## Abstract

More than 50 years ago, B. Lyot and later on I. Solc introduced new types of optical filters called birefringent filters. Such filters take advantage of the phase shifts between orthogonal polarization to obtain narrow band filters. It requires birefringent wave plates for introducing phase retardation between the two orthogonal components of a linearly polarized light that correspond to the fast and slow axes of the birefringent material. In this paper we present new methods and architectures that generalize the Lyot-Ohman and Solc filters for optimally synthesizing an arbitrary all-optical filter by defining an error metric and minimizing it with simulated annealing. We also suggest the use of the electro-optic effect for controlling the retardation of individual elements that make up the tunable filter. Such a filter could be used for instance for realizing a dynamically tunable optical add/drop multiplexer in a telecommunication system.

©2002 Optical Society of America

## 1. Introduction

All-optical filters are important for a large number of applications, as for instance, optical add/drop multiplexers, tunable modulators and gain flattening filters, which are used for adjusting the intensities of different wavelength channels coming out of an optical amplifier.

When faced with the problem of generating all-optical birefringent narrowband filters, known solutions are provided by the Lyot-Ohman [1–5] filter family or the Solc filters [6,7]. The latter has two implementations, known as “fan” and “folded”. Both Solc and Lyot families use different configurations of polarizers and retardation plates to create narrow-band (0.5-5Å) filters. These filters are composed of a stack of birefringent crystals, preferably with high birefringence (such as quartz or LiNbO_{3} crystals) interspersed with polarizers. Lyot developed his filter [1,2] (1933, 1944), using an alternating “plate-polarizer stage” configuration. Essentially, each stage multiplies the transmissivity of the previous stage. In the Lyot-Ohman configuration, depicted in Fig. 1, each plate whose “fast” and “slow” axes are oriented at 45° with respect to the polarizers, is twice thicker than that of the preceding plate, yielding the retardation of the k_{th} plate, Γ_{k} as:

The overall transmission of N stages is given by:

where T_{0} represents energy losses due to absorption and reflection.

Solc filters (either folded or fan configuration) require only two polarizers, located at the entrance and exit pupils of the optical system. The desired filter is achieved by using tilted retardation plates. When the incoming light is not polarized, a calcite can be used to separate the two orthogonal states of polarization, whereas each of the resultant rays can pass through the filter. A folded Solc filter (see Fig. 2) consists of alternating equal azimuth tilts. The angles in relation to the pre-determined x-axis are:

where k varies from zero to N-1 with N (an even number) being the number of retardation plates and ρ is an arbitrary angle, typically chosen as $\rho =\frac{\pi}{4N}$ for maximizing the transmitted energy. Making use of Jones matrix calculus and Chebyshev’s identity [8], we arrive at the expression for the transmittance of the folded Solc filter which is given by:

Where χ is defined through the equation:

L being the optical path length of the retardation plate.

The Fan Solc Filter is composed of the same basic configuration as the Folded Solc Filter, except that the angles do not alternate in sign, but increase in magnitude by a constant. Specifically, the angle of the k_{th} plate in a Fan Solc Filter in relation to the pre-determined x-axis is given by:

The transmittance of a Fan Solc Filter is identical to that of a Folded Solc Filter shifted by π in the phase retardation Γ.

This paper explores various architectures consisting of both retardation plates and polarizers, searching for the optimal performance of the filter according to a given criterion. Since analytical solutions are possible only in degenerate cases, a numeric approach that is based on simulated annealing [9,10] is therefore proposed.

The method of simulated annealing is used to find the global minimum of a function that has many local minima, which “hide” the global minimum. Here, the “Energy” is described as the error metric dependent on the free variables of the system. A “possible change” is defined and the new “Energy” is calculated, where the probability of an energy state E is given by the Boltzmann probability distribution:

Therefore, the energy function has a chance to jump to another level of energy. Since this probability decreases as the temperature decreases, the average changes decrease as temperature drops – this is evident if one notices T represents not only the temperature, but also be the average “step” (the average of an exponential distribution function). Depending on the difference between the energy of the new state and the last state, a statistical decision is made whether to accept the possible change or retain the old value. The probability of accepting the change is:

Notice that if ΔE is negative p is higher than one and the probability is set to one such that the step will always be taken. However, steps may also be taken when ΔE is positive for escaping local minima.

