## Abstract

We introduce a new general class of hybrid optical filters, which reduce to either transversal or lattice filters in particular limits, and are suitable for implementation as planar lightwave circuits. They can be used to synthesize arbitrary periodic transfer functions with finite impulse responses. Design tradeoffs can be used to minimize insertion loss and optimize layout. Examples of filter synthesis are presented.

© 2002 Optical Society of America

## 1. Introduction

Planar lightwave circuit (PLC) technology is currently being used to make some filters for WDM systems, particularly wavelength multiplexers based on arrayed waveguide gratings (AWGs). Interleavers based on lattice filters with a few stages have been reported [1,2]. Increasing the number of stages in lattice filters may lead to high excess loss, due to the linear accumulation of coupler excess losses. Transversal filters with up to 32 taps have been demonstrated [3], but have not been pursued recently, because they may exhibit a high intrinsic insertion loss. In this paper we investigate a new family of filters, consisting of a transversal array of *U* lattice filters, and show that they can be used to synthesize arbitrary periodic transfer functions with finite impulse responses (FIR’s). They reduce to either transversal or conventional lattice filters in particular limits. Insertion loss can be minimized by varying *U* used for a given FIR. We provide a method for calculating the characteristics of all the 2X2 couplers and phase shifters required to synthesize a particular FIR. We then provide design examples showing that some high-order filters can exhibit insertion loss lower than for lattice filters.

## 2. Theory

It was recently shown that the insertion loss of transversal filters can be reduced by using a combiner which is the mirror image of the splitter [4]. Here we extend this approach to a parallel array of lattice filters. We are going to show that such a structure can be used to synthesize an arbitrary periodic transfer function *F*(*z*), which is a polynomial of order *N* in *z* = exp(*iϖτ*), where *ϖ* is frequency, and *τ* is the elementary time delay. This is done by partitioning the set of coefficients *B* = {*b*
_{0}, *b*
_{1}, *b*
_{2},..., *b*
_{N}} of *F*(*z*) into *U* subsets. There is a large number of ways that this can be done, namely the number of partitions of (*N* + 1), Π(*N* + 1). Among these possibilities, some are more interesting than others, as they exhibit some regularity, and make efficient use of couplers. Here we confine ourselves to the case where (*N* + 1) = *U*(*V* + 1), where *U* and *V* are integers, and we partition *B* into *U* subsets of size (*V* + 1), each subset corresponding to consecutive powers of *z*. Thus we rewrite *F*(*z*) as

where the *F _{u}*(

*z*) ’s are polynomials of degree

*V*.

Based on this decomposition, we propose to synthesize *F*(*z*) by means of *U* lattices in parallel, each synthesizing an *F _{u}*(

*z*), and preceded by a delay (

*u*- 1)(

*V*+ 1)

*τ*to obtain the correct powers of

*z*. The resulting structure is shown in Fig.1. From the theory of lattice filters, we know that each

*F*(

_{u}*z*) can be synthesized, within a multiplicative normalization constant

*D*, by a lattice of 2X2 couplers, all separated by the same delay

_{u}*τ*[5]. If the target function is

*F*(

_{u}*z*), the transfer function actually implemented by the finished lattice filter is

*F*(

_{u,a}*z*) =

*F*(

_{u}*z*)/

*D*, where

_{u}*D*=

_{u}*Max*{|

*F*(

_{u}*z*)|}, for

*z*on the unit-radius circle [5]. To complete the overall design of

*F*(

*z*), we must find the coupling fractions in the splitter/combiner, so that the scaled versions of the

*F*(

_{u}*z*)’s will add up with correct magnitudes and phases to reconstruct

*F*(

*z*) as per Eq.(1). Let

*F*(

_{a}*z*)denote the actual transfer function of the finished filter; it must be of the form

*F*(

_{a}*z*) =

*F*(

*z*)/

*D*, where

*D*is the normalization constant for the whole network. In Fig. 1,

*w*denotes the field amplitude at the output of the

_{u}*u*th output of the splitter when a unit-amplitude field is present at its input. We must have

$$=D\sum _{u=1}^{U}\frac{{\left({w}_{u}\right)}^{2}{z}^{\left(u-1\right)\left(V+1\right)}{F}_{u}\left(z\right)}{{D}_{u}}=\sum _{u=1}^{U}{z}^{\left(u-1\right)\left(V+1\right)}{F}_{u}\left(z\right).$$

Eq.(2) implies that *D*(*w _{u}*)

^{2}/

*D*= 1, or (

_{u}*w*)

_{u}^{2}=

*D*/

_{u}*D*, all

*u*. Assuming a splitter with no excess loss, the fraction of power going from the input to the

*u*th output is (

*w*)

_{u}^{2}, and we $\sum _{u=1}^{U}}{\left({w}_{u}\right)}^{2}=1$. This leads to the expression for

*D*

From *D* we can calculate the maximum magnitude of the transfer function of the realized filter, which is

Clearly this new family of filters contains the well-known transversal filters and lattice filters as limiting cases, respectively for *U* = *N* + 1, and *U* = 1. For intermediate values of *U*, we have new filters which can be used for synthesizing arbitrary transfer functions given by Eq.(1). For values of *N* which are highly composite, there may be several suitable values of *U*. For example, for *N* = 23, we could use *U* = {1,2,3,4,6,8,12,24}, i.e. have 8 possible hybrid implementations. It is generally not possible to determine a priori which ones will have the lowest theoretical insertion loss, and each case must be studied numerically to do so.

