1. Introduction

The continued integration of photonic devices into multi-functional chips is one of the most important drivers for the modern photonics industry [1]. Integration offers increased reliability and decreased costs in the same fashion as integrated electronics enabled the computation revolution. Notably, the robustness of integrated photonics has also enabled new generations of quantum logic devices, which are particularly sensitive to environmental fluctuations and device irregularities [2].

Coherent adiabatic optical devices are gaining interest as they afford robust and controllable frameworks that are resilient to wavelength, realisation, or disorder in the design processes. Here we focus on the adiabatic three-state transfer method of SAP (Spatial Adiabatic Passage) [3–7] a spatial analog of the well-known STIRAP (STImulated Raman Adiabatic Passage) [8]. Much work has been done to describe the properties of effective three-state systems under certain idealised conditions that neglect effects due to, amongst other things: digitization error, unequal propagation constants and couplings beyond nearest neighbour. In the adiabatic limit, the robustness against design imperfections means that many of these complications can be subsumed in the effective coupling or some loss property.

Recent work has looked into the design of adiabatic systems that use piecewise or “digital” control schemes instead of continuous parameter variation [9–13]. We stress that the concept of adiabaticity is formally inapplicable in such cases, as adiabatic following is only possible with continuous variation in the control parameters. Despite this, digital adiabatic passage mimics the behaviour and robustness of typical adiabatic devices. Such a design pathway opens up possibilities for systems with inherent digitisation or non-continuous devices, such as are typically found with maskless lithographic write processes.

An important technique for rapid-prototyping of integrated waveguide devices is the femtosecond laser direct write (FLDW) approach [14,15]. This approach uses a focused intense laser to modify the refractive index of a glass material to generate core-cladding type waveguides. The write pattern is controlled in three dimensions, allowing highly novel devices to be achieved, including for example tritters [14] and exotic geometries [16]. One issue with FLDW is that day-to-day reproducibility of the write power is difficult, which in turn affects the refractive index variation between the core and cladding. To overcome this limitation, typically large arrays of devices with systematically varying properties are fabricated to identify the optimal device. It is thus attractive to study device architectures that show increased robustness to such device variability.

Here we study theoretically the properties of Digital Adiabatic Passage (DAP) applied to FLDW integrated photonic circuits. We consider Gaussian profile circular guides which have already shown to be able to generate functioning adiabatic devices [17], operating in the weakly guiding regime. This design has been chosen because of its structural simplicity and (semi-)analytical coupling function but neither are requirements for this method. An illustration of a digital device is shown in Fig. 1. We generate effective tight-binding models whose couplings are verified by rigorous full-wave descriptions of these systems. These descriptions are calculated using a custom EigenMode Expansion (EME) tool [18]. We show that despite digitisation, these devices operate with high fidelity with robustness to both operating wavelength and refractive index contrast. We consider the devices here to be suitable forerunners and valid benchmarks for future novel digital systems. We study the experimental implementation of our designs in the following paper [19].

 

Fig. 1 (a) Structure for digital waveguide adiabatic passage showing the segmented waveguide with circular geometry. The counter-intuitive coupling sequence is achieved by light propagating in the z-direction entering at the bottom left waveguide, and exiting via the top right, with the coupling mediated in the x-direction by the central waveguidelets (shown colored). Figures (b), (c) and (d) show the refractive index profiles for the red, green and blue cases from (a), demonstrating the additive nature of a continuous refractive index profile. The red lines are the refractive index of each element independently, and the blue lines show the sum of the refractive indices. When the central waveguidelet is closest to one of the outer waveguides, the independent waveguide approximation breaks down. The last two waveguidelet images are mirror images of the first two and are not shown for brevity. Note that these images are purely for illustrative purposes only; the particular device parameters can be found in Table 1.

Download Full Size | PDF

DAP is closely related to piecewise adiabatic passage (PAP) [9, 20]. Both schemes use discontinuous coupling pulses to effect adiabatic-like transport. PAP was envisaged with laser pulses applied in a STIRAP-like protocol, and therefore the couplings are smoothly varied in controlled ratios in each pulse. Such an approach would not be practical in a waveguide setting and hence we explore the digital scheme with abrupt changes in coupling.

