## Hybrid transversal-lattice optical filters

Optics Express, Vol. 10, Issue 21, pp. 1190-1194 (2002)

http://dx.doi.org/10.1364/OE.10.001190

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### 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

1. M. Oguma, T. Kitoh, K. Jinguji, T. Shibata, A. Himeno, and Y. Ibino, “Flat-top and low-loss WDM filter composed of lattice-form interleave filter and arrayed-waveguide gratings on one chip,” in OSA *Trends in Optics and Photonics (TOPS) Vol. 54, Optical Fiber Communication Conference*, Technical Digest, Postconference Edition (Optical Society of America, Washington, D.C., 2001), pp. WB3-1-WB3-3.

2. T. Chiba, H. Arai, K. Ohira, H. Nonen, H. Okano, and H. Uetsuka, “Novel architecture of wavelength interleaving filter with Fourier transform-based MZIs,” in OSA *Trends in Optics and Photonics (TOPS) Vol. 54, Optical Fiber Communication Conference*, Technical Digest, Postconference Edition (Optical Society of America, Washington, D.C., 2001), pp. WB5-1-WB5-3.

3. E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamoto, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” IEE Electron. Lett. **32**, 113–114 (1996). [CrossRef]

*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

*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

*F*(

_{u}*z*) ’s are polynomials of degree

*V*.

*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

5. K. Jinguji and M. Kawachi, “Synthesis of coherent two-port lattice-form optical delay-line circuit,” J. Lightwave Technol. **13**, 73–82 (1995). [CrossRef]

*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

5. K. Jinguji and M. Kawachi, “Synthesis of coherent two-port lattice-form optical delay-line circuit,” J. Lightwave Technol. **13**, 73–82 (1995). [CrossRef]

*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*(

*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

*D*

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

*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.

*b*of Eq.(1) are real and positive.. In that case it is clear that

_{n}*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

5. K. Jinguji and M. Kawachi, “Synthesis of coherent two-port lattice-form optical delay-line circuit,” J. Lightwave Technol. **13**, 73–82 (1995). [CrossRef]

## 3. Simulations

*N*= 23 [5

**13**, 73–82 (1995). [CrossRef]

*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.

*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.

*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.]

*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.

3. E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamoto, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” IEE Electron. Lett. **32**, 113–114 (1996). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | M. Oguma, T. Kitoh, K. Jinguji, T. Shibata, A. Himeno, and Y. Ibino, “Flat-top and low-loss WDM filter composed of lattice-form interleave filter and arrayed-waveguide gratings on one chip,” in OSA |

2. | T. Chiba, H. Arai, K. Ohira, H. Nonen, H. Okano, and H. Uetsuka, “Novel architecture of wavelength interleaving filter with Fourier transform-based MZIs,” in OSA |

3. | E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamoto, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” IEE Electron. Lett. |

4. | M. E. Marhic, “Parallel optical filters,” in ICT98 - International Conference on Telecommunications, F. N. Pavlidou, ed. (Thessaloniki, Greece : Aristotle Univ. Thessaloniki, 1998), pp. 503–508. |

5. | K. Jinguji and M. Kawachi, “Synthesis of coherent two-port lattice-form optical delay-line circuit,” J. Lightwave Technol. |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 16, 2002

Revised Manuscript: October 11, 2002

Published: October 21, 2002

**Citation**

Michel Marhic, "Hybrid transversal-lattice optical filters," Opt. Express **10**, 1190-1194 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-21-1190

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### References

- M. Oguma, T. Kitoh, K. Jinguji, T. Shibata, A. Himeno and Y. Ibino, "Flat-top and low-loss WDM filter composed of lattice-form interleave filter and arrayed-waveguide gratings on one chip," in OSA Trends in Optics and Photonics (TOPS) Vol. 54, Optical Fiber Communication Conference, Technical Digest, Postconference Edition (Optical Society of America, Washington, D.C., 2001), pp. WB3-1-WB3-3.
- T. Chiba, H. Arai, K. Ohira, H. Nonen, H. Okano and H. Uetsuka, "Novel architecture of wavelength interleaving filter with Fourier transform-based MZIs," in OSA Trends in Optics and Photonics (TOPS) Vol. 54, Optical Fiber Communication Conference, Technical Digest, Postconference Edition (Optical Society of America, Washington, D.C., 2001), pp. WB5-1-WB5-3.
- E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamoto and Y. Ohmori, "Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter," IEE Electron. Lett. 32, 113-114 (1996). [CrossRef]
- M. E. Marhic, "Parallel optical filters," in ICT'98 - International Conference on Telecommunications, F. N. Pavlidou, ed. (Thessaloniki, Greece : Aristotle Univ. Thessaloniki, 1998), pp. 503-508.
- K. Jinguji and M. Kawachi, "Synthesis of coherent two-port lattice-form optical delay-line circuit," J. Lightwave Technol. 13, 73-82 (1995). [CrossRef]

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