Spectral measurement of birefringence using particle swarm optimization analysis Maciej Kraszewski,* Marcin Strąkowski, Jerzy Plucinski, ´ and Bogdan B. Kosmowski Gdansk ´ University of Technology, Faculty of Electronics, Telecommunications and Informatics, G. Narutowicza 11/12, Gdansk, ´ 80-233, Poland *Corresponding author: [email protected] Received 14 July 2014; revised 14 November 2014; accepted 20 November 2014; posted 21 November 2014 (Doc. ID 216918); published 23 December 2014
Measurement of birefringence is useful for the examination of technical and biological objects. One of the main problems, however, is that the polarization state of light in birefringent media changes periodically. Without knowledge of the period number, the birefringence of a given medium cannot be reliably determined. We propose to analyze the spectrum of light in order to determine the birefringence. We use a particle swarm optimization algorithm for an automatic processing spectra of light transmitted through birefringent material for two orthogonal states of polarization. We have tested the described algorithm on a liquid crystal cell with varying effective birefringence. The proposed method can be used for the measurement of uniaxial positive birefringence without knowing the number of retardation periods or an approximate value of the measurement result. This fact makes the proposed method useful for automatic measurements, when hundreds or thousands of spectra need to be analyzed. © 2014 Optical Society of America OCIS codes: (120.5410) Polarimetry; (160.3710) Liquid crystals. http://dx.doi.org/10.1364/AO.54.000076
1. Introduction
Birefringence is an optical property of many materials. The simplest type of this property, but with a large number of practical applications, is uniaxial birefringence, which is the existence of two different refractive indices for two orthogonal states of linear light polarization. Since birefringence depends on structural properties of the material on a molecular level, it can provide useful information about many classes of objects. A major industrial application of this phenomenon is photoelastic stress analysis, which allows us to visualize stress in polymer materials [1]. In biomedical optics, biological tissue birefringence allows us to examine burn depth [2] or caries lesions [3]. The examination of birefringence is also important for evaluation of various optical elements such as liquid crystal devices or optical fibers. Research presented in this paper was focused on
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analysis of uniaxial birefringent media, particularly on the measuring difference between refractive indices corresponding to medium main optical axes. Such measurements are a key element of the abovementioned applications. Birefringence measurement is based on the measurement of the retardation angle, which is defined as the phase delay between ordinary and extraordinary light rays. The retardation angle can be determined by the measurement of the polarization state of light, e.g., using the setup based on a linear polarizer and quarter-wave plate, Babinet–Soleil compensator, or rotating retarder [4]. Precise retardation angle measurements are also possible using various interferometric techniques [5,6], ellipsometry [7], or spectropolarimetry [8]. During propagation in a birefringent material, the polarization state of light changes periodically. Therefore, it is useful to express the retardation angle as Γ Γ0 2mπ; where Γ0 h0; 2πi and m is the retardation order.
(1)
The polarization state of light depends on Γ0 , so if the retardation order is unknown, the measurement of birefringence is ambiguous [9–11]. Moreover, many measurement methods rely on the measurement of cos Γ [4]; therefore, Γ0 can be determined only in the range from 0 to π. Other methods, such as polarization-sensitive optical coherence tomography (PS-OCT), measures the value of j tan Γj based on the backscattered light from the sample [12], which limits the measurement range to h0; π∕2i. This measurement range is insufficient for many applications such as the analysis of stress in polymer foils, the examination of liquid crystal cells and materials, or testing optical fibers. However, this difficulty can be overcome by analyzing the retardation angle as a function of a wavenumber. The retardation angle of the homogeneous medium depends on the wavenumber and birefringence according to the equation: Γ kdΔn;
(2)
where k 2π∕λ is the wavenumber, λ is the wavelength in vacuum, Δn is the birefringence, and d is the thickness of the birefringent material. Processing the spectrum of the polarized light can provide information on the birefringence without the necessity of knowing the retardation order. Such an analysis has recently been demonstrated for the examination of optical fibers [13]. In this application, it is possible to obtain high orders of retardation by using fibers with a sufficient length. However, the examination of polymer foils or liquid crystal materials require a processing method that would work well for both high and low retardation order. In this paper, we present the application of a particle swarm optimization (PSO) algorithm [14–17] for processing retardation angle spectra. We have tested its performance on liquid crystal cells with voltagecontrolled birefringence. Our experiments show that PSO can be applied to the automation of birefringence measurements, which makes it valuable for processing a large number of spectra, as in automatic inspection or spectral polarization imaging [18,19]. 2. Problem Statement and Measurement Principles
In this contribution, the issue of direct determination of birefringence has been addressed. For this purpose, all considerations have been limited to the uniaxial birefringent medium having positive birefringence (Δn). The other cases are not in the topic of this contribution and will be investigated during further research. The retardation Γ introduced by a given medium can be expressed as a function of the light wavenumber (Fig. 1, upper plot). However, the only phenomenon that can be measured is the polarization state of the light guided through the birefringent medium. The state of monochromatic light polarization depends on the Γ0 described in Eq. (1), which varies from 0 to 2π (Fig. 1, central plot). Moreover, the
Fig. 1. Example of retardation changes over the light spectrum for birefringence Δn equal to 0.3, light wavelength in the range from 1.25 to 1.65 μm, and the thickness of the birefringent medium d equal to 55 μm. Γ, the exact value of the retardation introduced by the medium; Γ0 , the factor that has the most significant influence on the polarization state of light; γ, the measurable value of retardation angle.
