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.
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
Emerald Publishing Limited
Copyright © 2018, Abdelhak Boukharouba
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
Optical filters are most commonly used in modern optical telecommunication systems  and also used for biomedical spectral imaging [2–4] such as Raman chemical imaging  and fluorescence microscopy applications . 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 . 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  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. . 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  that allows the realization of impulse response having complex coefficients is detailed by Ammann and Yarborough . 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 . 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  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  where arbitrary filter characteristics corresponding to nth-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  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. represents the angle between the fast axis of the kth retarder and the y-axis which is the same angle between the slow axis of the retarder and the x-axis, while is the angle between the transmission axis of the output analyzer and the x-axis. After the input polarizer, the polarized light comes through the retarder stack where each retarder separates the input light into two components along its fast and slow axis respectively, and each component acts as the input of the next one [14,17].
The output of the optical filter must give the desired impulse response, .
L is the length of each retarder, represents the difference of their refractive indices, and 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, , is determined by Jones matrix [16,20].
The main idea of this algorithm is to transfer the matrix–vector product into tow polynomials and .
2.1 Forward algorithm
Let’s set , and . Consequently is expressed as follows:
For two stages has the following expression:
By induction, each component of the is a polynomial of z for . We set and and for stages, may be written as:
For n stages is given by:
We can simply prove that:
From Eq. (14) and knowing that the frequency responses of the two complementary filters are defined by and respectively. Thus, and , for
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 is given by and the coefficients of the order filter are given by and . But for the other coefficients, we must solve the system of Eqs. (15) and (16) where three determinants must be formed; the denominator determinant is and consequently the and have the following expressions for .
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 . However, in , each retarder is a birefringent plate composed of a section of nominal length L and a section with variable length that acts as an optical compensator, see Figure 2. The identical lengths introduce a unitary delay whereas the variable lengths (optical compensators) introduce variable phase shifts between slow-axis and fast-axis components. The expression of is the same as in (8) except the delay operator becomes where the phase shift is introduced by the variable length .
3.1 Forward algorithm
Following the same steps as in the Section 2.1, we can prove that:
where , , , and and can be calculated by the following equations for and .
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 is given by and the coefficients of the order filters are given by and . But for the other coefficients and from the Eq. (21) and (22) and have the following expressions for and .
As shown in Section 3.1, must be real thus we need to add an optical compensator in the front of the analyser by choosing and the filter coefficients at the analyser output become , . In practice, the final compensator may be removed from the filter structure because the transmittance with and without final compensator differ from each other by only a constant phase factor, and hence the final compensator can be ignored.
Moreover, must be also real and consequently, we can take and as solution especially for calculating the phase shift . If we count the optical compensator in the front of the analyser, we also have n compensators (). Note that if , a compensator is not required for this stage. The coefficients are in general complex numbers; we can note that if is a solution of (7), then is also a solution . Knowing that is also a real number then we can choose the coefficients of the complementary component as , where to make real. The Algorithm 2 explains the backward algorithm.
4. Complementary component calculation
Assume that and therefore the desired have been chosen. We must next find which is polarized perpendicular to and therefore stopped by the output analyzer.
Here must be chosen greater than, or equal to, the maximum value of . Having chosen , we can calculate from using the method given in  with further improvements. Letting , the expression of (25) becomes as follows:
Note that has got the same roots as which is a polynomial of degree .
Notice that the all above algorithms are developed for . However, if has an arbitrary value, the output of the optical filter calculated using the orientation angle found by the forward algorithm must be multiplied by to have the desired impulse response of the optical filter. Multiplying by , which represents an amplification/attenuation, has no effect on the shape of the filter response. For all the following examples the chosen value of is (). Consequently, the output of the filter must be multiplied by to have the desired impulse response.
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 is a seventh-order filter, and its coefficients are calculated in . However, are calculated using the algorithm stated in Section 4, and the relative angles are found by the backward algorithm of the Section 2 where the filter coefficients are real. Once the orientation angles are found, we can calculate the coefficients of the optical filter using the forward algorithm and compare them with the desired ones. Table 1 shows the filter parameters obtained by the proposed algorithms.
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 . In this case, the optical filter is synthesized to select three signals with frequencies of and from eight FDM signals. Thus, the transmittance values at the three frequency points of must be 1, and the transmission values at the other frequency points must be 0 where the number of expansion coefficients is set at 16. The inverse discrete Fourier transform is used to obtain the filter coefficients. As the coefficients are complex numbers, we must use the algorithms of the Section 3 to calculate the parameters of the filter. Table 2 shows the obtained opto-geometrical parameters of the designed filter. The phase shifts are non-zero, because the expansion coefficients are complex numbers. Once the relative angles and phases shifts are obtained using the backward algorithm, we can calculate the output of the optical filter using the forward algorithm where the calculated coefficients match exactly the desired coefficients , which confirms the accuracy of the proposed approach.