## 2. Tunable birefringent filter consisting of arbitrary retardation plates

The first architecture proposed in this paper closely resembles the Solc (Folded and Fan) family of filters. It consists of a stack of birefringent elements that are situated at different angles with respect to the x-axis. In this section, both the angles of the retardation plates and their optical path lengths (or birefringence) are considered as adjustable parameters. The goal is thus to choose the angles and optical path lengths (or birefringence) of each retardation-plate such that the resultant filter will be as close as possible (in the sense of a predefined error metric) to a desired filter.

In principle, for some error metrics there might be an analytical solution when the number of stages (wave-plates) is small. However, as the number of stages increases, it becomes extremely difficult to find an analytical result and the solution might be found numerically with the help of some iterative techniques such as simulated annealing.

When applying the simulated annealing algorithm one defines an initial temperature and then decreases it as the number of iteration increases. In the proposed architecture, each iteration consists of running over the 2N (N angles + N optical path lengths) variables M times (M should be defined for each case separately). Each time a variable is encountered, a random change (or “step”) is applied to it. Note that the other parameters are left unchanged, so that a step consists of changing one variable only. The decision whether to accept the change is based on a chosen error metric (e.g. mean square error). If the error due to the change is reduced, then the change is accepted. If the error increases, then according to the Boltzman probability distribution (Eq. 7) there is still a chance to accept it (this probability decreases as the temperature decreases or when the error difference increases). Thus, in each iteration M ∙ 2N possible changes are considered, where N is the number of stages (2N is the number of free parameters). At the end of each iteration, the temperature is decreased according to the cooling rate chosen.

Once the parameters are found, the proposed filter may become dynamically tunable by utilizing the electro-optic effect. For example, let us consider a LiNbO_{3} crystal rod with its input and output faces normal to the y-axis. The side surfaces are normal to the x and z axes and the electric field is applied along the z direction. The phase retardation of the x and z polarizations propagating along the y direction is thus:

where L is the length of the plate, d is its thickness, r_{33} and r_{13} are constants equal to: r_{33} =30.8pm/V and r_{13} =8.6pm/V and V is the applied voltage. The electrodes are at distance d from each other.

Applying the same induced retardation to all wave-plates will merely cause the pass-band of the optical filter to shift. Meaning that a tunable filter, whose band pass region can be dynamically varied, is obtained. The birefringent material limits the tuning speed. For the case of a LiNbO_{3} crystal, the tuning time is in the order of several nanoseconds.

In order to test the performance of the proposed algorithm, a set of computer simulations was carried, whereby the optical path lengths of all retardation-plates were fixed and only the azimuths of each retardation-plate were considered free parameters. The lengths of the retardation plates were all set equal and predefined as odd multiples of λ/2 plate:

Taking a LiNbO_{3} crystal as an example yields birefringence of approximately Δn=0.1, which results in L=2.71mm for λ_{0}=1.55μm. The choice of the number 301 is arbitrary. It led to a realistic thickness for L and the desired filter. These basic parameters are used throughout the simulations in this paper.

At first, a desired filter was defined. We defined the target to be the creation of a top-hat 100GHz filter around the central wavelength (1.55μm). Naturally, the resultant filter is periodic in $\frac{1}{\lambda}$. In this example, it is required that the period will not go below 1THz. As a first “guess” one might use the folded Solc filter with L being given in Eq. (10).

The number of elements, N was chosen to be 20. The cooling rate formula as a function of the iteration “i” is:

This is a common choice for the cooling rate in simulated annealing problems.

Note that “iteration” refers to observing M possible changes (M=10 in our case) in every one of the N free parameters. Thus each iteration consists of observing M^{*}N=200 possible changes and making a decision about each one.

The metric was chosen to be the MSE (mean square error) over a single period (since the spectrum is periodic). Other metrics fitting to various applications might be considered as well.

Figure 3 shows the desired filter (dashed-line) along with the initial guess that was chosen to be the folded Solc filter.

Note that this yields an (MSE) error of 0.0251. Figure 4 shows the performance of a filter achieved after 15 iterations of the algorithm. The MSE is 0.0067, yielding an improvement of $\frac{0.0251-0.0067}{0.0251}=73\%$. It was found that after 15 iterations of the simulated annealing process, the performance improvement was negligible.

Figure 4 shows that the resultant filter closely resembles the desired filter, especially in the pass-band region where the filter is almost flat. This comes at the expense of added side-lobes at frequencies far into the band-stop region. Note that the overall improvement is apparent both in the MSE achieved as well as in the shape of the filter itself. We present, for visualization purposes, the resultant filter tilt angles in Fig. 5. Note that in the resultant filter the first retardation plates are indeed tilted in angles of alternating signs, but the last retardation-plates are all at a positive angle. This suggests that the desired target filter requires a different structure than the one suggested by the Folded Solc Filter. Note that depolarization and scattering effects of the wave-plates may limit the extinction ratio of the filter to around 30dB. The extinction ratio of polarizers reaches much higher values and is not typically a limiting factor.

