20].
Fig. 1. (a) Scheme of an angled 1D array of waveguides with an angle $\alpha$, the nearest-neighbor coupling $c_1$ and the next-nearest-neighbor coupling $c_2$. The distance $d_x$ is the size of a unit cell, $L$ is the propagation length and $d_1$ and $d_2$ are the distances to the nearest and next-nearest-neighbor, respectively. (b) Coupling coefficients calculated by non-orthogonal coupled mode equations for differently angled arrays of waveguides with the parameters of the sample used in the experiment (see Fig. 3). The coefficients $c_1$, $c_2$, $c_3$ and $c_4$ are the couplings to the 1st, 2nd, 3rd and 4th neighbor, respectively. (c) Microscope image of the input facet of the fabricated structure with five arrays with $\alpha = {45}^{\circ}, {30}^{\circ}, {0}^{\circ}, {40}^{\circ}, {50}^{\circ}$, respectively. (d) Photo of the fabricated waveguide structure with 1 cm as reference.
The common tight-binding approximation for waveguide structures is based on the assumption that the eigenmode and eigenenergy is mostly determined by the individual waveguide, due to the relatively strong binding by the higher refractive index at the core. The presence of other waveguides nearby is treated as a small perturbation to the original approximation, accounted for by a coupling term that causes the individual eigenenergies to hybridize. This simplification allows us to describe the evolution of the field by the amplitudes $\vec {a}(z)=(\cdots,a_{p-1},a_p(z),a_{p+1}(z),\ldots )^{T}$ of the eigenmodes in each waveguide $p$ by a system of coupled mode equations
2.1 Non-orthogonal tight-binding approximation
To accurately describe the system by coupled mode equations, when the distances between waveguides are small, it is important to take into account the non-orthogonality of the original basis states of the uncoupled waveguides. The original basis states, which are given by Eq. (3), in many cases have a spatial overlap small enough to approximate them to be orthogonal. For small distances, when the NNN-coupling can no longer be neglected, also the increasing non-orthogonality has to be taken into account, as stated in Ref. [22,23]. This is the reason why the original coupled mode equations of Eq. (1) do not correctly give the coefficient for coupling to waveguides more distant than the NN. To correct this, the non-orthogonality has to be accounted for [21]
The diagonal elements of $\kappa$ are 1 and the other elements are between 0 and 1 (implying that $E_q$ and $E_p$ are positive and normalised functions). Each vector in $\kappa$ is longer than 1 and they are not orthogonal to each other, i.e., $\kappa$ is not normalized. Our goal is to eliminate $\kappa$ by changing into a basis where $\kappa$ is the identity matrix. By doing so, all new basis modes are normalised and orthogonal to each other.
The transformation into an orthogonal basis of localized functions $\vec {b}\left ( {z}\right )=\underline {T}\vec {a}\left ( {z}\right )$ can be done via the transformation matrix $\underline {T}$, which is the square root of $\underline {\kappa }=\underline {T}^{\dagger }\underline {T}$. After the transformation, the coupled mode equations return to the simpler form of Eq. (1) [24]
It is also worth mentioning that the new orthogonal basis states exponentially decay over the nearest sites and have an oscillating shape with a period roughly twice the distance $d_1$ to the next neighbor (see Fig. 2). The new basis functions share these features with the Wannier-functions, which are the Fourier-transformation of the Bloch-functions, and as such also form an orthogonal basis of localized functions. When these new orthogonal basis states are used for $E_q$ and $E_p$ instead of the unperturbed eigenmodes to calculate the overlap integral in Eq. (2), it becomes more intuitive why the NNN-coupling can have a negative sign with respect to the NN-coupling. In contrast to the original basis states from Eq. (3) which are purely positive and where the overlap can thereby only be positive, the new basis states have areas where they are negative so that the overlap can be negative. Due to the oscillating shape, the sign of the overlap also alternates with a period of roughly $2d_1$. For $\alpha = {0}^{\circ}$ where the NNN is at the distance of $2d_1$ and the NN is at the distance $d_1$, the overlap integral to those waveguides have opposite signs. This is similar to Ref. [6], where the oscillating shape of the defect mode allows the observation of positive and negative coupling constants dependent on the number of waveguides between the defects.
Fig. 2. Normalized spatial field density of one new orthogonal basis state (of $\vec {b}$) for differently angled arrays. In contrast, the original basis states (of $\vec {a}$), which are given by Eq. (3) are purely positive and are not affected by nearby waveguides. The shown spatial field densities of the new orthogonal basis are calculated by the superpositions of the original eigenmodes weighted by the matrix elements of $\underline {T}$. The spatial overlap of these new states is zero and the coupling integral can be positive or negative.
