We perform a simple sensitivity analysis of a W1 waveguide bend in a photonic crystal (PhC) where we use the information obtained to optimize the PhC bend’s frequency response. Within a single optimization step we already achieve very low power reflection coefficients over almost the entire frequency range of the photonic bandgap (PBG), i.e., an achromatic bend. A further analysis shows that there is a single critical rod in the optimized bend structure that exhibits an extraordinary high sensitivity at a given frequency. Hence power reflection becomes tunable from 0% up to 100% involving only small changes in the critical rod’s properties. This opens the door to novel topologies for compact switches and sensor applications.
©2003 Optical Society of America
Photonic crystal (PhC) structures [1–3] offer a powerful scheme for the realization of ultra-compact and multi-functional devices for high-density integrated optics . It is well known that simple line defects in such PhC structures form very effective optical waveguides, that provide extremely narrow waveguide bends with nearly perfect power transmission [5–7]. In the framework of conventional waveguiding (using high index-contrast materials) various alternative approaches for sharp bend designs have been theoretically studied in [4, 8], including, e.g., low-Q resonant cavities in combination with corner mirrors. Besides the high transmission performance (up to 98%) within such 2D simulations, bending schemes with waveguide corner mirrors have already been realized earlier allowing power transmissions of 60% up to 90% . Nevertheless, according to its compactness, its efficiency, and the low power reflection and radiation loss sharp bend design relying on PhC defect waveguides turns out to be still one of the most promising approaches that have to be further investigated.
In the following we analyze simple 90° bends as depicted in Fig. 1, which are based on W1 defect waveguides (one line of vacancies). Similar to  the underlying structure consists of a 2D PhC with circular dielectric rods arranged in a square lattice and embedded in air. The lattice data are as follows: the radius of each dielectric rod is r=0.18·a (with a=1 µm being the lattice constant), and the rod’s dielectric constant is ε=11.56. For TM polarized light the PhC’s first photonic band gap (PBG) appears between ω·a/(2·π·c)=0.30 and ω·a/(2·π·c)=0.44, whereas the fundamental defect waveguide mode shows a cutoff at ω·a/(2·π·c)=0.312. The spectral response of the power reflection (or reflectance R) and power transmission (or transmittance T) is also shown in Fig. 1 including the Poynting field for the frequency where R reaches its maximum value.
All simulations are carried out in the frequency domain with the multiple multipole (MMP)  solver contained in the MaX-1 software package [11, 12]. The proper MMP analysis of perfect PhCs , PhC defect waveguides  and the simulation of compact PhC devices  using the connection concept [10, 11, 13] for the broadband analysis of the latter two cases has already been published.
As one can see from Fig. 1, zero reflection is merely obtained for frequency values around ω·a/(2·π·c)=0.35, whereas at higher frequencies power reflection peaks towards 10%. Referring to power conservation reflectance R and the transmittance T should sum up to unity as long as propagating modes are concerned . Therefore power conservation is only warranted if the waveguides are sufficiently long (more than one effective wavelength ) in order to omit evanescent field coupling between the waveguide bend discontinuity and the corresponding device port where the values for R and T are retrieved. As indicated in Fig. 1 power conservation is provided for mostly the entire (first) PBG except when approaching the upper or lower band edge as well as the cutoff frequency of the fundamental waveguide mode. It has already been shown in  that the MMP model can easily be improved if the size of the underlying PhC is increased accordingly. Since it is not reasonable to operate PhC waveguides near the band edges of the PBG, we accept these slight inaccuracies in order to avoid longer computation times required for more accurate models [14, 15]. The reason for favoring small simulation domains (inevitably providing short computation times) lays in our aim to extensively analyze and optimize bend structures on a simple personal computer.
In the remainder of the paper we report on the design of an achromatic W1 waveguide bend. A first attempt to obtain low power reflectivity (around 5%) has already been described in , where a phenomenological picture based on a simple 1D analysis of a conventional waveguide discontinuity has led to an angle cutting of the sharp bend shape providing PhC bends with significantly larger “bending radii”. From an engineering point of view, increasing the size of the bend while maximizing its power transmission becomes somehow controversial because it mimics the characteristics of conventional waveguide bend designs. Thus we investigate a sharp W1 waveguide bend along its proper bending region: It seems obvious, that the rods which are confining the W1 waveguide in the bending area are more likely to impact the bend’s performance than those rods which lay apart. However, it is not clear at all, how these rods will affect the properties of the overall structure. Furthermore, it is important to quantify the influence of fabrication tolerances. In the following, we therefore analyze the influence of the rod size variation on the power reflectivity. Afterwards, we evaluate this information in order to obtain a fully transparent, achromatic W1 waveguide bend.
2. Sensitivity analysis
In a first step we perform a sensitivity analysis of the power reflectivity according to a 10% rod size variation for a limited area of the bend structure. As depicted in Fig. 2 the domain of interest contains 29 rods, thus leading to the sequential computation of the power reflectivity R for 29 readily different models, each having one out of 29 rods slightly modified. For clarity reasons we first decrease the radius of each rod by an amount 10% as shown in Fig. 2 (left) and then increase the rod size by the same value as given in Fig. 2 (right). In order to keep the computation time as small as possible, we only consider the frequency ω·a/(2·π·c)=0.42, where R peaks towards its maximum.
