## Abstract

A unified method was proposed to reduce the beat length of a multimode interference (MMI) coupler. By properly adjusting the phase difference of the N-fold images, the mode evolution is changed to generate self-images at a much shorter distance. The effect of adjusting the phase difference can be regarded as dividing the original MMI coupler into multiple sub-MMI couplers. Such an effect can be applied for both symmetric- and paired-interference cases. We applied the principle to design compact optical splitters operating at dual wavelength bands. The simulation shows that excellent performance with reduced coupler length can be obtained for splitters operating at both 1.3 and 1.55 µm bands.

©2006 Optical Society of America

## 1. Introduction

Integrated optical components (IOCs) are very promising for various applications due to their potential for integration with photonic and electronic components. Multimode interference (MMI) couplers, Y-branches [1–4], and directional couplers [5–7] are among the popular building blocks for IOCs. Because of their compact size and relative large tolerance on fabrication fluctuation, MMI couplers are very attractive for realizing planar lightwave circuits (PLCs) and various photonic modules. There have been a great number of research reports on the design and fabrication of MMI based devices, e.g., [8–12] and references therein.

For many applications, the IOCs need to work for multiple wavelength bands. Due to the wavelength dependency, the components based on MMI couplers usually require the device length to be a common multiple of the beat lengths for the wavelength bands. This results in difficulties for designing the MMI-based circuits when the wavelength bands are far apart. For instance, due to the weak wavelength dependency for MMI couplers, it takes a large common multiple and needs a long device for constructing a multi-band power splitter or multiplexers [9].

In this paper, we provide a new concept for reducing the beat length of MMI couplers. We propose to apply phase-shift regions in a MMI coupler for adjusting the phase difference between the localized images. Adjusting the phase difference changes the modal expansion and thus the wave evolution along the MMI region. The self-imaging spots within a MMI coupler can be regarded as new light sources to generate the subsequent interference patterns. When the adjacent images are out-of-phase, a new self-imaging effect with a reduced beat length can be generated.

The effect of adjusting the phase difference can also be regarded as dividing the original MMI coupler into multiple sub-MMI couplers. The effective width of the sub-MMI is reduced to a fraction of the original MMI width. The principle of phase adjustment and the effect of reduced beat length will be discussed for both symmetric- and paired-interference cases. Moreover, the effect of reduced beat length is applied to design MMI splitters operating at dual wavelength bands.

## 2. Operation principles

#### 2.1 Field representation in MMI

For easy reference, some of the basic formulas for MMI couplers are summarized below. The operation of MMIs device is based on the self-imaging effect of multimode interference [13]. From the superposition of the guided modes of a multimode waveguide, as shown in fig. 1, the local field distribution at any position along a MMI device can be given by:

where *c _{m}* denotes the expansion coefficient for the m-th eigenmode of which the field distribution is

*φ*and the propagation constant is

_{m}*β*. The coordinates of

_{m}*x*and

*z*stand for the transverse and longitudinal directions, respectively.

*β*can be expressed as:

_{m}where *λ*
_{0} and *n _{r}* denote the free-space wavelength and the refractive index of the core waveguide, respectively.

*W*is the effective width that includes the Goos-Hanschen shifts at the waveguide boundaries. The beat length

_{e}*L*is defined as [8]:

_{π}The beat length depends on the effective width and the wavelength of light. The modal field distribution *φ _{m}* can be approximated as [14]:

Therefore, the field profile given by Eq. (1) can be rewritten by replacing *φ _{m}* with Eq. (4).

#### 2.2 Principles of reduced beat length

Since the beat length is squarely proportional to the MMI width, it can be reduced by separating the MMI into multiple isolated regions along the transverse plane. With the approximate mode fields given in Eq. (4), a MMI coupler can be treated as if it is bounded by perfect conductors. Thus, inserting a perfect conductor at the middle of the MMI will turn it into two isolated half-width MMIs. With this approach, the self imaging occurs in a quarter of period than the original one.

The effect of inserting a perfect conductor plane at the middle of a MMI can be obtained instead by changing the phase of the images to support only the anti-symmetric (odd) modes. This can be performed at the longitudinal positions (*z*=0) where two self images are formed. Assume that one of the two images is represented as Φ(*x*,0), as given in Eq. (5). The two images can be treated as the point sources for generating the following wave propagation. Therefore, the field evolution can be regarded as the superposition of responses from the two point sources. The relative phases of the N-fold images can be derived for the general or symmetric-interference cases [16, 17]. By applying a phase change to one image such that the two images are out-of-phase, the field distribution can be written as:

Equation (5) was derived by using the properties of even and odd modes. Adding a π-phase shift can alter the modal expansion of even modes to that of odd modes, though the intensity profile remains the same. The odd modes vanish at *x*=0, as shown in Fig. 1. A superposition of purely odd modes results in a null along the center of waveguide(*x*=0), which is equivalent to the case that a mirror (or a perfect conductor) is placed along the center. The field evolution in the MMI becomes the propagation in two sub-MMI couplers with a half of width. This can be verified by setting *x*′=*x*-*W _{e}* 4and

*m*=2

*m*′+1. Eq. (5) becomes:

where the expansion coefficient is given by

It is clear that the modal expansion in Eq. (7) is equivalent to that for a MMI with half a width. Furthermore, since only the odd modes are supported, the propagation constant given in Eq. (2) can also be written as

Therefore, the propagation constants are separated by multiples of *πλ*
_{0}/4*n _{r}*(

*W*′)

_{e}^{2}with

*W*′=

_{e}*W*/2. Eqs (6) and (8) prove that the net effect of applying the π-phase shift is to divide the original MMI into two isolated regions. Since the effective width is halved, the beat length becomes

_{e}*L*π

_{′}=

*L*

_{π}/4. The above principles can be applied to general types or restricted-interference types of MMI couplers. The phase-shift scheme can also be applied to the longitudinal positions where the

*N*-fold (

*N*>2) images are formed.

In the case of *N*=3, if phase changes are applied to the local images so that the adjacent images are out-of-phase, the resultant wave evolution can be treated as if the MMI is divided into three isolated regions with an effective width of *W _{e}*′=

*W*/3. This can be easily verified for the case with three equally-spaced images. Assume that the images appear at

_{e}*x*=-

*W*/3, 0,

_{e}*W*/3, respectively. When phase changes are applied to the three images to provide relative phase shifts of π, 0, and, π, respectively, the resultant field can be expressed as:

_{e}By substituting Φ(*x*,0) with Eq. (1) and (4) for *z*=0, it is easy to show that the expansion coefficient is not vanishing only if *m*=2, 5, 8, … Let *m*=3*m*′+2, Eq. (9) can be rewritten as:

It can also be easily shown that, with *m*=3*m*′+2, the propagation constant can be written as:

Eqs. (10) and (11) are the general form of the modal expansion for a MMI coupler with one third of the effective width. Therefore, with the phase shifts the resultant wave evolution along the MMI is like the propagation in three separate couplers. This is equivalent to the effects of inserting mirrors at *x*=-*W _{e}*/6 and

*W*/6, respectively. The beat length is reduced to

_{e}*L′*=

_{π}*L*/9.

_{π}Similar derivations can be applied to the cases of *N*=4 or above to show the same effects from the phase shifts. Therefore, with appropriate phase shifts on the *N*-fold images, the effective width and beat length can be reduced to *W _{e}/N* and

*L*

_{π}/N^{2}, respectively.

The phase change can be applied by varying the refractive index of the waveguide around the imaging spots [18]. For a phase shift region, the phase change can be written as

where *L _{m}* is the length of the phase shift region and Δ

*n*is the change in the refractive index. The size and location of the phase shift region could affect the wave evolution in a MMI. In our design, the phase shift region is placed on the self-imaging spots. Therefore, the design is not sensitive to the size variation as long as the region covers the spot and does not overlap with other self-imaging spots [18]. The tolerance on the location depends on the beat length of the MMI coupler.

_{r}In the following analysis, the effect of phase shifts will be applied to both paired- and symmetric-interference types of MMI couplers [8]. The beam propagation method (BPM) developed by RSoft, Inc. is used to simulate the wave evolution in a MMI coupler. Compared to other numerical methods, the BPM method is very robust for different device structure and can be calculated with less computation time. All the simulations are under the transparent boundary condition. The designs are based on the typical silica waveguide structure. The waveguide cross-section is 6×6µm^{2} for the single mode access waveguide and the refractive index difference between the core and the cladding is 0.75%.

## 3. Restricted interference cases

#### 3.1. Modified symmetric interference

For symmetric interference cases, the expansion coefficients can be expressed as *c _{m}*=0 for

*m*=1, 3, 5…., and the periodic distance is

*L*=3

_{c}*L*[8]. At the two-fold self-imaging positions, the two images have zero phase difference. By applying a phase change of

_{π}*π*to one image, the modal expansion is modeled as the superposition of two out-of-phase point sources. Thus, the field becomes the summation of purely odd modes. The resultant effective width is equal to one half of the original one and the periodic distance is reduced by a factor of 1/2

^{2}. In general, the periodic distance is reduced to

*L′*=

_{c}*L*when appropriate phase shifts are applied to the

_{c}/N^{2}*N*-fold images. Figure 2 depicts the light evolution along the MMI coupler for the standard symmetric-interference case and the ones with phase shifts on two- to five-fold images. The rules for applying the phase shifts are summarized in Table 1. It is clear that the effects of applying phase shifts on the

*N*-fold images are like adding mirrors to divide a MMI to

*N*sub-MMI couplers.

