## Abstract

Fast iterative algorithms for designing birefringent filters with any specified spectral response are proposed. From the Jones formalism, we derive two polynomials representing the transmitted and rejected response of the filter, respectively. Once the coefficients of the filters are obtained, the orientation angle of each birefringent section and the phase shift introduced by each compensator can be determined by an iterative algorithm that gives an efficient solution to the birefringent filter design problem. Afterward, some design examples are presented to demonstrate the effectiveness of the proposed approach. In comparison with results reported in the literature, this approach provides the best performance in terms of accuracy and time complexity.

## Keywords

## Citation

Boukharouba, A. (2021), "Fast iterative algorithms for birefringent filter design", *Applied Computing and Informatics*, Vol. 17 No. 2, pp. 250-263. https://doi.org/10.1016/j.aci.2018.08.004

## Publisher

:Emerald Publishing Limited

Copyright © 2018, Abdelhak Boukharouba

## License

Published in *Applied Computing and Informatics*. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

## 1. Introduction

Optical filters are most commonly used in modern optical telecommunication systems [1] and also used for biomedical spectral imaging [2–4] such as Raman chemical imaging [5] and fluorescence microscopy applications [6]. Optical filters are essentially based on optical interleavers that separate an incoming spectrum into two complementary set of periodic spectra or combine them into a composite spectrum [7]. Most interleavers are based on Michelson interferometers, Mach–Zehnder interferometer (MZI) or birefringent filter principles [8–10]. The two classical designs of birefringent filters are Lyot filter [11] and Solc filter [12,13], which both types of filters use different configurations of polarizers and retardation plates to create narrow-band filters. Interference birefringent filters are optical finite impulse response (FIR) filters based on the changes induced in the state of polarization of light by birefringent materials. They are composed of a stack of retardation plates of birefringent material placed between a polarizer and an analyser. A wide range of filters can be achieved by orienting birefringent elements in an appropriate way. They play an important role in dense wavelength division multiplexing (DWDM) systems, as in gain equalization, dispersion compensation, prefiltering, and channels add/drop applications. The basic idea for the birefringent filter design is proposed by Harris et al. [14]. The desired output spectrum is first developed into the finite terms of a Fourier series and then the relative angles of both retarders and analyzer are determined according to the backward transfer method. A generalization of the procedure presented in [14] that allows the realization of impulse response having complex coefficients is detailed by Ammann and Yarborough [15]. The resulting network consists of n stages between an input and output polarizer, with each stage containing a birefringent crystal and optical compensator. Afterwards, the main problem of optical filter design becomes a problem of designing FIR filters using the Fourier series expansion of the desired frequency response. This technique is modified and improved by using a windowing technique to improve the shape of the frequency response. Classical optimization methods such as weighting least square sense and Parks-McClellan method are also used for designing digital FIR filters. To improve the performance of the classical methods, many researchers have utilized heuristic evolutionary optimization algorithms such as Genetic Algorithm (GA), Differential Evolution (DE), and Swarm Optimization (SO). For example, an optical finite impulse response (FIR) filter design methods based on crystal birefringence to produce arbitrary spectrum output are presented where a typical example of a green/magenta filter used in a liquid crystal on silicon projection display is synthesized [16,17]. An example of birefringent equalizing filter suitable for dispersion compensation in wavelength division multiplexed (WDM) communication systems is presented in [18]. A backward recursion of the transfer matrix is used to calculate the parameters of an optical filter that has an impulse response with complex coefficients. In [7] a general synthesis method for designing asymmetric flat-top birefringent interleavers is reported using a combination of digital signal processing approach and computational optimization by GAs.

The birefringent filter structure may be synthesized using a different technology based on coherent optical delay-line circuit with a two-port lattice-form configuration [19] where arbitrary filter characteristics corresponding to *n*th-order complex FIR digital filters can be realized by n cascaded two-port lattice-form optical delay-line circuits.

