1. Introduction

Photonic integration is used to reduce the number of fiber interconnections between components, but the integrated system still faces the daunting task of interfacing with the rest of the network. Input/output coupling remains an issue for most active devices with small dimensions. A vertical coupler (VC) provides a relatively simple way to realize a mode-size transformer needed to minimize the coupling loss, and has been utilized in several applications [1–5]. Consider, for example, a waveguide electroabsorption modulator (EAM) with a typical cross section of 2 µm×0.5 µm and a length as small as 100 µm. These dimensions make it badly suited for coupling to the outside world and difficult to handle. These drawbacks are at once removed, without any effect on the active volume, if a compact vertical coupler is integrated at each end of the EAM to serve as a spot-size transformer to facilitate low-loss coupling with standard fibers.

A common shortfall of these VC is the polarization dependent behavior caused by the polarization-sensitive coupling between the asymmetric waveguides, resulting in the whole device being polarization-dependent. Previous works dealt mostly with the VC as an integrated part of the semiconductor laser [1–5], which produces highly polarized output. Hence, the polarization independence was not the main consideration. Polarization dependency of the taper waveguide alone had been studied [6], but so far no systematic approach for designing polarization-independent VC has been developed. Therefore, we present, for the first time, a reasonably systematic way to design polarization-independent VC, which is of great importance in broadening its usefulness as a building block for minimizing insertion loss in photonic integrated circuits.

2. Concept of polarization-independent vertical coupler

2.1 Vertical coupler structure and figures of merit

A vertical coupler (VC) is generally formed by two asymmetric waveguides one on top of another, separated by thin spacer layers. The upper waveguide core has higher refractive index, but smaller thickness and width than the underlying waveguide. The core material is usually predetermined by the device application. The material system is assumed to be InP/InGaAsP, with InP forming the claddings and spacer layers. The upper waveguide is laterally tapered, and light transfers up or down at the transfer regions. Each transfer region is sandwiched by two steep taper regions. The schematic of the VC is shown in Fig. 1. The taper profile over the VC is shown as piecewise linear, but may have other shapes. The lower waveguide is shown as having a uniform width across the whole length, but may have other lateral profiles depending on the dimensions chosen.

 

Fig. 1. Schematic of an active device with a vertical coupler at each end that acts as mode size transformer.

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Many devices have a p-i-n diode structure, for which the doping profile needs to be specified. To the first order, the doping profile does not affect the design of the vertical coupler and hence it will not be further considered.

The efficacy of the VC is measured by the transfer efficiency from the lower waveguide to the upper waveguide for an input of arbitrary polarization. The overall insertion loss of the device is reduced if the transfer loss in the VC is more than compensated by the reduction in coupling loss in the underlying waveguide. The transfer loss and coupling loss are not independent, and the design of VC is a tradeoff between minimizing the transfer loss and maximizing the fiber coupling efficiency. The latter depends also on the fiber used, which provides an extrinsic control for the coupling loss. As such, we will focus only on the transfer section and discuss how to minimize its transfer loss. Besides the transfer efficiency, other figures of merit may include good separation between upper and lower modes outside the taper and compactness (total length of the taper). In some cases, limitations in fabrication may dictate that the critical width at the point of transfer is not smaller than a specified value, and we will show how this can be controlled.

2.2 Polarization-independent resonant coupling vs. adiabatic transfer

The idea of using adiabatic mode crossing to transform the size of input mode relies on the assumption that one can vary the effective indices of the two participating waveguides sufficiently slowly. In our case, this means slow variation in the width of the taper as a function of the propagation distance. True adiabatic limit, however, requires much too long a taper than is practical for the compact devices of interest here.

We rely on the resonant coupling mechanism in the design of the VC. The resonant coupling occurs when the effective indexes of the two dissimilar waveguides become nearly identical. For a given combination of core index and core thickness, the effective index of the upper waveguide is a function of width only, as shown in Fig. 2. The effective index of the lower waveguide is shown to be constant. The width of the tapered upper waveguide at which the effective index crosses that of the lower waveguide is defined as the resonant width. The resonant width is generally different for different polarizations, which means that the maximum transfer occurs at different points along the taper, resulting in different transfer efficiencies for TE and TM. If these two critical points are far apart, then their inclusion within the taper will require a longer taper, or a taper with greater slope, which degrades the transfer efficiency. Hence, it is advantageous to design a taper waveguide that has the same resonant width for both TE and TM.