For any digital variation in nearest neighbor couplings it is possible to determine a compensated scheme, explained below, where the lengths of the piecewise waveguide segments, which we term waveguidelets, are varied so as to optimise the transport [12]. This optimisation method is compatible with any other system that can be described with (or approximated by) a tight-binding basis inter alia strip waveguides, (hybrid) ridge waveguides, planar waveguides, multi-core fibres, and may be useful for non-photonic systems [21] opening up more new potential design opportunities.

This paper is organised as follows: we begin with an analysis of the general Hamiltonian for three-state digital adiabatic passage. Next, we use realistic writing parameters and material properties to generate the effective tight-binding model for our systems of interest. Using these parameters we present designs for three-state digital adiabatic passage devices, we analyse some of the expected design limitations and their effects on performance, including next nearest neighbour coupling and non-uniformity in the waveguide effective refractive indices. Finally, using these techniques we design two adiabatic power dividers with different morphologies.

2. Hamiltonian

Three-state adiabatic passage is described by the following generic Hamiltonian:

The physics of any Hamiltonian can be described by solving for its eigenvalues and eigen-vectors. To solve for the eigenvalues of a 3 × 3 Hamiltonian we solve the cubic characteristic equation,

3. Tight-binding Hamiltonian

The previous section assumed a three-state solution with arbitrarily controllable parameters. In practice, all of the parameters are a function of the write geometry and are not completely independent. While having perfect control over device lengths is a more direct comparison to [12] which focused on length-dependent effects, the more physically relevant variable for FLDW guides is that the designer likely has a high level of precision in the control of length, separation and wavelength but may have systematic imprecision in the parameters controlling magnitude of coupling and wavelength dependence. However, the DAP approach provides significant robustness, and we show that devices can be used for operation across different wavelength regimes, an advantage for practical devices. Here we show how to calculate the tight-binding parameters from the usual waveguide modelling data.

To account for wavelength dependent refractive index we let the cladding index n_cl vary according to the Sellmeier equation of silica (SO₂) glass [22] (with λ expressed in μm):

Equation (15) gives the nearest-neighbour couplings where we have included the more commonly cited exponential approximation (16) for completeness. To arrive at (16) an asymptotic series of the modified Bessel function is taken. This asymptotic series leads to an over-estimate of couplings at all separations. However, exponentially large coupling corresponds to very short distances; at such length scales the guides are no longer optically separate. Therefore both coupling functions can be used in the well-separated regime. A comparison of the analytically and numerically obtained coupling values can be found in Fig. 2.

 

Fig. 2 (a) Effective change to propagation constant due to the presence of another guide. (b) Numerically (solid) and semi-analytically (dashed) obtained coupling of Gaussian index fibers. The minimum separation is 2ρ so that the waveguides are clearly distinguishable. Device parameters are given in Table 1.

Download Full Size | PDF

4. Device design

The DAP device is realised by digitising the central waveguide of standard waveguide adiabatic passage into several parallel piecewise continuous waveguidelets. For any digital variation in Ω_ab and Ω_bc, it is possible to determine a compensated scheme where the lengths of the waveguidelets are varied so as to optimise the transport [12]. The reason for this is that when discretely shifted out of equilibrium, the state of the system oscillates between nearby eigenstates with an angular frequency directly proportional to the difference in their eigenvalues ω_ij = (E_i − E_j), if the coupling values or energies are known, we can compensate against the effect. The identity operation is completed per integer (n) rotation L_n = 2πn/ω_ij, and thus no transfer occurs. Conversely, if the waveguidelet lengths are set at half integer increments L′_n = 2π(n − 1/2)/ω_ij maximal transfer between states occurs. We refer to the first solution L′₁ as the optimal segment length. The compensated scheme is robust to variations in the operating wavelength. For ideal systems with equal propagation terms or no direct next nearest neighbour (a–c) coupling, the effective a–c coupling rate [12] the ideal segment length is $L_{\text{opt}}=\pi {\left({\mathrm{\Omega }}_{ab}^2+{\mathrm{\Omega }}_{bc}^2\right)}^{-1/2}$.