polarization state of the light is determined by using the trigonometric relations between the measured quantities, which were mentioned in the introduction. Therefore, the value of retardation, which can be obtained directly, is measurable in the range from 0 to π (Fig. 1, bottom plot). The problem has been presented in Fig. 1; however, the details of the whole analysis based on particular examples are explained in the following part of the paper. The birefringence of liquid crystal cells were measured with the setup presented in Fig. 2(a). The radiation from the broadband, supercontinuum source (Menlo TB-1550) with the light spectrum spanning from 1200 to 1700 nm was collimated and transmitted through the linear polarizer L1. Then it was transmitted through the homogenous (HG), planar liquid crystal cell LC with 50 μm thickness [20]. The optical axis of the cell was oriented under 45° from the axis of polarizer L1. After that, light was transmitted through the polarizer L2 into the optical spectrum analyzer (Anritsu, MS9740A). The liquid crystal cell was driven by alternate voltage (square wave with 50% duty cycle) with a frequency of 100 Hz and the root mean square (rms) varying from 0 to 5 V. For this frequency, the electrical anistropy of the used liquid crystal material was positive (Δε > 0). For each value of the voltage, two spectra were collected: one with the optical axis of L2 in parallel with the axis of L1 and second with the axis of L2 orthogonal to the axis of L1. The liquid crystal material presents substantial birefringence on the order of 0.3. Driving the cell with the voltage 1 January 2015 / Vol. 54, No. 1 / APPLIED OPTICS
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changes the orientation of liquid crystal molecules, which results in the drop of effective birefringence of the liquid crystal layer. The reference measurement of birefringence has been performed using the setup presented in Fig. 2(b). Two superluminescent diodes with peak wavelengths 1560 nm (Superlum, Broadlighter S1550) and 1290 nm (Superlum, Broadlighter S1300) and spectral width (FWHM) 50 and 45 nm, respectively, have been used as a light source. The polarizer L2 and the spectrum analyzer have been replaced by a polarimeter (Thorlabs, PAX5710). Such a setup allows us to determine the retardation angle of the liquid crystal cell at the single wavelength, but does not allow us to determine the order of retardation directly. To overcome this problem, the cell was initially driven by the voltage with rms equal to 5 V. For this voltage, the first order of retardation was assumed. This assumption agrees with the spectral measurement of the cell birefringence (Fig. 6). After that, the value of the applied voltages was being gradually decreased with steps, allowing us to obtain four data points for each retardation order. This allowed us to unwrap measured values of the retardation angle Γ0 and calculate the birefringence of the cell. Jones vectors of light coming to the spectrum analyzer for two orientations of the polarizer L2 are given by p I 0 k 1 1 e−iΓk J⃗ v k ; 1 2
(3)
p I 0 k 1 1 − e−iΓk ; J⃗ h k −1 2
(4)
and
where k 2π∕λ is the wavenumber, J v is the Jones vector in case of polarizers L1 and L2 aligned in parallel, J h is the Jones vector in case of polarizers L1
and L2 crossed, Γ is the phase retardation introduced by the liquid crystal cell, and I 0 is the mean intensity of incident light. Corresponding intensities are given by I v k J⃗ v k · J⃗ v k I 0 k1 cos Γk;
(5)
I h k J⃗ h k · J⃗ h k I 0 k1 − cos Γk:
(6)
Cosine of the retardation angle Γ can be calculated based on the measurements of I v and I h as cos Γk
I v k − I h k ; I v k I h k
(7)
which leads to γk arccos
I v k − I h k ; I v k I h k
(8)
where γk is the retardation angle Γk wrapped in the range from 0 to π: γk
Γ0 2π − Γ0
if Γ0 < π; if Γ0 ≥ π;
(9)
where Γ0 is defined by Eq. (1). Further in this paper, γk denotes function based on the measurement of I v k and I h k and calculated with Eq. (8). We use additional symbol γ t k as a theoretical value of function γk. While γk is the result of the experiment, γ t k is calculated assuming certain values of thickness and birefringence of the liquid crystal cell and can be calculated as γ t k arccoscos kdΔn;
(10)
where d and Δn are the thickness and effective birefringence of the cell, respectively. 3. Numerical Analysis of Birefringence Spectra A. Problem of Varying Order of Retardation
Fig. 2. Experimental setup for birefringence measurement using (a) the proposed spectral method and setup for (b) the reference measurement. L1 and L2, linear polarizers; LC, liquid crystal cell. 78
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The examples of spectra γk based on the measurement of I v k and I h k and calculated with Eq. (8) for the cell driven with rms voltages 0 and 2 V are presented in Fig. 3. The periodicity of γk is visible in both spectra; however, each of them requires a different approach for determination of Δn. If at least a few different orders of retardation are present in the spectrum, the distance between peaks in γk can be used to calculate the birefringence. It is also possible to use the base frequency of γk calculated with standard tools such as Fourier transform. More orders of the retardation provide higher accuracy of this processing method. When the effective birefringence is small, the spectrum γk may contain only one order of the retardation. In such a case, the birefringence can
Fig. 3. Measured spectra of wrapped retardation angle for the liquid crystal cell driven by different voltages.
be calculated based on the slope of γk. However, due to wrapping γk in the range from 0 to π, it is difficult to automatically find a proper part of the spectrum for the slope calculation. The described methods cannot be reliably applied for low and high effective birefringence of the cell. It could be possible to choose one of two methods depending on the value of birefringence, but making such a choice automatically could raise new difficulties. Instead, an optimization algorithm can be used for processing spectra regardless of the value of the birefringence. B.
Application of Optimization Algorithms
Low and high birefringence of a cell can be reliably determined by fitting nonlinear function γ t k to measurement result γk using optimization algorithms. In this approach, the value of Δn is searched within a specified range at Δn space based on certain optimization criteria. The most common criterium is the minimization of a mean square error (MSE): sP M−1 2 m0 γ t km − γkm MSE PM−1 ; 2 m0 γkm
(11)
where km is the wavenumber of mth point in the spectrum. Simulated MSE for different values of real and fitted birefringence are presented in Fig. 4. Presented results show that the cell’s birefringence can be determined by finding such a value of Δn for which the value of MSE takes its global minimum. The simplest optimization methods are based on gradient descent algorithms. Example of such a method is the Levenberg–Marquadt algorithm [21,22], which is widely used in spectroscopic ellipsometry [22–24]. However, gradient descent methods are susceptible to local minima of MSE. They provide an accurate solution only if the initial point of the algorithm is close to the measured value of Δn. The problem of local minima is important when values of Δn may vary within several orders of magnitude. In such a case, an approach different than the gradient descent method is needed. Examples of alternative optimization algorithms are the simplex method [25], genetic algorithms [26], simulated annealing [27], and PSO [14–17]. In this study, we
Fig. 4. Mean square errors between two phase retardation spectra for different values of sample birefringence. Matching the real and fitted birefringence by MSE minimization with gradient descent algorithms is likely to fail due to many local minima of MSE. Ideally, the optimization algorithm should find a point on the line marked with black points regardless of the initial guess of the birefringence.
have chosen the PSO because of its simplicity and possibility of easy implementation of parallel computing. C.