For a majority of the rods we observe that extending their size causes an increase of the power reflectivity R whereas a decrease of the rod radius is correspondingly related to a similar lowering of R. Even though this behavior may imply a linear dependence for most of the rods, there is still a small group in the proper corner area, which behaves opposite while having one rod amongst them showing a highly pronounced sensitivity towards radius variations. Thus, increasing the size of e.g. two rods at the inner waveguide corner (brown in Fig. 2 (left)) and simultaneously reducing the radius of, e.g., five rods in the outer corner (green and blue in Fig. 2 (left)) may be viewed as an intuitive strategy to lower the power reflectivity of the bend at the given frequency. Furthermore, we see that the most sensitive rod (blue in Fig. 2 (left)) becomes the most critical one for fabrication purposes, because small radius variations will considerably affect R.
One should keep in mind that the sensitivity analysis shown in Fig. 2 is far from being complete. The analysis has been restricted to one single frequency as well as to the radius variations only. A more complete optimization should consider the entire frequency range and it has to deal with the influence of small displacements, including deformations of the rod’s shape, and changes in the material properties. Such investigations are usually very time-consuming. However, the intuitive assumption that a distinct set of rods in the vicinity of the waveguide corner turns out to be most influential is confirmed. This brings us towards a first optimization step where we try to minimize R at the given frequency ω·a/(2·π·c)=0.42.
3. Minimization of the reflection coefficient
The sensitivity analysis has already proven successful to decrease power reflection R at a single frequency such as e.g. ω·a/(2·π·c)=0.42. In order to enter proper optimization one has to define the rods that should be increased or decreased as well as the corresponding amount of size variation. Moreover it is still not known how this procedure will affect R at other frequencies. For simplicity reasons and while underpinning the fabrication point of view, we select two reduced sets of rods in the close vicinity of the waveguide corner (as depicted in Fig. 1). Following our sensitivity analysis we choose the inner group at the waveguide corner to undergo a simultaneous increase of the rod’s radius whereas the radius of the rods in the proper corner angle will be decreased by the same amount. Furthermore we now care for the entire frequency response, where a first example is shown in Fig. 3 (left) for ±10% radius variation of the 10 rods (as marked Fig. 1). By linear extrapolation one can estimate the radius variation for vanishing R to be Δr=±13.27%. The associated frequency response is also depicted in Fig. 3 (right). The residual power reflectivity of less than 1% over almost the entire PBG actually contradicts the assumption of linearity, but since we have performed just a very rough single optimization step, this still exceeds all our expectations. From an engineering point of view, we have already obtained an achromatic bend with almost zero reflection.
One could now easily move to a numerical optimization scenario but in order to obtain more insight into the PhC bend’s peculiarities, we now perform a subsequent sensitivity analysis starting from the optimal structure where the two local maxima of the power reflection (at the two frequencies as indicated in the right part of Fig. 3) are observed. Since the power reflection R is already small at these two frequencies, the radius variation is chosen to be only 2%. One should be aware that when tackling such subtle modifications of the PhC bend model an accurate numerical method becomes strictly mandatory. Since the MMP method is always close to analytic solutions, we can easily meet the required accuracy , .
The results of the second sensitivity analysis are shown in the Figs. 4 and 5. As one can see, there is only one critical rod left which allows one to reduce R simultaneously at both frequencies ω·a/(2·π·c)=0.367 and ω·a/(2·π·c)=0.417. The same figures indicate that the critical rod is the only one, that maintains a synchronous behavior regarding the sense of radius variation. Radius variations at other rods will always increase R at the lower frequency and decrease R at the higher frequency or vice versa. Although this would allow us to obtain even a better flattening of the residual power reflectivity’s frequency response R(ω), it is not worth trying because the results are already very convincing.
As a final optimization step we estimate a - 30% radius variation of the single critical rod allowing a further reduction of R at both frequencies. Figure 6 displays a result that is quite surprising. As one can see, the power reflection has now drastically increased up to 100% within a frequency range that lies below the lower local maximum of R (narrowing slightly the PhC bend’s operation bandwidth) but still retains a pass band with almost zero reflection. Providing an almost 100% onset of power reflection from virtually nothing while undergoing a radius variation of - 30% prove this single rod to be extraordinarily sensitive. Even if this may imply an ultimate precision for the underlying fabrication technology it mainly opens the door towards interesting applications. First of all, since the variation of the radius has a similar influence as the modification of the rod’s permittivity, this critical rod may become a valuable candidate for tuning purposes. Tuning just a single rod is expected to work much faster than the tuning of large areas within a PhC device as required, e.g., in ,  for PhC switching devices. Apart from active tuning, such highly sensitive rod may also form the basis for novel sensor schemes. However, to explore the potentialities of such a pronounced local sensitivity, further research on corresponding PhC topologies (including, e.g., the impact of various loss contributions associated with a planar device realization) is required.
While performing a simple sensitivity analysis based on the MMP code, we have obtained an efficient method for improving the frequency response of PhC devices. In particular we have applied this technique to a simple 90° W1 waveguide bend emerging from an underlying PhC with square lattice symmetry. A single optimization step has already obtained nearly zero reflection over almost the entire PBG, i.e., an achromatic bend has been achieved. This technique can easily be extended to other PhC types and crystal symmetries using various other PhC properties for optimization purposes. Nevertheless, one should be aware that when performing sensitivity analysis at low parameter variation levels accurate numerical modeling methods become strictly mandatory. We have discovered that highly sensitive (or critical) areas within a PhC may emerge. These domains may turn out to be just as small as one single rod. Albeit the implications on a fabrication technology, especially, such pronounced local sensitivity may also become very promising for the design of ultra-compact switches and novel sensors topologies.
This work was supported by the Swiss National Science Foundation in the framework of the project NFP-2000-065102.01/1 and the research initiative NCCR Quantum Photonics.
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