The advantage of having this choice among several possible filters is that we can use it to optimize several key aspects of PLC filters. First is insertion loss, which in realistic filters is a function of excess loss of 2X2 couplers, and waveguide loss. Also important is the fact that different hybrid filter implementations will lead to different layouts for the delay lines, couplers, etc,. which can then be optimized to reduce overall filter area, to equalize end-to-end loss of various paths, etc. Thus this new family of filters provides an important new degree of freedom for designing PLC filters.

An important class of filters corresponds to the case where the coefficients *b _{n}* of Eq.(1) are real and positive.. In that case it is clear that

*F*(

*z*), and all of its parts

*F*(

_{u}*z*), reach their maximum values for

*z*=1, and that these maximum values are all real and positive. Then the absolute values can be removed in Eq.(3), and this indicates that the same

*D*will be obtained for any partition of

*F*(

*z*). Hence, for this class of filters, exactly the same transfer function can be realized with any hybrid implementation. In particular, they will all exhibit 100% maximum transmittance, like the lattice filters [5].

## 3. Simulations

As an example of application, we have applied the method to the design of a Chebyshev filter with *N* = 23 [5]. Table I list the values of *U* and the corresponding losses. The intrinsic losses are calculated by assuming that the 2X2 couplers have no excess loss; then the only loss is due to the normalization constant *D*, calculated for each case as discussed above.

The excess loss is due to the realistic loss of 2X2 couplers. In all such PLC filters it has been found necessary to use tunable couplers, since it not possible to fabricate fixed couplers with the required accuracy. Tunable 2X2 couplers have an excess loss of the order of 1.5 dB [6]. This figure has been used to calculate the excess loss for each configuration, obtained by counting the number of couplers encountered in the splitter, lattice, and combiner in each case. Note that when *U* is not a power of 2, the branches of the splitter and combiner do not all have the same number of couplers; in that case we have used a number of couplers between input and output which is close to the mean number for all possible paths, for particular splitter and combiner structures. The fact that the number of couplers is not a constant will have another effect, namely that the excess loss will be path-dependent. This will in turn affect the amplitudes of all terms at the output, and will distort the resulting transfer function. In such cases, path loss equalization may be necessary, possibly by such means as the insertion of dummy couplers or other loss mechanisms.

Table I shows that the intrinsic losses range from 0 dB (*U* =1 or lattice filter ) to 2.96 dB (*U* = 24 or transversal filter), and that the variation between these extreme values is not monotonic. [Note that the cases *U* =12 and *U* = 24 have exactly the same intrinsic insertion loss and total loss. It can be shown that this is always true, because one can always take the 2 lattice couplers of the *U* =12 case, or the single coupler of the *U* = 24 case, and incorporate them into the splitter and combiner, thereby obtaining the same classical transversal filter.]

Thus, if excess loss was negligible, the lattice implementation would be better than all others, exhibiting a few dB less loss. When coupler excess loss is included however, we see that the conclusion is reversed: the lattice exhibits a loss of 36 dB, which is prohibitive. On the other hand, the transversal version has a loss of 16. 46 dB, i.e. 20 dB lower, which is a considerable improvement We also see that there is very little difference in total loss for the values *U* = 4,6,8,12,24 . Hence one could choose among these possible solutions by using some other criterion, such as aspect ratio of the overall layout, utilization of wafer area, etc.

Other types of filters have been modeled. In the case of a Gaussian filter [3], all Fourier coefficients are real and positive. As explained at the end of section 3, this implies that all hybrid implementations exhibit 0 dB intrinsic insertion loss; then a transversal implementation is best, because the number of couplers cascaded in each branch is minimized, and so is the total coupler excess loss. This has been verified by simulations.

## 4. Conclusion

We have introduced a new class of hybrid transversal-lattice filters, which reduce to the well-known transversal and lattice filters in particular limits. This enlarged family of filters can synthesize the same transfer functions as these classical filters, and the procedure for doing so is straightforward once the overall filter normalization constant is calculated as shown here. This family of filters enlarges the number of degrees of freedom available in PLC filter synthesis, and may lead to configurations optimum from the point of view of total loss and/or topology, which are not available in the conventional filter families.

## Acknowledgments

The support of Intel Corporation, Photonics Technology Operations, is gratefully acknowledged.

## References and links

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