Particle Swarm Optimization Algorithm
PSO is an optimization method similar to genetic algorithms, which can search a broad region of parameter space for an optimal solution [14–16]. This makes it much less sensitive to the local minima of MSE than gradient descent methods. The PSO relies on searching the best value of Δn by scanning an entire search space with a set of “particles.” Each particle is moves through search space by changing the corresponding value of Δn by a value denoted as “velocity.” The initial velocities of all particles are set to 0 and are modified in each iteration of the algorithm. The velocity of each particle in the next iteration is the weighted sum of its previous velocity, direction to the best solution already found by this particle, and direction to best solution found by the entire particles set. The algorithm is finished after finding Δn with the value of MSE below a specified threshold or after a maximal number of iterations. Symbols used in the description of the presented algorithm are listed in Table 1. The algorithm, illustrated in Fig. 5, consists of following steps: 1. Set the initial positions of particles randomly in the search space: Δni ∈ hΔnmin ; Δnmax i. 2. Calculate the values of MSE for each particle and set them as the initial values of MSEl;i. 3. Find the best value of birefringence Δngl and the error MSEg from the entire set of particles. 4. Draw uniform random values a1 ∈ h0; A1 i, a2 ∈ h0; A2 i, and a3 ∈ h0; A3 i. 5. Update particle velocities according to the equations: vi ←a1 vi a2 vl;i a3 vg;i ; 1 January 2015 / Vol. 54, No. 1 / APPLIED OPTICS
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Table 1.
List of Symbols Used for Description of the PSO Algorithm
Parameter
Symbol
Number of particles Measured spectrum Initial birefringence range Iterations limit Mean square error threshold Velocity weight limits Birefringence of ith particle Velocity of ith particles Best birefringences found by ith particle Smallest error obtained by ith particle Best birefringence found by all particles Smallest error obtained by all particles
of measured and fitted spectra of retardation angle are presented in Fig. 6. D.
N ~ Γk Δnmin , Δnmax K max MSEmax A1 , A2 , A3 Δni vi Δnl;i MSEl;i Δng MSEg
vl;i ←Δnl;i − Δni ;
(13)
vg;i ←Δng − Δni :
(14)
6. Update the particle positions Δni ←Δni vi. 7. If any particle is beyond the limits of searching space, Δni > Δnmax or Δni < Δnmin, it obtains a new random position and velocity. 8. Calculate errors MSEi associated with the position of each particle. 9. Update the values of the best positions Δnl;i , Δng , and the errors MSEl;i and MSEg . 10. If MSEg < MSEmax or the maximal number of iterations was exceeded, finish the algorithm; otherwise, return to Step 5. 11. After the algorithm is finished, the value Δng is taken as the measured value of birefringence. To test the PSO algorithm, measurement of the birefringent layer with thickness 50 μm was made. The algorithm used 20 particles and velocity coefficients: A1 1, A2 2.5, and A3 1.5. The error and number of iterations were limited to MSEmax 0.02 and K max 50, respectively. The exact value of birefringence varied from 0.001 to 0.5. The algorithm was run 30 times for each value of the birefringence. For each case, the birefringence value was determined with accuracy better than 0.0021. Examples
Fig. 5. Illustration of the PSO algorithm in the case of two particles. In each algorithm iteration, particle positions (Δni ) are modified by their velocities vi . Each particle possesses information on its best position (Δnl;i ); thus, the best position of the entire particle set (Δng ) that defines vectors vl;i and vg;i . These vectors are used for the modification of particle velocities in each algorithm iteration. 80
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Modeling Dispersion of Birefringence
Unequal dispersion of the refractive index for ordinary and extraordinary rays may cause spectral variations of the birefringence. These variations may lead to false results of optimization algorithms. However, by proper modeling, the changes of birefringence can be included in the PSO algorithm. In such a case, a multidimensional optimization is required, where one of the dimensions is the value of the birefringence; others are the parameters of the assumed dispersion model. The PSO algorithm can be easily extended for this purpose by representing the positions and velocities of particles as vectors in a multidimensional space. In this paper, we have assumed a quadratic variation of birefringence as a function of a wavenumber: Δnk Δn0 ak − kl bk − kl 2 ;
(15)
where a and b are the model parameters, k 2π∕λ, kl 2π∕λl , and λl is the longest wavelength in the recorded spectrum. For simplicity, a normalized version of the model parameters was used in order to restrict the searching space of the algorithm: an aks − kl ;
(16)
bn bks − kl 2 ;
(17)
where ks 2π∕λs , and λs is the shortest wavelength in the recorded spectrum. The reason for using an and bn instead of a and b is that the limits of an and bn are equal to the maximal change of the birefringence in the entire spectrum introduced by the corresponding term in Eq. (15). The value of the birefringence in the presented model is given by
Fig. 6. Comparison of measured and preprocessed spectra (points) of wrapped retardation angle with the spectra fitted by the PSO algorithm (lines). Measured spectra were downsampled for reducing computation time.
Δnk Δn0 an
k − kl k − kl 2 bn : ks − kl ks − kl 2
(18)
The searching space was restricted by two conditions: an;min ≤ an ≤ an;max ;
(19)
jan j ≤ ra Δn0 ;
(20)
where an;min , an;max , and ra are algorithm parameters. Analogous conditions were set to values of bn. Sample results of the PSO algorithm applied to the retardation angle spectrum of the liquid crystal cell without driving voltage are presented in Fig. 7. The algorithm has been tested for three cases: without modeling dispersion, with the linear model of dispersion (0 ≤ an ≤ 0.015, bn 0, ra 0.05), and with the quadratic model (0 ≤ an ≤ 0.015, 0 ≤ bn ≤ 0.015, ra 0.05, rb 0.05). Even though the values of MSE are similar in each case, and no significant improvement of fitting is visible, the value of birefringent at 1550 nm in the case of modeling dispersion is much closer to the result of the reference measurement obtained with the procedure described in Section 2, which was equal to 0.296. Values of MSE for the cell without driving voltage and linear dispersion model (bn 0) are presented in Fig. 8. There are a few different points in the (Δn0 ; an ) plane corresponding with the local minima
Fig. 8. Values of the mean square error between the phase retardation spectrum of a nondriven LC cell and theoretical spectra. The theoretical spectra were calculated using the linear dispersion model with varying values of Δn0 and an . Different runs of the PSO algorithm lead to different local minima of MSE (marked with two black dots). Using the model without taking into account birefringence dispersion (an 0) leads to a different algorithm result (marked with a triangle and indicated by an arrow).