## 3. Tunable birefringent filter in a folded architecture

Naturally, it is desired to attenuate the side-lobes and to reduce the number of retardation-plates. The side-lobe attenuation may be obtained by using a weight function on the MSE criterion at the expense of peak uniformity or by increasing the number of stages.

The number of stages can be effectively increased if the optical system is folded (see Fig. 6). The beam is rotated back and goes through the system again. The path of the optical beam may be reversed by a corner roof prism, which has the effect of reversing the y and z axes of the propagating beam. The effective azimuth angles therefore now have a reverse sign, but the same magnitude, and they come in reverse order. Adjacent to the corner roof prism one may insert a polarizer for buffering between the two passages through the wave-plates (which results in multiplication of the transmissivity). Without a polarizer, this architecture is the equivalent to designing a filter of 2N stages with a constraint of anti-symmetry for the azimuth angles. Notice that the number of free variables is still N while the number of stages is now 2N. Furthermore, note that a Folded Solc Filter can be implemented with this architecture as it has alternating azimuth angles and obeys the anti-symmetric constraint.

The folded architecture was tested where the number of wave-plates was chosen to be 10. Since the wave-plates were situated in a folded architecture, the optical beam passes through each wave-plate twice. The polarizer that should have been placed after the 10^{th} plate was removed in order to be comparable with a Folded Solc Filter leaving a single polarizer in the system, located at the entrance, functioning both as entrance and exit polarizer.

The starting guess here is a Folded Solc Filter. Meaning that the stages are positioned such that after passing through the folded system, the light goes through the equivalent of a folded Solc filter of 2N=20 stages.

Figure 7 presents the resultant filter behavior. The mean square error of the 10 stage folded-architecture filter using Simulated Annealing was 0.008343, yielding a 67% improvement with respect to a folded Solc filter of 20 stages. In addition the improvement (with respect to the folded Solc filter) in the suppression of the side-lobes is noticeable. As expected, the simulation result of Section 2, where all 20 wave-plates were arbitrarily positioned, exhibits a slightly better result. Clearly, the advantage of this method over the one suggested in section 2 is the use of only half the number of plates needed for the Folded Solc Filter, and the suppressed sidelobes.

## 4. Tunable birefringent filters in combined polarizers-retardation plates architecture

Another degree of freedom when designing tunable birefringent filters is the incorporation of a number of polarizers in the filter’s architecture such as schematically shown in Fig. 8. One can view this kind of architecture as a midway design that resides between the Solc and Lyot-Ohman filters. It takes advantage of the fact that insertion of polarizers between the stages results in some sort of buffering that causes multiplication of the transmissivity of the cascaded filters. Such architecture might be advantageous when trying to suppress the side-lobes of the filter and use minimal number of wave-plates. As a first step, one may start by inserting a single polarizer into the system somewhere in midway. After optimization, if the side-lobes are still large, one may try to insert additional polarizers into the system and repeat the procedure.

In the following simulation, the number of wave-plates was chosen to be 12. Here, an additional polarizer (except for the ones at the entrance and exit pupils) was inserted after the 6^{th} stage. The starting guess here was chosen as the same tilt angles for a Folded Solc Filter of 12 stages. The iterations number was increased to 20 due to the fact that this design kept showing an improvement even after the 15^{th} iteration. The result is shown in Fig. 9. The MSE was found to be 0.00931. As expected, the mean square error is worse than the previous optimized filters since the number of stages was reduced. However, there is still 63% improvement with respect to the 20 stages Folded Solc Filter. The filter uses almost half as many stages as in the Folded Solc Filter and still provides a dramatically improved result, as far as side-lobes suppression and MSE are of interest.

## 5. Conclusions

This paper presents several architectures and design methods that implement all-optical tunable birefringent filters. At first, the paper explores a configuration that is identical to the folded (or fan) Solc filter but exploits the free parameters of the system. Then, a folded architecture is considered, where the optical beam passes more than once through the system, yielding a finer filter with a reduced number of components. Finally, a more general and hybrid architecture is considered. There, additional polarizers are gradually introduced into the optical system, such that one obtains a filter that for some target filter and error metric may converge to a Solc-like configuration and for other target filters may become a Lyot-Ohman architecture.

## References and links

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