To most strongly demonstrate digital adiabatic passage, the waveguidelets are separated in the longitudinal direction so that any excitation left in the central waveguide at the end of each segment will scatter. Note that because the outer waveguides are so well separated, the exponential term in (16) is orders of magnitude smaller than the smallest coupling observed. Any appreciable influence on the transport would require a spacing between waveguidelets many orders of magnitude larger than those considered. Hence, we set Ω_ac = 0 with very little-to-no effect on the dynamics. As observed by Shapiro et al. [9, 20], where there is no coupling, there is no evolution, and the discontinuities just remove excitation. Therefore the spacings between waveguidelets in the transport direction have no effect on the evolution. We also assume the propagation constants to be equal, i.e. β_a = β_b = β_c, this equality does not always hold in general and is discussed further in the following section. As the ratio of coupling values determines the instantaneous eigenstates we choose coupling values (and hence positions) such that equal excitation is transported each step. The device parameters are shown in Table 1 and their resultant final state excitations in Fig. 3.

Table 1. Device geometry and parameters used in all calculations regarding the three-state coupler. DAP is from |a〉 to |c〉, and the central waveguide, |b〉, is split into 5 waveguidelets, |b〉₁ to |b〉₅. Propagation occurs in the z-direction and all segments are aligned at y = 0. |a〉, |b〉₁, |c〉 all begin at z = 0. Segment |b〉_i+1 is connected to the end of segment |b〉_i. The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings.

View Table | View all tables in this article

 

Fig. 3 (a) Pseudo-colour plot showing the final state population (colour axis) as a function of δ and λ.(b) Pseudo-colour plot showing final state population as a function of λ and device length, L. In both cases note the wide wavelength range over which devices provide high-fidelity transport. The fidelity is periodic, and we have highlighted only one period here. The dark patch in the top right of the length subfigure is actually a pessimal resonance [12] with 90% in the initial state despite being designed for a completely different length and wavelength. All other parameters not being varied are the same as they are in Table 1.

Download Full Size | PDF

The parameters in Table 1 show a vast robustness to operating wavelength and variations in the local refractive index difference as shown in Fig. 3, where the large bright regions indicate high fidelity adiabatic transport (> 90%) over a broad 100 nm wavelength range about the optimal parameters, and indeed showing similar bandwidths away from its designed optimal range. Evidently, when one parameter deviates from its intended value, the coupling values (and hence device lengths) are no longer optimized. Despite this there are still regions of optimality. This can be explained by Eq. (16), where a positive increase in δ, hence Δ, leads to a decreased coupling, and increases in wavelength lead to increased coupling. Despite there not being a one-to-one relationship between the coupling deviations of wavelength or refractive index, the parameters shown herein are only marginally different and result in only a small decrease in peak efficiency away from the chosen parameters (∼ 1%). Indeed, a similar plot exhibiting the same features could be made for δ versus L.

We have modelled our devices from a range of around L = 30–70mm. This was to ensure clear adiabatic results and to work with the diffusivity of the FLDW write process. Other designs with close waveguide spacings (e.g. ridge waveguides) and hence higher couplings, would result in commensurately shorter total device length.

5. Non-dissipative physical design loss mechanisms

As discussed earlier, residual population in the central waveguidelets at the end of each segment will be scattered, reducing the overall transmission from |a〉 to |c〉 and acting as an effective source of loss. We now discuss the possible loss mechanisms originating from reintroduction of population into |b〉 from two important effects: the next-nearest neighbour coupling and the difference in propagation constants. We stress that these are not design errors, but unavoidable consequences of realistic device geometries. That is, even when the written device has perfectly tuned L_opt, there will still be loss. In this section we discuss the worst case scenario, where all population is lost at each discrete step, this results in loss that increases with the number of segments, N. Indeed, if we connect the waveguidelets, only a certain portion of the residual population will scatter (inversely with the state overlap between sucessive waveguidelets) and thus would decrease with N, which is not discussed here. As stated earlier we focus on the losses originating from next-nearest neighbour coupling and a difference in the propagation constants. Each of these perturbations will shift the optimal waveguidelet length and the following derivations are derived with respect to that point; fabrication error in device length or structure is not considered. The following perturbations are cumulative, if Ω_ac, β_diff ≪ Ω; the change to the population in the central state is the sum of each contribution.