 

Fig. 2. Effective indexes of the coupled waveguide for TE and TM modes at various taper widths

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To fulfill the above requirement, we have to utilize a width where the waveguide has the same effective index for both polarizations. At this point, the waveguide is said to be polarization-independent. Figure 2 shows that the effective index curves for the upper waveguide cross at the critical width [7]. Hence, for the taper waveguide to have same resonant width for both TE and TM, it is clear that the critical width must coincide with the resonant width. This represents an important requirement in designing a polarization-independent VC. In the next section, the detailed design of the various parts of the VC will be presented. The operating wavelength is assumed to be 1.55 µm. The 3-D semi-vectorial finite-difference beam propagation method (FD-BPM) [8] is used in all simulations.

3. Design of polarization-independent vertical coupler

3.1 Optimum spacer thickness

The upper waveguide and the lower waveguide are separated by a spacer layer. It ensures that the upper mode is better confined and spatially well separated from the lower waveguide, but it does reduce the coupling. The criterion for optimum spacer thickness can be deduced from the coupled-mode theory by considering the VC as a kind of co-directional coupler, where the outputs a₁(z) and a₂(z) of guide1 and guide2 respectively are given by [9]

where

The VC requires that the transfer efficiencies (lower to upper and upper to lower) are equally high. From Eq. (1), we can see that these require δ≈0 and s≈κ₁₂≈κ₂₁. While a smaller spacer thickness will increase κ₁₂ and κ₂₁, it does not ensure that κ₁₂≈κ₂₁. Thus, the optimum spacer is one that makes κ₁₂≈κ₂₁. The actual spacer thickness is found by BPM simulation, and it is found to occur around 0.2 µm.

3.2 Taper waveguide design

The next important step in designing the polarization-independent VC is the determination of the upper waveguide structure that would match the critical width with the resonant width. Figure 3 shows the critical widths and the corresponding effective indexes of the upper waveguide obtained by BPM simulations for various combinations of core thickness and refractive index. These figures are used to deduce the core thickness-refractive index pair that is able to yield a desired critical width with the required modal effective index. The lower waveguide is then designed to match its effective index with the effective index at the critical width (i.e., matching the critical width and the resonant width). Note that the larger the critical width, the lesser the combinations that are available.

 

Fig. 3. (a) Critical widths, and (b) effective indexes for the different combinations of core thickness and refractive index for the upper waveguide. The color bands represent the different ranges of critical width or effective index

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The lateral and longitudinal structure of a VC is defined by lithographic patterning, and is centered about the critical width. The upper waveguide must start with a reasonably small initial width to avoid a non-adiabatic jump in the effective index of the lower waveguide due to the presence of the upper waveguide. This width can then be increased rapidly to the initial width of the taper over an initial buffer region, which should be as short as possible in order not to affect the light transfer. Nonetheless, it must not be too short due to the fabrication consideration. Hence, it is set as 30 µm while maintaining the compactness of the VC in our design. Within the transfer section the taper width is increased slowly to the final width over a transfer length, which can be as small as 150 µm. Afterwards, the waveguide width is increased rapidly to the width of the active device over another buffer region, which is also 30 µm based on the same consideration as the initial buffer region.

The most critical dimension is the length of the transfer section, along with the initial width and the final width of the taper. Maximum resonant power transfer must occur over this taper for both TE and TM modes. A very gradual slope is required to maximize the transfer efficiency. At the same time, a very compact taper is often desirable.

The design of initial width and final width is an iterative process, starting with an initial guess for one while varying the other to determine the optimum value for maximum transfer efficiency. Simulations show that, for a given taper length, power transfer from the lower waveguide begins some distance (typically within 0.3 µm) before the critical width and reaches the maximum a short distance (typically within 0.05 µm) after the critical width, with the transfer occurring earlier and faster for the TM mode. This is because the TM mode has a greater coupling coefficient than the TE mode, even though the effective index may be the same at the critical width.

In one of our designs, the upper waveguide has a material index of 3.30 and thickness of 0.7 µm, which then lead to a critical width of about 1.30 µm with the corresponding effective index of 3.18. For this case, assuming a taper length of 150 µm, the transfer efficiencies for the TE and TM modes are shown as a function of initial width and final width in Fig. 4(a) and (b), respectively. The end result of this iterative process is the selection of the initial width to be 1.05 µm, near the point where the TE and TM curves intercept, and the final width to be 1.35 µm where the transfer efficiencies for TE and TM are nearly equal and above 90%. Using these values, we then vary the taper length around the nominal point of 150 µm. The results show that the transfer efficiency and the polarization dependence are quite insensitive to the taper length from 150 to 200 µm. If the taper length is greater, the polarization dependency increases. When the taper length is smaller, the transfer efficiencies decrease. Thus, a taper length of 150 µm appears to be the optimum and most compact possible for the above waveguide structure. The simulation result for this particular VC will be discussed in Section 4.