#### 3.2. Modified paired-interference

For the paired-interference case, the expansion coefficients *c _{m}*=0 for

*m*=2, 5, 8…; and the periodic distance is

*L*. By applying a

_{c}=L_{π}*π*/2-phase shift to one of the two-fold images and make them to have

*π*phase difference, the modified paired-interference can also be described by Eq. (6). The net effect looks as if the MMI is divided into two sub-MMI’s. The wave evolution in each sub-MMI possesses the property of paired-interference. When the same principle is applied to the

*N*-fold images,

*L′*.

_{c}/L_{c}/N^{2}Figure 3 depicts the light propagation for the original and the modified paired-interference cases. Again, the net effect of applying the phase shifts is like adding mirrors to separate the MMI to *N* sub-MMI’s. The above schemes of applying phase shifts result in point sources with alternating 0 and *π* phases. The out-of-phase sources generate anti-symmetric wave evolution patterns that vanish at the intermediate planes. Another interesting arrangement of the phases is to make the *N*-fold images to have symmetric phases such that the paired-interference property is turned into the symmetric-interference case. This changes the self-imaging patterns and the periodic distance. A simulation example is shown in Fig. 3(e) by using the phase shifts listed in Table 2. The periodic distance is reduced to *L _{c}*/4.

Several combinations of the restricted interference types can be realized by properly adjusting the phase shifts. For example, the phases of the *N*-fold images can also be adjusted to include both symmetric- and paired-interference cases. Figure 3(f) depicts the wave evolution when the 4-fold images are adjusted to have two pairs of phases that have a phase difference of *π*, as listed in Table 2. The results reveal that the periodic interval can be reduced to *L _{c}*/16 though the effective width is only reduced to one half. From the above results, there are a variety of phase shift combinations to alter the periodic distance of a MMI coupler. Since the beat lengths for the symmetric-type and paired-type excitations are different, the proposed method offers flexibility in choosing the periodic distance for a MMI coupler.

## 4. Applications

The method of reducing the beat length of a MMI is applied to design dual-band optical splitters. Figure 4 shows the device schematic. In principle, 1×N optical splitters can easily be realized with MMI couplers. From Eq. (3), the beat length of a MMI coupler depends on the wavelength. To operate simultaneously at two wavelengths (*λ*
_{1} and λ_{2}), the total length of the MMI coupler should satisfy:

where *L _{c,λ1}* and

*L*are the beat lengths for the λ

_{c,λ2}_{1}and λ

_{2}wavelengths, respectively.

*q*and

_{1}*q*are integers with no common divisor with

_{2}*N*. Using a conventional MMI structure to realize a dual-band 1×N power splitter would typically require a long device, because the beat length is only weakly wavelength dependent. The difference between

*L*and

_{c,λ1}*L*is usually small, so the values

_{c,λ2}*q*and

_{1}*q*are large.

_{2}By applying the scheme of reduced beat length, the periodic distance can be shortened. There may be several options to achieve this goal. Here, we proposed to adjust the phase shift for one wavelength band but keep the phase of the other wavelength unchanged. By doing so, it is possible to obtain a smaller common multiple of the coupler lengths such that uniform power splitting can be achieved for both wavelengths with a shorter MMI.

As shown in fig. 4, the MMI coupler is divided into two sections and *L _{MMI}=L_{d1}+L_{d2}*. When the length of the first region

*L*satisfies:

_{d1}the *N*-fold images of λ_{1} light and the single image of λ_{2} light will appear at the same z position. *s _{1}* and

*s*are integers that have no common divisor with

_{2}*N*. Under such a condition, applying the phase shift on the

*N*-fold images for λ

_{1}except the single-image spot for λ

_{2}can affect only the phase difference of λ

_{1}light. Therefore, the

*N*-fold images of λ

_{1}light will repeat with a periodic distance of

*L*, while the periodic distance of λ

_{c,λ1/N}_{2}light remains unchanged. When the length of the second part

*L*obeys this relation:

_{d2}the *N*-fold images of λ_{1} light and the *N*-fold images of λ_{2} light will appear at the same z position. Under such a condition, 1×N power splitting can be achieved for both wavelengths.

Similar equations as Eqs. (14) and (15) should work when λ_{1} and λ_{2} are interchanged. Thus, the alternative expressions for *L _{d1}* and

*L*are:

_{d2}$${L}_{d2}={s}_{3}\xb7\left(\frac{{L}_{c,\lambda 2}}{{N}^{2}}\right)={s}_{4}\xb7\left(\frac{{L}_{c,\lambda 1}}{N}\right).$$

where *s _{3}* and

*s*are integers that have no common divisor with

_{4}*N*.