5.1. Next nearest neighbour coupling

In many device designs, next-nearest neighbour coupling (here coupling between the outer waveguides) is taken to be zero for convenience. This is typically acceptable as coupling is often negligible because three-state adiabatic passage is robust against small direct left-right coupling, and a simple heuristic for determining when such coupling is important can be found [24]. We now consider cases where this coupling is non-zero and the implications for digital processes. We plot both approximate and analytic forms of these errors in Fig. 4. The overlap of |b〉 with |E₂〉 is analytically described by (5), which, to first order in Ω_ac is:

 

Fig. 4 P_b = |〈E₂|b〉|² as a function of scaled perturbative parameters using analytical (solid) and approximate (dashed) forms for Ω_ac (left) and β_diff (right). Values are symmetric with respect to Ω_ab ↔ Ω_bc, to represent data in reduced units we divide through all parameters by Ω_ab. We also divide the perturbative parameters by Ω_bc, as any value exceeding Ω_bc/2 would no longer be a perturbation. The functions each have a local maximum at ${\mathrm{\Omega }}_{bc}={\mathrm{\Omega }}_{ab}/\sqrt{5}$ and ${\mathrm{\Omega }}_{bc}={\mathrm{\Omega }}_{ab}/\sqrt{2}$ (see (17) and (19) respectively) and so values are linearly spaced up to those points. These data show that the approximations are good over a wide range of possible values with deviations strongest at the turning point.

Download Full Size | PDF

5.2. Propagation constant mismatch

The derivation of the tight-binding parameters relies on the waveguides and their modes being optically separable, i.e. one can clearly distinguish when one ends and the other begins. In Fig. 1 we can see how designing a device by additive diffusive profiles can instigate an effective change to the propagation constants of the Hamiltonian and the independent waveguide approximation breaks down, thereby intertwining the two waveguides and their modes. The following derivation assumes that only two of neighbouring guides’ propagation constants are approximately equal, and the third different, for example β_b ≈ β_c ≠ β_a, i.e. the two nearest guides are strongly affecting each other but are only weakly affected by the next-nearest guide, in our design this corresponds to the waveguides confining modes |b〉 and |c〉 being close to each other, the solution also corresponds to β_a ≈ β_b ≠ β_c by symmetry but the derivation itself works in general. Letting β_diff be the difference between the strongly and weakly coupled guides, then the approximate on-site term for the central state is given by:

6. Digital adiabatic power dividers

In sections 4 and 5 we demonstrated the robustness of digital adiabatic passage for a three-state based optical coupler. In this section we will show that the principles of digital adiabatic passage can be extended to geometries other than this simple three-state set up. We will focus on two new devices: a planar five-state 50:50 power divider modelled on the designs in [25] based on the three-state device in section 2, and a triangular four-state arbitrary power divider, based on the designs in [26–28]. Simple illustrations of these geometries can be found in Figs. 5 and 7.

 

Fig. 5 Illustrations of the mirrored five-state device considered in this subsection. (left) schematic demonstrating the couplings of the device and (right) illustration demonstrating the mirror geometry, each color corresponds to a set of waveguides described in Table 2. Light is injected in |c〉 and coupled into a superposition of |a〉 and |e〉. For simplicity, we will only consider when Ω_bc = Ω_cd and Ω_ab = Ω_de resulting in a equal division of power. The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings. All waveguides are placed such that no next-nearest neighbour coupling is possible.

Download Full Size | PDF

6.1. Five-state symmetric adiabatic passage

In section 4 we demonstrated how to design a three-state based coupler using an input state, an intermediate state and an output state. To design a five-state power divider (Fig. 5) we effectively “mirror’ the design about the input waveguide. This yields a device with two intermediate states and two outputs. The Hamiltonian for this device is:

Table 2. Device geometry and parameters used in transport calculations for five-state based 50:50 splitter. DAP is from |c〉 to |a〉 and |e〉, and the intermediate waveguides, |b〉, |d〉, are split into 5 waveguidelets each, |b〉₁, |d〉₁ to |b〉₅, |d〉₅. Propagation occurs in the z-direction and all segments are aligned at y = 0. |a〉, |b〉₁, |c〉, |d〉₁, |e〉 all begin at z = 0. Segment |b〉_i+1 (|d〉_i+1) is connected to the end of segment |b〉_i (|d〉_i). The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings. Recall that the device is symmetric about |c〉, and the positions of |a〉, |e〉 and |b〉, |d〉 are related by x → −x. Properties ρ, δ, λ_opt are the same as Table 1.