 

Fig. 4. Change of transfer efficiency with (a) initial width with final width set at 1.35 µm, and (b) final width with initial width set at 1.05 µm

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3.3 Underlying waveguide design

The typical case is that the upper waveguide structure is predetermined by device requirements. So, it is the lower waveguide modal index that is tuned to match the upper waveguide modal index at the critical width. Simulations show that the critical width of the upper waveguide is insensitive to the dimension of the lower waveguide. The thickness of the lower waveguide, however, does impact the transfer efficiency. Intuitively, the greater the thickness, the more the light beam would have to shift its mode centroid vertically. However, the small thickness will affect the fiber coupling. Thus, as a compromise, 3 µm is chosen as a value that gives sufficient overlap with the fiber mode and also with the upper mode. Likewise, the effective index is quite insensitive to the lower waveguide width between 3 to 5 µm, and hence, it is fixed at 4 µm. The next design parameter is the material index that gives the effective index equal to that of the critical width, and it is found to be 3.19.

Complete energy transfer between the two waveguides can be achieved theoretically only if there are no more than two modes that interact in the mode conversion process. In our design, the lower waveguide is made transversely single-mode because the index difference between the core and cladding is sufficiently small. It could be made laterally single-mode if the etching through the waveguide is sufficiently shallow as to form a rib waveguide. If the waveguide is multimode, the higher-order modes will have lower effective indices. They can still couple resonantly to the upper waveguide but with lower efficiency at different locations along the taper, and so the taper will have to be either longer or steeper. These degrade the transfer efficiency for the fundamental modes. Therefore, it is desirable to prevent higher-order modes from being excited in the first place.

4. Simulation results

Figure 5 shows the simulation results of a vertical coupler based on the example described so far. It is summarized as follows: The lower waveguide is single-mode and has a core index of 3.19, and a cross section of 4×3 µm. The substrate and claddings have an index of 3.17. The matching upper waveguide has a core index of 3.30, a thickness of 0.7 µm, and a critical width of about 1.30 µm. At the critical width, the effective index of both the upper and the underlying waveguide is about 3.18. The transfer region is tapered from 1.05 µm to 1.35 µm over a length of only 150 µm, and is followed by a steep taper to 2 µm. The transfer efficiency is obtained using the overlap integral between the propagating mode and the eigenmode in the upper waveguide. The full transparent boundary condition [8] is used.

The transfer efficiencies for both TE and TM are more than 90%. The tolerance in the operating wavelength is about ±10 nm. The small mismatch in the efficiencies is because the lower waveguide still has some birefringence. Note that the upward coupling occurs more rapidly for the TM mode. This is because the TM mode has a larger coupling coefficient. In Fig. 5(b), the small oscillation in the lower waveguide is due to the back-coupling that occurred at the active region. The oscillation periods for both polarizations are calculated to be around 50 µm, which match the simulated oscillating periods. The similar oscillation does not appear in the upper waveguide because the power oscillation is of a minute fraction of the total power, thus, it is only visible at the lower waveguide where the power is very small after the light transfer. Likewise, it will not be clearly visible in the upper waveguide where the power is high.

The above example is used because the critical width is reasonably large. Many other polarization-independent designs are possible, depending on the desired critical width. Generally, to obtain a larger critical width, the upper waveguide needs to have a smaller core refractive index, but with a larger core thickness to maintain strong optical confinement.

 

Fig. 5. (a) Contour plots along Y-Z axis for TE and TM propagation, and (b) the transfer efficiency of the polarization independent vertical coupler as a function of propagation distance. The transfer region begins at z=50 µm.

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5. Summary

In conclusion, we have presented a systematic design approach for compact polarizationindependent mode-size transformers based on tapered resonant vertical coupler with high transfer efficiency (greater than 90%). The key steps in designing a polarization-independent VC are summarized as follows:

(a) For a given underlying waveguide design (with a given effective index), find the corresponding upper waveguide design (i.e., core thickness, core index, and critical width) whose effective index, at the critical width, is identical for TE and TM (i.e., polarization independent), and is also the same as that of the underlying waveguide.

(b) Conversely, if an upper waveguide structure (i.e., core thickness and core index) is given, determine the critical width and the corresponding effective index, and design the matching underlying waveguide to have the same effective index.

(c) Using the critical width above as a starting point, design the dimension of the transfer region by making an educated initial guess of the initial width (or final width), and subsequently performing the optimization of the taper structure with respect to the final (initial) width and the taper length, so as to maximize the transfer efficiency.