of MSE with similar values. This may cause false results of the PSO algorithm and require more thorough setting algorithm parameters than the model without dispersion. In the case of the quadratic dispersion model, the PSO algorithm needs to search a 3D space. Finding the global minimum in such a case is more difficult than in the case of the linear dispersion model. This may be the cause of the slightly higher value of MSE in the case of quadratic dispersion in the results from Fig. 7. For the set of optimal parameters, the quadratic model should provide a value of MSE not greater than that of the linear model. In the presented results, the higher value of MSE for the quadratic model shows that the algorithm has found only one of local minima. 4. Measurement of Liquid Crystal Cell Voltage Characteristics
Fig. 7. Result of modeling dispersion in the PSO algorithm. The fitted spectra (lines) are compared with the measurement results (points). Modeling dispersion provides the value of the birefringence at 1550 nm closer to the value obtained by the reference measurement (0.296) than the algorithm without the dispersion model.
The reference measurements of the cell birefringence as a function of driving voltage were performed by the rotating wave-plate polarimeter in the setup presented in Fig. 2(b). Two series of measurements have been taken for wavelengths 1290 and 1560 nm, as described in Section 2. The spectra of the retardation angle were measured in the setup from Fig. 2(a). Spectra I v k and I h k were recorded for 250 points spanned uniformly in the wavelength range from 1200 to 1700 nm. Prior to calculating γk, the spectra were digitally filtered by 64th order finite impulse response filter with the cut-off frequency equal to 0.1 of the sampling frequency and a sinx∕x type kernel. The first and the last 50 samples of the spectrum were discarded due to low intensity of the signal and effects of the nonstationary state in the digital filter output. Each spectrum was processed for different dispersion models: zero dispersion (an bn 0), linear dispersion (an > 0, bn 0), and quadratic dispersion (an > 0, bn > 0). Since birefringence in 1 January 2015 / Vol. 54, No. 1 / APPLIED OPTICS
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liquid crystal materials drops with increasing the wavelength [28,29], only positive values of coefficients an and bn were assumed. Comparison of the results of the reference measurements at 1290 and 1560 nm confirms this assumption (Figs. 9 and 10). Values of constants A1 , A2 , and A3 were experimentally selected as 2.5, 2.5, and 1.25, respectively. For each dispersion model, the optimization algorithm was run 150 times in order to calculate statistical parameters of the proposed method. The following parameters were calculated: 1. Relative standard deviation σ of the measurement at each voltage. 2. Relative bias error for the entire voltage characteristic:
b
M X N Δnm;n − Δnm;ref 1 X ; Δnm;ref MN m1 n1
(21)
where m is the index of the voltage, n is the index of a single run of the PSO algorithm, M is the number of voltage points, N is the number of runs of the PSO algorithm, Δnm;n is the result of nth run of the PSO algorithm for mth voltage, and Δnm;ref is the result of reference measurement for mth voltage.
Fig. 10. Birefringence of the liquid crystal cell as a function of driving voltage. The solid line corresponds with the reference measurement by polarimeter. Points and bars indicate mean value and standard deviation of PSO optimization.
Table 2.
an 0 ≤0.015 ≤0.015
ra
bn
rb
Np
K max
b (%)
0 0.06 0.06
0 0 ≤0.008
0 0 0.03
20 50 50
20 20 50
4.2 −4.1 −6.2
Table 3.
an 0 ≤0.015 ≤0.015
Results of PSO Algorithm for 1290 nm
Results of PSO Algorithm for 1560 nm
ra
bn
rb
Np
K max
b (%)
0 0.06 0.06
0 0 ≤0.008
0 0 0.03
20 50 50
20 20 50
7.3 −3.1 −6.7
The measurement results are presented in Figs. 9 and 10. Statistical parameters of the results are presented in Tables 2 and 3. It is important to stress that the reference measurements were performed by changing the cell driving voltage in small steps; thus, the retardation order was known for each measurement point. As an opposite, values of the cell effective birefringence using our new method were obtained without any a priori information on the retardation order. Fig. 9. Birefringence of the liquid crystal cell as a function of driving voltage. The solid line corresponds to the reference measurement by polarimeter. Points and bars indicate mean value and a standard deviation of PSO optimization. 82
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5. Discussion and Conclusion
The presented results (Tables 2 and 3) indicate that the proposed method allows us to measure values of
birefringence with 7% accuracy. Birefringence dispersion appears to be the main factor limiting the performance of the presented method. Proper modeling of dispersion may reduce the bias error of the measurement and, thus, improve its accuracy. However, the application of such models requires performing multidimensional optimization. This leads to greater values of standard deviation of optimization algorithm results. Moreover, results presented in Tables 2 and 3 show that a too complicated model of dispersion may actually increase the bias error. Plotting the values of MSE for the linear dispersion model (Fig. 8) and comparing fitted spectra for different dispersion models (Fig. 7) reveal another important problem. Very similar spectra can be obtained by different sets of parameters Δn0, an , and bn . This ambiguity cannot be resolved by any modification of the optimization algorithm. Without additional information about the dispersion of birefringence, it is unlikely to obtain measurement accuracy better than a few percent. Despite described difficulties, the proposed method allows us to determine the birefringence of objects, introducing retardation with orders higher than zero. Moreover, the approximate value of birefringence does not have to be known prior to the measurement. The presented research concerns only uniaxial birefringent media. However, one of possible future research topics is the extension of the proposed method on other types of optical anisotropy; e.g., the extension of an optical setup by achromatic phase retarders may allow