In this paper, we present iterative algorithms to design a birefringent filter whose coefficients are real or complex numbers and having an arbitrary frequency response. Some design examples are also presented to demonstrate the effectiveness of the proposed algorithms. The paper is organized as follows: Section 2 presents a theoretical analysis of the proposed algorithms for synthesising an optical FIR filter with real coefficients. Next, an extension to an arbitrary frequency response where the filter coefficients are complex numbers is detailed in Section 3. Section 4 introduces further improvements in the method given in [15] to calculate the complementary component. Design examples and simulation results are discussed in Section 5. Discussions and performance comparisons against other existing methods are presented in Section 6 and finally some conclusions are exposed in Section 7.

## 2. Optical finite impulse response filter

First, we study an optical FIR filter which is composed of a stack of identical birefringent retarders with same length L placed between an input polarizer and output analyser as shown in Figure 1. The x-axis is chosen parallel to the transmission axis of the input polarizer while *s* and *f* represent respectively the slow and fast axis of the birefringent elements. The solid arrows represent the fast axes of birefringent retarders and the transmitted axis of the output analyser. *k*th retarder and the y-axis which is the same angle between the slow axis of the retarder and the x-axis, while

The output of the optical filter must give the desired impulse response,

*a*denotes the time intervals between the impulse series.

L is the length of each retarder, *c* is the velocity of light in vacuum, so the phase difference caused by each retarder is expressed by:

The frequency response of the optical filter is the Fourier transform of its impulse response (1).

The complementary component, which is the output along the perpendicular direction of the analyzer, can be expressed as:

We assume that the optical network is lossless, which means that the energy must be conserved at all points within the network independently of the optical frequency

The output,

The main idea of this algorithm is to transfer the matrix–vector product into tow polynomials

### 2.1 Forward algorithm

Let’s set

For two stages

By induction, each component of the *z* for

For *n* stages

We can simply prove that:

From Eq. (14) and knowing that the frequency responses of the two complementary filters are defined by

Once the relative angles are known, the forward algorithm gives the impulse responses of the optical filters. However, the main problem of the birefringent filter synthesis is how to calculate the relative angles from the coefficients of the desired impulse response. To do this, we use a backward algorithm where the relative angles are the solution of the set of equations of the forward algorithm.

### 2.2 Backward algorithm

From the forward algorithm, the value of the *nth* relative azimuth angle

The Algorithm 1 summarizes the whole recursive process:

## 3. Extension to an arbitrary frequency response

In this case the optical network consists of *n* retarders between a polarizer and an analyser where each retarder is composed of a birefringent crystal with equal length and optical compensator [15]. However, in [18], each retarder is a birefringent plate composed of a section of nominal length *L* and a section with variable length

### 3.1 Forward algorithm

Following the same steps as in the Section 2.1, we can prove that:

where

### 3.2 Backward algorithm

The backward algorithm determines the relative angle of each crystal, the retardation introduced by each compensator, and the relative angle of the analyser. From the forward algorithm, the value of the *nth* relative azimuth angle

As shown in Section 3.1,

Moreover, *n* compensators (

## 4. Complementary component calculation

Assume that

Here

Note that

Notice that the all above algorithms are developed for

## 5. Design examples and simulation results

In order to demonstrate the effectiveness of the proposed algorithms for optical filter design, some design examples are presented and discussed.

### 5.1 Flat-top birefringent interleaver filter

we study the case of an asymmetric flat-top birefringent interleaver synthesized using Parks-McClellan optimal equiripple FIR filter design algorithm. The resulting filter

We can notice that the calculated coefficients of the optical filter are exactly equal to the desired ones and consequently the desired filter and the obtained one have the same spectrum.