2.2 Experimental demonstration of negative NNN-coupling
To experimentally verify the design possibility of a negative NNN-coupling, we fabricate the corresponding waveguide structures, using the direct laser writing principle employed in [12] (for further details see the Materials and Methods section). We fabricate a sample with three arrays with $\alpha =0^{\circ}$, $40^{\circ}$, $50^{\circ}$ to show the effects for negative, almost zero and positive NNN-coupling, respectively. The negative sign of $c_2$ is not necessarily detectable from the intensity distribution alone, as it mostly affects the phase, but would be noticeable for example in the band structure of the 1D array [15,25,26]
Here $d_x$ is used as the spatial extent of an effective unit cell. The actual structure only repeats after $2d_x$ in the $x$-direction, but due to the fact that the two waveguides in the $2d_x$ unit cell are coupled in the same way to their neighbors, we can reduce the system to an effective unit cell with a thickness of $d_x$.
Since we do not have direct access to the band structure in our experiment, we instead measure the diffraction of a wave packet with momentum $k_x$ [27]. This allows us to gather information about the band structure, as the diffraction is the second order derivative of the band structure at $k_x$. The absolute value of second derivative of Eq. (7) $\left | {\partial ^{2} \beta /{\partial k_x^{2}}}\right |$ determines how much the wave packet spreads along $x$ during propagation through the array in $z$. One sees, that at $k_x=0$ the diffraction is proportional to $\left | {c_1+4c_2}\right |$ while at $k_x=\pi /d_x$ it is proportional to $\left | {c_1-4c_2}\right |$. Therefore, for positive $c_2$, the diffraction of a wave packet at the center of the Brillouin zone is greater than it is at the edge of the Brillouin zone, while for negative values of $c_2$ it is the opposite.
This effect in the diffraction is clearly visible in the intensity profiles at the output facet of the structure, shown in Fig. 3(a), which were captured by a CMOS camera in our experiment. Those images were taken for multiple $k_x$ in the first Brillouin zone and used to calculate the standard deviation, which quantifies the diffraction. These results are summarized in Fig. 3(b).
Fig. 3. (a) Measured intensity profiles at the output facet of the waveguide structure. The intensity profiles are normalized to their respective maximum value to increase visibility. A greater diffraction of a wave packet at the edge of the Brillouin zone at $k_x=\pi /d_x$ compared to one in the center $k_x=0$ points to a negative NNN-coupling. The radius of a waveguide is 1.3 µm, the center-to-center distance to the next waveguide is $d= {4}\,\mathrm{\mu}\textrm{m}$ and the propagation length is 2 mm. The refractive index contrast of the waveguide to the surrounding material is approximately $\Delta n=0.006$ and the used wavelength is 600 nm. (b)-(d) To quantify the diffraction or the spreading of a wave packet, the standard deviation is calculated for the intensity distribution after the propagation. A wave packet is coupled into an array of waveguides with $c_2<0$, $\alpha = {0}^{\circ}$ ; $c_2\approx 0$, $\alpha = {40}^{\circ}$ and $c_2>0$, $\alpha = {50}^{\circ}$, respectively. The standard deviation of the wave packet with momentum $k_x$ after a propagation distance of 2 mm is (b) extracted from the measured intensity profile, (c) calculated via the conventional tight binding approximation with matrix $\underline {c}$ and (d) calculated by accounting for the non-orthogonality with the matrix $\underline {\tilde {c}}$.
To compare the measurement with the theory, tight binding calculations were performed to predict the intensity profiles at the output. As input, the same state is used as in the experiment and the discrete output state of the tight binding calculations is mapped to a spatial intensity distribution using Eq. (3) as basis functions. The standard deviations were calculated as shown in Fig. 3(c,d). The results of the calculation plotted in (c) were based on the standard tight binding approximation where the coupling is always positive, and the results plotted in (d) were based on the corrected coupling constants $\underline {\tilde {c}}$. Note that for negative values of $c_2$ the maximum of the diffraction is at $k_x=\pi /d_x$ while it is at $k_x=0$ for positive values. This is a feature that only shows in the measurement and the non-orthogonal tight binding model, but is missing in the conventional one.
The measured refraction is slightly lower as the one predicted by the two tight-binding calculations, which we assume is due to a too low estimated refractive index contrast $\Delta n$. The measurement is also asymmetric which is most likely caused by fabrication imperfections during the polymerization process such as vignetting or the proximity effect [28]. The overall shape, however, of our measurements qualitatively agrees with the corrected coupling constants $\tilde {c}$ and with the negative value of the NNN-coupling.