characterizing the optical activity of the examined sample. Another important question is the alignment of sample main optical axes. The misalignment between sample and polarizers axes results in reduced amplitude of the spectrum oscillations. Perhaps this effect could be used to determine the orientation of the sample axis. Our future work will focus on the application of the described method in polarization-sensitive optical coherence tomography systems (PS-OCT). PS-OCT is an imaging technique that allows us to obtain spectrally resolved polarization images of scattering objects. The spectral analysis of the polarization state of light can greatly increase the number of PS-OCT applications. Another issue is increasing computation speed, which is crucial in the development of real-time imaging techniques. The applications of graphical processing units are promising for computational speedup. Such applications also will be a topic of further research. In conclusion, we have described the method for determination of birefringence of uniaxial birefringent media. The proposed method is based on spectral measurement of the phase retardation between ordinary and extraordinary light rays. The main advantage of this new method is its ability to determine the value of birefringence without knowing the phase
retardation order in a wide range of phase retardation values. The above features make it useful for measurement cases where the approximate value of birefringence is unknown, and it can change for more than one order of magnitude. Analysis of liquid crystal materials or photoelastic stress analysis of polymer materials are example applications of the proposed method. It also may useful in the field of biomedical optics for analysis of biological tissue birefringence. This research work has been supported by The National Centre for Research and Development (NCBiR), Poland, under grant no. LIDER/32/205/L3/11 and the DS program of Faculty of Electronics, Telecommunications and Informatics, Gdansk ´ University of Technology. References 1. T. Scharf, Polarized Light in Liquid Crystals and Polymers (Wiley, 2007). 2. H. B. Park, C. Saxer, J. F. de Boer, S. M. Srinivas, and J. S. Nelson, “In vivo burn depth determination by high-speed fiber-based polarization sensitive optical coherence tomography,” J. Biomed. Opt. 6, 474–479 (2001). 3. S. Shafi, J. D. B. Featherstone, D. Fried, J. Xie, T. M. Breunig, and C. Le, “Imaging caries lesions and lesion progression with polarization sensitive optical coherence tomography,” J. Biomed. Opt. 7, 618–627 (2002). 4. D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003). 5. S. J. Hwang, “Precise optical retardation measurement of nematic liquid crystal display using the phase-sensitive technique,” J. Disp. Technol. 1, 77–81 (2005). 6. M. H. Chiu, C. D. Chen, and D. C. Su, “Method for determining the fast axis and phase retardation of a wave plate,” J. Opt. Soc. Am. A 13, 1924–1929 (1996). 7. J. E. Hayden and S. D. Jacobs, “Automated spatially scanning ellipsometer for retardation measurements of transparent materials,” Appl. Opt. 32, 6256–6263 (1993). 8. D. B. Chenault and R. A. Chipman, “Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32, 3513–3519 (1993). 9. Z. Wang and E. Patterson, “Use of phase-stepping with demodulation and fuzzy sets for birefringence measurement,” Opt. Laser Eng. 22, 91–104 (1995). 10. B. Wang, “Linear birefringence measurement instrument using two photoelastic modulators,” Opt. Eng. 41, 981–987 (2002). 11. J.-F. Lin, “Measurement of linear birefringence using a rotating-wave-plate Stokes polarimeter,” Optik 121, 2144–2148 (2010). 12. J. F. De Boer and T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. 7, 359–371 (2002). 13. Z. Li, Q. Liu, L. Zheng, Y. Wang, and Z. Xu, “A new optical fiber birefringence measurement method based on polarization detection and wavelength scanning,” in Third International Conference on Digital Manufacturing and Automation (ICDMA) (IEEE, 2012), pp. 102–104. 14. M. Clerc, Particle Swarm Optimization, Vol. 243 (Iste, 2006). 15. R. Poli, J. Kennedy, and T. Blackwell, “Particle swarm optimization: an overview,” Swarm Intell. 1, 33–57 (2007). 16. J. Kennedy, “Particle swarm optimization,” in Encyclopedia of Machine Learning (Springer, 2010), pp. 760–766. 17. J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag. 52, 397–407 (2004). 18. M. Anastasiadou, S. B. Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” J. Eur. Opt. Soc. 2, 07018 (2007). 1 January 2015 / Vol. 54, No. 1 / APPLIED OPTICS
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19. M.-R. Antonelli, A. Pierangelo, T. Novikova, P. Validire, A. Benali, B. Gayet, and A. De Martino, “Mueller matrix imaging of human colon tissue for cancer diagnostics: how Monte Carlo modeling can help in the interpretation of experimental data,” Opt. Express 18, 10200–10208 (2010). 20. E. Nowinowski-Kruszelnicki, J. Kędzierski, Z. Raszewski, L. Jaroszewicz, R. Dąbrowski, M. Kojdecki, W. Piecek, P. Perkowski, K. Garbat, M. Olifierczuk, M. Sutkowski, K. Ogrodnik, P. Morawiak, and E. Miszczyk, “High birefringence liquid crystal mixtures for electro-optical devices,” Opt. Appl. 42, 167–180 (2012). 21. J. J. Moré, “The Levenberg-Marquardt algorithm: implementation and theory,” in Numerical Analysis (Springer, 1978), pp. 105–116. 22. M. Kraszewski and R. Bogdanowicz, “Laser reflectance interferometry system with a 405 nm laser diode for in situ measurements of cvd diamond thickness,” Metrol. Meas. Syst. 20, 543–554 (2013).
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23. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007). 24. R. Bogdanowicz, J. Ryl, K. Darowicki, and B. Kosmowski, “Ellipsometric study of oxide formation on Cu electrode in 0.1 m NaOH,” J. Solid State Electrochem. 13, 1639–1644 (2009). 25. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965). 26. M. Mitchell, An Introduction to Genetic Algorithms (MIT, 1998). 27. L. J. Fogel, A. J. Owens, and M. J. Walsh, Artificial Intelligence Through Simulated Evolution (Wiley, 1966). 28. J. Li, S.-T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (2005). 29. J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive indices of liquid crystals for display applications,” J. Disp. Technol. 1, 51–61 (2005).