### 5.2 Multi-channel selector

The second example is a multi-channel selector which is an optical frequency filter designed to select signals at certain frequencies from eight frequency-division multiplexing (FDM) signals [19]. In this case, the optical filter is synthesized to select three signals with frequencies of

### 5.3 Dispersion compensation

Birefringent equalizing filters are interesting examples of optical filters whose coefficients of the impulse response are complex numbers. They are suitable for dispersion compensation in wavelength division multiplexed (WDM) communication systems. The filter coefficients

We can also notice that the calculated coefficients

### 5.4 Particle Swarm Optimization (PSO)

The opto-geometrical parameters of the birefringent filter can also be calculated using the heuristic evolutionary optimization algorithms. The cost function to be minimized is expressed according to the optical filter output

The parameters to be determined are the relative angle *p* denotes the size of the swarm. In each generation *d* by referring to the personal best position

*w*is the inertia weight,

*m*iterations, the global best

*m*. Consequently, we can retain the global best

Compared to the proposed algorithms, the PSO with 50,000 iterations and for the simplest example it could not even find the exact coefficients of the desired response. As a result, the heuristic evolutionary optimization algorithms are not suitable for birefringent FIR filter design.

## 6. Discussion

As shown above in the Section 5, simulation results and comparisons with a state of the art methods show that the proposed algorithm is faster, easier and more accurate to calculate the optical structure of the birefringent filters. In [18,7] the procedure of the parameter calculation is complicated, not clear and needs a great number of basic arithmetic operations (addition, subtraction, multiplication and division) compared to our algorithm. For example the phase shift in [18] is expressed as a ratio between two quantities and needs four multiplications, two additions and one division, whereas in our algorithm only two multiplications and one addition are required as expressed above in the second backward algorithm. In addition, the algorithm propose by [7] need two recurrent equations to calculate the coefficients of the *kth* stage. However, in the proposed algorithm we have two coefficients (

## 7. Conclusion

In this paper, iterative algorithms for designing optical FIR filters with any specified spectral response have been presented. They have been tested using different examples and it is observed that they provide exact results in many applications such as asymmetric flat-top birefringent filter, multi-channel selector, and dispersion compensation in wavelength division multiplexed (WDM) communication systems. However, for PSO based birefringent filters, the algorithm must be run many times with a large number of iterations to obtain good results. Moreover, the evolutionary optimization algorithms are extremely sensitive to starting points and the objective function is multimodal and highly non-lineare, which make them very expensive in terms of execution time. In the proposed algorithms, such complicated problem is reduced to find only the roots of polynomial of degree

Finally, knowing that liquid crystal tunable filters are used in optical telecommunication systems and they are also used in multispectral and hyperspectral imaging systems because of their high image quality and rapid tuning over a broad spectral range. Consequently and as a future work, we will try to replace the variable sections of the filter structure with liquid crystal cells whose birefringence can be controlled and tuned with a small voltage. In this way and keeping the same filter structure, we propose to synthesize liquid crystal tunable filters by tuning only the birefringence of the liquid crystal using the same iterative algorithms.