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 are calculated in . The coefficients of the complementary component are calculated using the procedure described in Section 4 whereas the orientation angles, the phase shift introduced by each stage and the coefficients of the obtained impulse response are calculated using the algorithms described in Section 3. Table 3 illustrates the coefficients ; relative angles , the phase shifts and the coefficients of the obtained impulse response.
We can also notice that the calculated coefficients match exactly the desired 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 calculated by the forward algorithms and the desired output for both above cases.
The parameters to be determined are the relative angle of each crystal and the retardation introduced by each compensator . In this work, we can take as example the PSO algorithm, which is the famous one of the heuristic evolutionary optimization algorithms . Suppose that the search space is a n-dimensional space, then the ith particle can be represented by a n-dimensional vector, , and velocity , where and p denotes the size of the swarm. In each generation , particle i adjusts its velocity and position for each dimension d by referring to the personal best position and the swarm’s best position as follows :
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.
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  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  need two recurrent equations to calculate the coefficients of the stage from ones of the kth stage. However, in the proposed algorithm we have two coefficients ( and ) calculated as a ratio between two numbers in each stage and the two last coefficients ( and ) are also expressed as a simple ratio between two numbers as illustrated in the first backward algorithm. Moreover the coefficients are calculated using FIR filter design techniques in , which needs a heavy computation and lacks accuracy compared to the proposed algorithm. However, we have developed a concise and accurate algorithm to calculate the coefficients, , of the complementary component. The forward algorithm is used to simulate the optical filter by calculating the filter coefficients from the relative angles and the phase shifts obtained using the backward algorithm and compare them with the desired ones. It may be also used with the classical optimisation methods and the heuristic evolutionary optimization algorithms where the cost function to be minimized is a function of the filter parameters, which is easily calculated using the forward algorithm. The proposed approach is also compared with the PSO algorithm, which is the famous one of the heuristic evolutionary optimization algorithms. As illustrated in Table 4, the PSO with 50,000 iterations and for the simplest example it could not even find the exact coefficients of the desired response. Moreover, the evolutionary optimization methods are realized by multiple iterations of updating parameters until convergence. Consequently, they are also not competent in designing optical FIR filters with complex coefficients where the phase shifts are included due to their heavy computation and unguaranteed convergence. On the other hand, the proposed algorithm does well in designing optical FIR filters of any desired spectral shape and with any order using a simple iterative calculation with a minimum number of arithmetic operations.
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 and the exact solution is determined using iterative algorithms with only a few number of operations.
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.
Filter parameters obtained by the proposed algorithms, for .
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 , relative angles , the phase shifts and the obtained coefficients , for .
filter coefficient using PSO with different number of iterations, for .
|Iterations = 10,000||Iterations = 30,000||Iterations = 50,000|
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.
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.
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.
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.
H.R. Morris, C.C. Hoyt, P. Miller, P.J. Treado, Liquid crystal tunable filter raman chemical imaging, Appl. Spectrosc. 50 (1996) 805–811.
M. Dandin, P. Abshire, E. Smela, Optical filtering technologies for integrated fluorescence sensors, Lab Chip 7 (2007) 955–977.
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.
J. Zhang, Y. Wang, Synthesis of multifunction optical filter based on digital signal processing method, Opt. Quant. Electron. 49 (2017) 196.
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.
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.
B. Lyot, Le filtre monochromatique polarisant et ses applications en physique solaire, in: Annales d’astrophysique, vol. 7, 1944, p. 31.
J.W. Evans, Solc birefringent filter, JOSA 48 (1958) 142–145.
I. Šolc, Birefringent chain filters, JOSA 55 (1965) 621–625.
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.
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.
Y. Zhou, G. Zeng, F. Yu, H.S. Kwok, Study on optical finite impulse response filter, Opt. Eng. 42 (2003) 2318–2323.
Y. Zhou, G. Zeng, F. Yu, Particle swarm optimization-based approach for optical finite impulse response filter design, Appl. Opt. 42 (2003) 1503–1507.
R. Chu, G. Town, Synthesis of birefringent equalizing filters for wdm dispersion compensation, Opt. Commun. 214 (2002) 147–154.
K. Jinguji, M. Kawachi, Synthesis of coherent two-port lattice-form optical delay-line circuit, J. Lightwave Technol. 13 (1995) 73–82.
E. Ammann, Optical network synthesis using birefringent crystals. iii. some general properties of lossless birefringent networks, JOSA 56 (1966) 943–951.
Q. Bai, Analysis of particle swarm optimization algorithm, Comput. Inf. Sci. 3 (2010) 180.
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.
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.