Measurement of birefringence is useful for the examination of technical and biological objects. One of the main problems, however, is that the polarization state of light in birefringent media changes periodically. Without knowledge of the period number, the birefringence of a given medium cannot be reliably determined. We propose to analyze the spectrum of light in order to determine the birefringence. We use a particle swarm optimization algorithm for an automatic processing spectra of light transmitted through birefringent material for two orthogonal states of polarization. We have tested the described algorithm on a liquid crystal cell with varying effective birefringence. The proposed method can be used for the measurement of uniaxial positive birefringence without knowing the number of retardation periods or an approximate value of the measurement result. This fact makes the proposed method useful for automatic measurements, when hundreds or thousands of spectra need to be analyzed. © 2014 Optical Society of America OCIS codes: (120.5410) Polarimetry; (160.3710) Liquid crystals. http://dx.doi.org/10.1364/AO.54.000076
1. Introduction
Birefringence is an optical property of many materials. The simplest type of this property, but with a large number of practical applications, is uniaxial birefringence, which is the existence of two different refractive indices for two orthogonal states of linear light polarization. Since birefringence depends on structural properties of the material on a molecular level, it can provide useful information about many classes of objects. A major industrial application of this phenomenon is photoelastic stress analysis, which allows us to visualize stress in polymer materials [1]. In biomedical optics, biological tissue birefringence allows us to examine burn depth [2] or caries lesions [3]. The examination of birefringence is also important for evaluation of various optical elements such as liquid crystal devices or optical fibers. Research presented in this paper was focused on
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analysis of uniaxial birefringent media, particularly on the measuring difference between refractive indices corresponding to medium main optical axes. Such measurements are a key element of the abovementioned applications. Birefringence measurement is based on the measurement of the retardation angle, which is defined as the phase delay between ordinary and extraordinary light rays. The retardation angle can be determined by the measurement of the polarization state of light, e.g., using the setup based on a linear polarizer and quarter-wave plate, Babinet–Soleil compensator, or rotating retarder [4]. Precise retardation angle measurements are also possible using various interferometric techniques [5,6], ellipsometry [7], or spectropolarimetry [8]. During propagation in a birefringent material, the polarization state of light changes periodically. Therefore, it is useful to express the retardation angle as Γ Γ0 2mπ; where Γ0 h0; 2πi and m is the retardation order.
(1)
The polarization state of light depends on Γ0 , so if the retardation order is unknown, the measurement of birefringence is ambiguous [9–11]. Moreover, many measurement methods rely on the measurement of cos Γ [4]; therefore, Γ0 can be determined only in the range from 0 to π. Other methods, such as polarization-sensitive optical coherence tomography (PS-OCT), measures the value of j tan Γj based on the backscattered light from the sample [12], which limits the measurement range to h0; π∕2i. This measurement range is insufficient for many applications such as the analysis of stress in polymer foils, the examination of liquid crystal cells and materials, or testing optical fibers. However, this difficulty can be overcome by analyzing the retardation angle as a function of a wavenumber. The retardation angle of the homogeneous medium depends on the wavenumber and birefringence according to the equation: Γ kdΔn;
(2)
where k 2π∕λ is the wavenumber, λ is the wavelength in vacuum, Δn is the birefringence, and d is the thickness of the birefringent material. Processing the spectrum of the polarized light can provide information on the birefringence without the necessity of knowing the retardation order. Such an analysis has recently been demonstrated for the examination of optical fibers [13]. In this application, it is possible to obtain high orders of retardation by using fibers with a sufficient length. However, the examination of polymer foils or liquid crystal materials require a processing method that would work well for both high and low retardation order. In this paper, we present the application of a particle swarm optimization (PSO) algorithm [14–17] for processing retardation angle spectra. We have tested its performance on liquid crystal cells with voltagecontrolled birefringence. Our experiments show that PSO can be applied to the automation of birefringence measurements, which makes it valuable for processing a large number of spectra, as in automatic inspection or spectral polarization imaging [18,19]. 2. Problem Statement and Measurement Principles
In this contribution, the issue of direct determination of birefringence has been addressed. For this purpose, all considerations have been limited to the uniaxial birefringent medium having positive birefringence (Δn). The other cases are not in the topic of this contribution and will be investigated during further research. The retardation Γ introduced by a given medium can be expressed as a function of the light wavenumber (Fig. 1, upper plot). However, the only phenomenon that can be measured is the polarization state of the light guided through the birefringent medium. The state of monochromatic light polarization depends on the Γ0 described in Eq. (1), which varies from 0 to 2π (Fig. 1, central plot). Moreover, the
Fig. 1. Example of retardation changes over the light spectrum for birefringence Δn equal to 0.3, light wavelength in the range from 1.25 to 1.65 μm, and the thickness of the birefringent medium d equal to 55 μm. Γ, the exact value of the retardation introduced by the medium; Γ0 , the factor that has the most significant influence on the polarization state of light; γ, the measurable value of retardation angle.
polarization state of the light is determined by using the trigonometric relations between the measured quantities, which were mentioned in the introduction. Therefore, the value of retardation, which can be obtained directly, is measurable in the range from 0 to π (Fig. 1, bottom plot). The problem has been presented in Fig. 1; however, the details of the whole analysis based on particular examples are explained in the following part of the paper. The birefringence of liquid crystal cells were measured with the setup presented in Fig. 2(a). The radiation from the broadband, supercontinuum source (Menlo TB-1550) with the light spectrum spanning from 1200 to 1700 nm was collimated and transmitted through the linear polarizer L1. Then it was transmitted through the homogenous (HG), planar liquid crystal cell LC with 50 μm thickness [20]. The optical axis of the cell was oriented under 45° from the axis of polarizer L1. After that, light was transmitted through the polarizer L2 into the optical spectrum analyzer (Anritsu, MS9740A). The liquid crystal cell was driven by alternate voltage (square wave with 50% duty cycle) with a frequency of 100 Hz and the root mean square (rms) varying from 0 to 5 V. For this frequency, the electrical anistropy of the used liquid crystal material was positive (Δε > 0). For each value of the voltage, two spectra were collected: one with the optical axis of L2 in parallel with the axis of L1 and second with the axis of L2 orthogonal to the axis of L1. The liquid crystal material presents substantial birefringence on the order of 0.3. Driving the cell with the voltage 1 January 2015 / Vol. 54, No. 1 / APPLIED OPTICS