## Figures

Filter parameters obtained by the proposed algorithms, for

0.0197 | 0.0003 | 0.0169 | 0.0197 |

−0.0587 | 0.0019 | −0.0469 | −0.0587 |

0.0043 | −0.0027 | −0.0067 | 0.0043 |

0.5347 | −0.0183 | 0.4293 | 0.5347 |

0.5347 | 0.0321 | 0.4293 | 0.5347 |

0.0043 | 0.0572 | −0.0067 | 0.0043 |

−0.0587 | −0.2362 | −0.0469 | −0.0587 |

0.0197 | 1.1662 | −1.5539 | 0.0197 |

Obtained opto-geometrical parameters of the designed filter, for

0.1875 | −0.0081 + 0.0034i | −0.1396 | 2.7489 | 0.1875 |

0.0577 − 0.0239i | 0.0039 − 0.0039i | 0.0504 | 2.7489 | 0.0577 − 0.0239i |

−0.0442 + 0.0442i | −0.0009 + 0.0022i | 0.0494 | 2.7489 | −0.0442 + 0.0442i |

0.0239 − 0.0577i | −0.0000 + 0.0357i | 0.0481 | −0.3927 | 0.0239 − 0.0577i |

−0.1875i | 0.0085 + 0.0205i | 0.1524 | −0.3927 | −0.1875i |

−0.0239 − 0.0577i | −0.0130 − 0.0130i | 0.0519 | 2.7489 | −0.0239 − 0.0577i |

0.0442 + 0.0442i | 0.0134 + 0.0056i | 0.0519 | 2.7489 | 0.0442 + 0.0442i |

−0.0577 − 0.0239i | 0.0759 + 0.0000i | 0.0516 | −0.3927 | −0.0577 − 0.0239i |

−0.1875 | 0.0337 − 0.0140i | 0.1539 | −0.3927 | −0.1875 |

−0.0577 + 0.0239i | −0.0229 + 0.0229i | 0.0493 | 2.7489 | −0.0577 + 0.0239i |

0.0442 − 0.0442i | 0.0108 − 0.0261i | 0.0504 | 2.7489 | 0.0442 − 0.0442i |

−0.0239 + 0.0577i | −0.0000 − 0.1193i | 0.0512 | −0.3927 | −0.0239 + 0.0577i |

0.1875i | −0.0196 − 0.0474i | 0.1437 | −0.3927 | 0.1875i |

0.0239 + 0.0577i | 0.0335 + 0.0335i | 0.0434 | 2.7489 | 0.0239 + 0.0577i |

−0.0442 − 0.0442i | −0.0400 − 0.0165i | 0.0452 | −0.3927 | −0.0442 − 0.0442i |

0.0577 + 0.0239i | 1.3342 | 1.5240 | 0 | 0.0577 + 0.0239i |

Coefficients

−0.0027 | −1.5995 | |||

0.0128 | 1.5686 | |||

0.0449 | 1.5731 | |||

0.1196 | 1.5708 | |||

0.2378 | 1.5692 | |||

0.3239 | 1.5693 | |||

0.2429 | −1.5755 | |||

0.0270 | 1.5739 | |||

0.2468 | 1.5740 | |||

0.1082 | −1.5751 | |||

0.2125 | 1.5751 | |||

0.1082 | −1.5740 | |||

0.2468 | −1.5739 | |||

0.0270 | 1.5755 | |||

0.2429 | −1.5693 | |||

0.3239 | −1.5692 | |||

0.2378 | −1.5708 | |||

0.1196 | −1.5731 | |||

0.0449 | −1.5686 | |||

0.0128 | 1.5995 | |||

1.0099 | 1.5681 | 0.1573 |

filter coefficient using PSO with different number of iterations, for

Iterations = 10,000 | Iterations = 30,000 | Iterations = 50,000 | ||||
---|---|---|---|---|---|---|

0.0197 | 0.4315 | 0.0197 | 1.0926 | 0.0197 | 1.4735 | 0.0197 |

−0.0587 | −1.1422 | −0.0588 | 1.1563 | −0.0587 | 0.0669 | −0.0587 |

0.0043 | 0.6651 | 0.0039 | −0.5953 | 0.0042 | 0.3078 | 0.0042 |

0.5347 | 0.7994 | 0.5341 | −0.8517 | 0.5345 | −1.0761 | 0.5346 |

0.5347 | 0.8004 | 0.5341 | −0.8517 | 0.5345 | −1.0761 | 0.5346 |

0.0043 | 0.6637 | 0.0040 | −0.5953 | 0.0042 | 0.3079 | 0.0042 |

−0.0587 | −1.1424 | −0.0588 | 1.1563 | −0.0587 | 0.0670 | −0.0587 |

0.0197 | −1.1370 | 0.0198 | −0.4783 | 0.0197 | −0.0973 | 0.0197 |

## References

[1]I. Abdulhalim, G. Moddel, K. Johnson, High-speed analog spatial light modulator using a hydrogenated amorphous silicon photosensor and an electroclinic liquid crystal, Appl. Phys. Lett. 55 (1989) 1603–1605.

[2]A.J. Chaudhari, F. Darvas, J.R. Bading, R.A. Moats, P.S. Conti, D.J. Smith, S.R. Cherry, R.M. Leahy, Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging, Phys. Med. Biol. 50 (2005) 5421.