The negative sign of $c_2$ in a straight array can intuitively be explained by the increasing similarity to a free particle in the limit of short distances between the waveguides. As the coupling increases, light is less confined to a single waveguide and the dispersion relation approaches the parabolic shape typical for a free particle. The quadratic behavior with $k_x$ follows from the paraxial approximation where $k_z\gg k_x$, such that the dispersion in a homogeneous medium can be written as $\beta \left ( {k_x}\right )=\sqrt {k^{2}-k_x^{2}}\approx k-k_x^{2}/(2k)$ [3]. Bear in mind that due to the mathematical similarity between Schrödinger equation and paraxial Helmholtz equation, in our waveguide model system $\beta$ corresponds to the energy in solid state systems. This limit determines what the smallest value of $c_2/c_1$ for identical waveguides can be. The dispersion relation for an array with the coupling constants $c_m$ up to the $N$th neighbor is given by [25]
In the limit of zero spacing between waveguides, the coupling constants $c_m$ in Eq. (8) have to converge to values of the dispersion relation of a free particle, i.e., a parabolic shape in the first Brillouin zone. By using ${d_x/\pi }\cos \left ( {m k_x d_x}\right )$ with $m\in \mathbb {N}_{>0}$ as a normalized basis and the integral over the Brillouin zone as a scalar product in the space of the symmetric and $d_x$-periodic functions, the ratio of the coupling constants in this limit directly follows
The smallest possible value of $c_2/c_1$ therefore is -1/4, which indeed has not been undercut by any ratios extracted from our measurements and simulations, further pointing to -1/4 as the lower limit. However, if we are not restricting ourselves to waveguides with equal propagation constants only, this limit can be overcome by detuning the next neighbors, as it has been demonstrated in [6,8].
Also, this limit of a free particle can explain why at $\alpha = {60}^{\circ}$ $c_2$ is bigger than $c_1$ even though $d_1=d_2$ (see Fig. 1(b)). As the light is less confined to a single waveguide, the influence of the surrounding refractive index averages out, so that for the light the waveguide array becomes increasingly similar to a box potential which is elongated in the $x$-direction. The NNN-coupling in our zig-zag array acts (in contrast to the NN-coupling) only along the $x$-direction, which, in the limit of an elongated box potential, is the direction where the wavefunction is least hindered to spread. The NN-coupling therefore decreases as $\alpha$ increases, because it increasingly acts along the $y$-direction.
3. Materials and methods
3.1 Sample fabrication
The sample is fabricated in the negative photoresin IP-Dip (Nanoscribe) using a commercial direct laser writing (DLW) system (Nanoscribe Photonic Professional GT). The structure is written layer by layer, where each layer is stacked onto the other in the $z$-direction with a distance of 250 nm. Each layer, which looks like the input facet in Fig. 1(c), is written line by line with a line distance of 100 nm. The refractive index contrast of approximately $\Delta n=0.006$ is achieved by choosing a high laser intensity for the area of the waveguide (LaserPower 75%) and low intensity for the surrounding area (LaserPower 32%) close to the polymerization threshold (LaserPower 30%) at a writing speed of 20 mm s−1. Thereby, LaserPower 100% refers to a laser intensity of 55 mW before the 63$\times$ focusing objective of the DLW system. The radius of a waveguide is 1.3 µm, the center-to-center distance to the next waveguide is $d_1= {4}\,\mathrm{\mu}\textrm{m}$ and the propagation length is $L= {2}\,\textrm{mm}$. After the writing process, excess photoresist on top of the sample, that would cause distortions at the measurement, is removed by dipping the top into PGMEA (Propylene glycol methyl ether acetate) for a minute. As a last step, the sample is exposed to UV light from an Omnicure S2000 with 95% iris opening for 60s to cure excess photoresist.
3.2 Measurement
For the measurement, laser light at a wavelength of 600 nm from a white light laser (NKT photonics) and a VARIA filter box is used. The light is linearly polarized, expanded and sent onto a spatial light modulator (SLM). Afterwards, all light besides the first diffraction order of the blazed grating from the SLM is blocked. With a 20$\times$ objective (NA=0.4), the Fourier transformed hologram from the SLM is imaged onto the input facet of the waveguide sample. The intensity and phase profile on the sample’s input facet can be tuned by adjusting the hologram. To couple a wave packet with momentum $k_x$ into the waveguide array, light is focused into seven neighboring waveguides. For each waveguide, light is coupled in with a Gaussian-shaped intensity-envelope and a phase difference to the central focus corresponding to the $k_x$ we want to measure. The overall envelope of the seven foci has a Gaussian shape with a standard deviation of $d_x$, which corresponds to a wave packet in momentum space with a standard deviation of $1/(2d_x)$. The intensity of the light from the output facet of the waveguide structure is finally imaged via a second 20$\times$ objective onto a CMOS camera (Thorlabs DDC1545M).
4. Conclusion
In conclusion, we explained the counterintuitive negative sign of the NNN-coupling and demonstrated its implications experimentally. We showed that we are able to tune $c_2/c_1$ to zero or even below simply by arranging the waveguides in a zig-zag shape with the corresponding angle. We showed that the simple assumption that the coupling just decreases with distance is not necessarily valid and the non-orthogonality should be accounted for whenever couplings to more distant waveguides become relevant. With these results in mind, we can decrease the distance between the waveguides to achieve higher nearest-neighbor coupling, without introducing unwanted effects caused by NNN-coupling. On the other hand, we can look specifically at model systems where these counterintuitive negative coupling values have been dismissed so far. Our results can also be interesting for other discrete systems in which the tight-binding model is often used, as, e.g., in cold gases in optical lattices.
Funding
Alexander von Humboldt-Stiftung (within the Feodor-Lynen Fellowship program); Deutsche Forschungsgemeinschaft (CRC/Transregio 185 OSCAR (project No. 277625399)).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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