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changes the orientation of liquid crystal molecules, which results in the drop of effective birefringence of the liquid crystal layer. The reference measurement of birefringence has been performed using the setup presented in Fig. 2(b). Two superluminescent diodes with peak wavelengths 1560 nm (Superlum, Broadlighter S1550) and 1290 nm (Superlum, Broadlighter S1300) and spectral width (FWHM) 50 and 45 nm, respectively, have been used as a light source. The polarizer L2 and the spectrum analyzer have been replaced by a polarimeter (Thorlabs, PAX5710). Such a setup allows us to determine the retardation angle of the liquid crystal cell at the single wavelength, but does not allow us to determine the order of retardation directly. To overcome this problem, the cell was initially driven by the voltage with rms equal to 5 V. For this voltage, the first order of retardation was assumed. This assumption agrees with the spectral measurement of the cell birefringence (Fig. 6). After that, the value of the applied voltages was being gradually decreased with steps, allowing us to obtain four data points for each retardation order. This allowed us to unwrap measured values of the retardation angle Γ0 and calculate the birefringence of the cell. Jones vectors of light coming to the spectrum analyzer for two orientations of the polarizer L2 are given by p I 0 k 1 1 e−iΓk J⃗ v k ; 1 2
(3)
p I 0 k 1 1 − e−iΓk ; J⃗ h k −1 2
(4)
and
where k 2π∕λ is the wavenumber, J v is the Jones vector in case of polarizers L1 and L2 aligned in parallel, J h is the Jones vector in case of polarizers L1
and L2 crossed, Γ is the phase retardation introduced by the liquid crystal cell, and I 0 is the mean intensity of incident light. Corresponding intensities are given by I v k J⃗ v k · J⃗ v k I 0 k1 cos Γk;
(5)
I h k J⃗ h k · J⃗ h k I 0 k1 − cos Γk:
(6)
Cosine of the retardation angle Γ can be calculated based on the measurements of I v and I h as cos Γk
I v k − I h k ; I v k I h k
(7)
which leads to γk arccos
I v k − I h k ; I v k I h k
(8)
where γk is the retardation angle Γk wrapped in the range from 0 to π: γk
Γ0 2π − Γ0
if Γ0 < π; if Γ0 ≥ π;
(9)
where Γ0 is defined by Eq. (1). Further in this paper, γk denotes function based on the measurement of I v k and I h k and calculated with Eq. (8). We use additional symbol γ t k as a theoretical value of function γk. While γk is the result of the experiment, γ t k is calculated assuming certain values of thickness and birefringence of the liquid crystal cell and can be calculated as γ t k arccoscos kdΔn;
(10)
where d and Δn are the thickness and effective birefringence of the cell, respectively. 3. Numerical Analysis of Birefringence Spectra A. Problem of Varying Order of Retardation
Fig. 2. Experimental setup for birefringence measurement using (a) the proposed spectral method and setup for (b) the reference measurement. L1 and L2, linear polarizers; LC, liquid crystal cell. 78
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The examples of spectra γk based on the measurement of I v k and I h k and calculated with Eq. (8) for the cell driven with rms voltages 0 and 2 V are presented in Fig. 3. The periodicity of γk is visible in both spectra; however, each of them requires a different approach for determination of Δn. If at least a few different orders of retardation are present in the spectrum, the distance between peaks in γk can be used to calculate the birefringence. It is also possible to use the base frequency of γk calculated with standard tools such as Fourier transform. More orders of the retardation provide higher accuracy of this processing method. When the effective birefringence is small, the spectrum γk may contain only one order of the retardation. In such a case, the birefringence can
Fig. 3. Measured spectra of wrapped retardation angle for the liquid crystal cell driven by different voltages.
be calculated based on the slope of γk. However, due to wrapping γk in the range from 0 to π, it is difficult to automatically find a proper part of the spectrum for the slope calculation. The described methods cannot be reliably applied for low and high effective birefringence of the cell. It could be possible to choose one of two methods depending on the value of birefringence, but making such a choice automatically could raise new difficulties. Instead, an optimization algorithm can be used for processing spectra regardless of the value of the birefringence. B.
Application of Optimization Algorithms
Low and high birefringence of a cell can be reliably determined by fitting nonlinear function γ t k to measurement result γk using optimization algorithms. In this approach, the value of Δn is searched within a specified range at Δn space based on certain optimization criteria. The most common criterium is the minimization of a mean square error (MSE): sP M−1 2 m0 γ t km − γkm MSE PM−1 ; 2 m0 γkm
(11)
where km is the wavenumber of mth point in the spectrum. Simulated MSE for different values of real and fitted birefringence are presented in Fig. 4. Presented results show that the cell’s birefringence can be determined by finding such a value of Δn for which the value of MSE takes its global minimum. The simplest optimization methods are based on gradient descent algorithms. Example of such a method is the Levenberg–Marquadt algorithm [21,22], which is widely used in spectroscopic ellipsometry [22–24]. However, gradient descent methods are susceptible to local minima of MSE. They provide an accurate solution only if the initial point of the algorithm is close to the measured value of Δn. The problem of local minima is important when values of Δn may vary within several orders of magnitude. In such a case, an approach different than the gradient descent method is needed. Examples of alternative optimization algorithms are the simplex method [25], genetic algorithms [26], simulated annealing [27], and PSO [14–17]. In this study, we
Fig. 4. Mean square errors between two phase retardation spectra for different values of sample birefringence. Matching the real and fitted birefringence by MSE minimization with gradient descent algorithms is likely to fail due to many local minima of MSE. Ideally, the optimization algorithm should find a point on the line marked with black points regardless of the initial guess of the birefringence.
have chosen the PSO because of its simplicity and possibility of easy implementation of parallel computing. C.
Particle Swarm Optimization Algorithm
PSO is an optimization method similar to genetic algorithms, which can search a broad region of parameter space for an optimal solution [14–16]. This makes it much less sensitive to the local minima of MSE than gradient descent methods. The PSO relies on searching the best value of Δn by scanning an entire search space with a set of “particles.” Each particle is moves through search space by changing the corresponding value of Δn by a value denoted as “velocity.” The initial velocities of all particles are set to 0 and are modified in each iteration of the algorithm. The velocity of each particle in the next iteration is the weighted sum of its previous velocity, direction to the best solution already found by this particle, and direction to best solution found by the entire particles set. The algorithm is finished after finding Δn with the value of MSE below a specified threshold or after a maximal number of iterations. Symbols used in the description of the presented algorithm are listed in Table 1. The algorithm, illustrated in Fig. 5, consists of following steps: 1. Set the initial positions of particles randomly in the search space: Δni ∈ hΔnmin ; Δnmax i. 2. Calculate the values of MSE for each particle and set them as the initial values of MSEl;i. 3. Find the best value of birefringence Δngl and the error MSEg from the entire set of particles. 4. Draw uniform random values a1 ∈ h0; A1 i, a2 ∈ h0; A2 i, and a3 ∈ h0; A3 i. 5. Update particle velocities according to the equations: vi ←a1 vi a2 vl;i a3 vg;i ; 1 January 2015 / Vol. 54, No. 1 / APPLIED OPTICS
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Table 1.