[3]O. Aharon, A. Safrani, R. Moses, I. Abdulhalim, Liquid crystal tunable filters and polarization controllers for biomedical optical imaging, in: Proc. SPIE, vol. 7050, 2008, pp. 70500.

[4]P.J. Miller, C.C. Hoyt, Multispectral imaging with a liquid crystal tunable filter, in: Optics in Agriculture, Forestry, and Biological Processing, volume 2345, International Society for Optics and Photonics, 1995, pp. 354–366.

[5]H.R. Morris, C.C. Hoyt, P. Miller, P.J. Treado, Liquid crystal tunable filter raman chemical imaging, Appl. Spectrosc. 50 (1996) 805–811.

[6]M. Dandin, P. Abshire, E. Smela, Optical filtering technologies for integrated fluorescence sensors, Lab Chip 7 (2007) 955–977.

[7]P.J. Pinzón, C. Vazquez, I. Pérez, J.S. Pena, Synthesis of asymmetric flat-top birefringent interleaver based on digital filter design and genetic algorithm, IEEE Photonics J. 5 (2013) 7100113–7100113.

[8]J. Zhang, Y. Wang, Synthesis of multifunction optical filter based on digital signal processing method, Opt. Quant. Electron. 49 (2017) 196.

[9]H. Han, Y. Liu, Y. Kong, F. Ansari, T. Taylor, The influence of stress on the quartz birefringent optical filter, Opt. Quant. Electron. 50 (2018) 28.

[10]N. Zhou, S. Zheng, Y. Long, Z. Ruan, L. Shen, J. Wang, Reconfigurable and tunable compact comb filter and (de) interleaver on silicon platform, Opt. Exp. 26 (2018) 4358–4369.

[11]B. Lyot, Le filtre monochromatique polarisant et ses applications en physique solaire, in: Annales d’astrophysique, vol. 7, 1944, p. 31.

[12]J.W. Evans, Solc birefringent filter, JOSA 48 (1958) 142–145.

[13]I. Šolc, Birefringent chain filters, JOSA 55 (1965) 621–625.

[14]S. Harris, E. Ammann, I. Chang, Optical network synthesis using birefringent crystals. i. synthesis of lossless networks of equal-length crystals, 54 (1964) 1267–1279.

[15]E. Ammann, J. Yarborough, Optical network synthesis using birefringent crystals. v. synthesis of lossless networks containing equal-length crystals and compensators, JOSA 56 (1966) 1746–1754.

[16]Y. Zhou, G. Zeng, F. Yu, H.S. Kwok, Study on optical finite impulse response filter, Opt. Eng. 42 (2003) 2318–2323.

[17]Y. Zhou, G. Zeng, F. Yu, Particle swarm optimization-based approach for optical finite impulse response filter design, Appl. Opt. 42 (2003) 1503–1507.

[18]R. Chu, G. Town, Synthesis of birefringent equalizing filters for wdm dispersion compensation, Opt. Commun. 214 (2002) 147–154.

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

[20]E. Ammann, Optical network synthesis using birefringent crystals. iii. some general properties of lossless birefringent networks, JOSA 56 (1966) 943–951.

[21]Q. Bai, Analysis of particle swarm optimization algorithm, Comput. Inf. Sci. 3 (2010) 180.

[22]Y. Shi, R. Eberhart, A modified particle swarm optimizer, Evolutionary Computation Proceedings, 1998. The 1998 IEEE International Conference on IEEE World Congress on Computational Intelligence, IEEE, 1998, pp. 69–73.

## Acknowledgements

Publishers note: The publisher wishes to inform readers that the article “Fast iterative algorithms for birefringent filter design” was originally published by the previous publisher of *Applied Computing and Informatics* and the pagination of this article has been subsequently changed. There has been no change to the content of the article. This change was necessary for the journal to transition from the previous publisher to the new one. The publisher sincerely apologises for any inconvenience caused. To access and cite this article, please use Boukharouba, A. (2021), “Fast iterative algorithms for birefringent filter design”, *Applied Computing and Informatics*. Vol. 17 No. 2, pp. 250-263. The original publication date for this paper was 04/09/2018.