After searching the solutions of Eqs. (14) to (16) and obtaining the smallest set of the s-integers, a compact dual-band 1×N optical power splitter can be realized. The uniformity of power splitting can be evaluated by using the power imbalance parameter:

where *PO _{max}* and

*PO*are the maximum and minimum powers at the output waveguides.

_{min}#### 4.1 1×2 dual-band splitter

Without loss of generality, let λ_{1} and λ_{2} be 1.3 µm and 1.55 µm, respectively. For obtaining compact devices, the s-integers and the MMI width should be optimized. Though a smaller MMI width is desired to shorten the beat length, the width should be large enough to support as many modes as required for self-imaging. From numerical simulation, we limit the MMI width to *W _{M}*≧38µm. Moreover, the solution for the shortest MMI does not necessarily occur at the smallest

*W*. After calculation, the optimal solution for each case is listed in Table 3.

_{M}The 1×2 splitter is designed with the paired-interference type of couplers. The symmetric-interference type can also be used. For the standard design, the best values of q_{2} and q_{1} are 43 and 37, respectively, when *W _{M}*=38µm. It requires a long MMI section (

*L*=45045 µm). By using the phase-shift scheme listed in Table 2, the shortest device is found while (

_{MMI}*s*)=(7, 3) and

_{3}, s_{4}*W*=45 µm. The MMI coupler is 15057 µm long under this condition. Figure 5(a) and (b) shows the light propagating pattern along the MMI coupler for the shortest solution. Power splitting can be achieved for the two wavelengths. Figure 6(a) illustrates the simulated imbalance and excess loss as a function of the deviation in device length (ΔL

_{MMI}_{MMI}). The solid and dashed lines show the performance for the TE and TM polarization, respectively. The imbalance is below 0.15 and 0.25 dB for the 1.3- and 1.55-µm wavelengths, respectively. The excess loss is below 0.1 dB for both wavelengths. Among the solutions listed in Table 3, the one with the shortest device has better performance in terms of the excess loss and imbalance.

The three-dimensional BPM simulation indicates very low polarization-dependent loss (PDL) for the design at the optimal length. The low PDL is due to our design in the waveguide cross-section and needs to be further verified experimentally. The same simulation method has been applied to verify the performance of a polarization converter [20]. The results indicate that the splitting function has a large tolerance on device length. Figure 6(b) shows that the wavelength tolerance is about 20 nm for an excess loss <0.5dB. The design is more sensitive to the variation of MMI width [19]. For ±0.2 µm of width fluctuation, the imbalance and excess loss rise to 0.8 dB and 1.8 dB, respectively.

#### 4.2. 1×3 dual-band splitter

For the design of 1×3 power splitters, the symmetric-interference type of MMI coupler is used as the example. The optimal parameters for the conventional and modified device designs are calculated and summarized in Table 4.

For the conventional design, the shortest device occurs when (*q _{2}, q_{1}*)=(43, 37) and

*W*=38 µm, which gives a MMI length of 23100 µm for constructing a dual band splitter. By using the phase-shift scheme listed in Table 1, The best solution is obtained when (

_{MMI}*s*)=(7, 2) and

_{3}, s_{4}*W*=52 µm, which gives the shortest MMI coupler (

_{MMI}*L*=8802µm). The MMI with phase shifts can be much shorter than the conventional one. Figure 7 shows the power evolution along the MMI coupler for the two wavelength bands. The 1×3 power splitting function can be obtained for both wavelengths. Figure 8 illustrates the simulated imbalance and excess loss as a function of the device length (Δ

_{MMI}*L*) and wavelength. The imbalance is below 0.01 and 0.04 dB for the 1.3- and 1.55-µm wavelengths, respectively. The excess loss is below 0.06 dB for both wavelengths. The wavelength tolerance for less than 0.5dB of excess loss is larger than 30 nm. Again, from Table 4, the solution with the shortest device has better performance in terms of the excess loss and imbalance.

_{MMI}## 5. Conclusions

A novel scheme of reducing the beat length of MMI couplers is demonstrated in this work. With proper phase shifts to the self-imaging spots, the original MMI waveguide becomes one having the properties of multiple sub-MMI’s. The effect is like inserting mirrors in the MMI couplers. It can be applied for both symmetric-interference and paired-interference types of couplers. Such concept is applied to design the 1×N optical power splitters operated at dual bands. It helps to shorten the device length. The simulation of 1×2 and 1×3 power splitters shows very good performance with much shorter device. The devices have also very large tolerance on the deviation of device length tolerance and operation wavelength. More splitting ports can be designed with the same principle or by cascading the designed 1×2 and 1×3 power splitters.

## Acknowledgments

This work was supported in part by Ministry of Economic Affairs, R.O.C. under contract No. 91-EC-17-A-07-S1-0011 and by the National Science Council under contract no NSC94-2213-E011-016.

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