View Table | View all tables in this article

 

Fig. 6 Populations P_i = |〈ψ|i〉|² during the five-state digital adiabatic 50:50 power division protocol on a regular (left) and log (right) scale. Faint dashed lines show the end/beginning of waveguidelets. It can be seen that the population traces P_a, P_e and P_b, P_d are directly on top of each other and that population in the intermediate waveguides goes to zero at the end of each waveguidelet. Coupling values were chosen to transfer equal population per step and device parameters can be found in Table 2.

Download Full Size | PDF

 

Fig. 7 (left) Illustration demonstrating couplings in the triangular four-state system considered in this subsection. (right) Schematic to-scale side-view of the parameters found in Table 3, waveguide sizes are decreased for distinguishability and the inset provided demonstrates the relatively small vertical movement. Light is injected into |a〉 and is brought into a controlled superposition of |c〉 and |d〉 by controlling the ratio $\alpha ={\mathrm{\Omega }}_{bc}/{\mathrm{\Omega }}_{bd}=\sqrt{P_c/P_d}$. The x-coordinate describes the centre of the waveguide. The x and y-coordinates describe the centre of the waveguides. Segments are varied in the x and y-directions to vary the couplings while maintaining a constant ratio of couplings $\alpha ={\mathrm{\Omega }}_{bc}/{\mathrm{\Omega }}_{bd}=\sqrt{P_c/P_d}$. All waveguides are placed such that no next-nearest neighbour coupling is possible.

Download Full Size | PDF

6.2. Four-state controlled ratio splitter

A more flexible geometry for beam splitting is a four-state scheme [26] Now we consider a set up similar to the original three-state based coupler with two output states instead of one. Positioning our outer waveguides far enough apart so that the coupling is zero, we arrive at the following Hamiltonian:

 

Fig. 8 Populations P_i = |〈ψ|i〉|² during the four-state adiabatic 1/3:2/3 power division protocol on a regular (left) and log (right) scale. Faint dashed lines show the end/beginning of waveguidelets. It can be seen that the population in the intermediate waveguides goes to zero at the end of each waveguidelet. Coupling values were chosen to transfer equal population per step and device parameters can be found in Table 3.

Download Full Size | PDF

It can be seen in Table 3 that the four-state power splitter is twice as long as the three-state adiabatic coupler, and this is due to the increased distance between the central waveguidelets and the outer waveguides required to minimize cross-talk. One could decrease these distances and get a trade-off between coupling between outer states and a smaller device but these distances were chosen to be internally consistent within this paper.

Table 3. Device geometry and parameters used in transport calculations for four-state based 1/3:2/3 splitter. DAP is from |a〉 to |c〉 and |d〉, and the intermediate waveguide, |b〉, is split into 5 waveguidelets, |b〉₁ to |b〉₅. Propagation occurs in the z-direction. |a〉, |b〉₁, |c〉, |d〉 all begin at z = 0. Segment |b〉_i+1 is connected to the end of segment |b〉_i. The x and y-coordinates describe the centre of the waveguides. Segments are varied in the x and y-directions to vary the couplings while maintaining a constant ratio of couplings ${\mathrm{\Omega }}_{bc}/{\mathrm{\Omega }}_{bd}=\sqrt{P_c/P_d}$. Properties ρ, δ, λ_opt are the same as Table 1. Distances between |a〉, |c〉, and |d〉 are all 21μm to minimize cross-talk.

View Table | View all tables in this article

7. Conclusion

Our results indicate that digital adiabatic processes are potentially a useful new technique to be employed in the design of photonic circuits. Properties not commonly discussed such as the shift in propagation constants due to adjacent guides were also introduced and discussed. We have shown that despite digitisation, devices give high fidelity transport with broadband spectral response. Our modelling also highlights the fact that digital approaches should work for other adiabatic process, in particular should be able to assist the design of adiabatic quantum gates [8,28,29]. This approach to spatial adiabatic passage opens new design rules and hence the potential for new/more complicated photonic devices. In the following paper [19] we show fabrication of waveguide DAP devices that confirm our predictions.