List of Symbols Used for Description of the PSO Algorithm
Parameter
Symbol
Number of particles Measured spectrum Initial birefringence range Iterations limit Mean square error threshold Velocity weight limits Birefringence of ith particle Velocity of ith particles Best birefringences found by ith particle Smallest error obtained by ith particle Best birefringence found by all particles Smallest error obtained by all particles
of measured and fitted spectra of retardation angle are presented in Fig. 6. D.
N ~ Γk Δnmin , Δnmax K max MSEmax A1 , A2 , A3 Δni vi Δnl;i MSEl;i Δng MSEg
vl;i ←Δnl;i − Δni ;
(13)
vg;i ←Δng − Δni :
(14)
6. Update the particle positions Δni ←Δni vi. 7. If any particle is beyond the limits of searching space, Δni > Δnmax or Δni < Δnmin, it obtains a new random position and velocity. 8. Calculate errors MSEi associated with the position of each particle. 9. Update the values of the best positions Δnl;i , Δng , and the errors MSEl;i and MSEg . 10. If MSEg < MSEmax or the maximal number of iterations was exceeded, finish the algorithm; otherwise, return to Step 5. 11. After the algorithm is finished, the value Δng is taken as the measured value of birefringence. To test the PSO algorithm, measurement of the birefringent layer with thickness 50 μm was made. The algorithm used 20 particles and velocity coefficients: A1 1, A2 2.5, and A3 1.5. The error and number of iterations were limited to MSEmax 0.02 and K max 50, respectively. The exact value of birefringence varied from 0.001 to 0.5. The algorithm was run 30 times for each value of the birefringence. For each case, the birefringence value was determined with accuracy better than 0.0021. Examples
Fig. 5. Illustration of the PSO algorithm in the case of two particles. In each algorithm iteration, particle positions (Δni ) are modified by their velocities vi . Each particle possesses information on its best position (Δnl;i ); thus, the best position of the entire particle set (Δng ) that defines vectors vl;i and vg;i . These vectors are used for the modification of particle velocities in each algorithm iteration. 80
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Modeling Dispersion of Birefringence
Unequal dispersion of the refractive index for ordinary and extraordinary rays may cause spectral variations of the birefringence. These variations may lead to false results of optimization algorithms. However, by proper modeling, the changes of birefringence can be included in the PSO algorithm. In such a case, a multidimensional optimization is required, where one of the dimensions is the value of the birefringence; others are the parameters of the assumed dispersion model. The PSO algorithm can be easily extended for this purpose by representing the positions and velocities of particles as vectors in a multidimensional space. In this paper, we have assumed a quadratic variation of birefringence as a function of a wavenumber: Δnk Δn0 ak − kl bk − kl 2 ;
(15)
where a and b are the model parameters, k 2π∕λ, kl 2π∕λl , and λl is the longest wavelength in the recorded spectrum. For simplicity, a normalized version of the model parameters was used in order to restrict the searching space of the algorithm: an aks − kl ;
(16)
bn bks − kl 2 ;
(17)
where ks 2π∕λs , and λs is the shortest wavelength in the recorded spectrum. The reason for using an and bn instead of a and b is that the limits of an and bn are equal to the maximal change of the birefringence in the entire spectrum introduced by the corresponding term in Eq. (15). The value of the birefringence in the presented model is given by
Fig. 6. Comparison of measured and preprocessed spectra (points) of wrapped retardation angle with the spectra fitted by the PSO algorithm (lines). Measured spectra were downsampled for reducing computation time.
Δnk Δn0 an
k − kl k − kl 2 bn : ks − kl ks − kl 2
(18)
The searching space was restricted by two conditions: an;min ≤ an ≤ an;max ;
(19)
jan j ≤ ra Δn0 ;
(20)
where an;min , an;max , and ra are algorithm parameters. Analogous conditions were set to values of bn. Sample results of the PSO algorithm applied to the retardation angle spectrum of the liquid crystal cell without driving voltage are presented in Fig. 7. The algorithm has been tested for three cases: without modeling dispersion, with the linear model of dispersion (0 ≤ an ≤ 0.015, bn 0, ra 0.05), and with the quadratic model (0 ≤ an ≤ 0.015, 0 ≤ bn ≤ 0.015, ra 0.05, rb 0.05). Even though the values of MSE are similar in each case, and no significant improvement of fitting is visible, the value of birefringent at 1550 nm in the case of modeling dispersion is much closer to the result of the reference measurement obtained with the procedure described in Section 2, which was equal to 0.296. Values of MSE for the cell without driving voltage and linear dispersion model (bn 0) are presented in Fig. 8. There are a few different points in the (Δn0 ; an ) plane corresponding with the local minima
Fig. 8. Values of the mean square error between the phase retardation spectrum of a nondriven LC cell and theoretical spectra. The theoretical spectra were calculated using the linear dispersion model with varying values of Δn0 and an . Different runs of the PSO algorithm lead to different local minima of MSE (marked with two black dots). Using the model without taking into account birefringence dispersion (an 0) leads to a different algorithm result (marked with a triangle and indicated by an arrow).
of MSE with similar values. This may cause false results of the PSO algorithm and require more thorough setting algorithm parameters than the model without dispersion. In the case of the quadratic dispersion model, the PSO algorithm needs to search a 3D space. Finding the global minimum in such a case is more difficult than in the case of the linear dispersion model. This may be the cause of the slightly higher value of MSE in the case of quadratic dispersion in the results from Fig. 7. For the set of optimal parameters, the quadratic model should provide a value of MSE not greater than that of the linear model. In the presented results, the higher value of MSE for the quadratic model shows that the algorithm has found only one of local minima. 4. Measurement of Liquid Crystal Cell Voltage Characteristics
Fig. 7. Result of modeling dispersion in the PSO algorithm. The fitted spectra (lines) are compared with the measurement results (points). Modeling dispersion provides the value of the birefringence at 1550 nm closer to the value obtained by the reference measurement (0.296) than the algorithm without the dispersion model.
The reference measurements of the cell birefringence as a function of driving voltage were performed by the rotating wave-plate polarimeter in the setup presented in Fig. 2(b). Two series of measurements have been taken for wavelengths 1290 and 1560 nm, as described in Section 2. The spectra of the retardation angle were measured in the setup from Fig. 2(a). Spectra I v k and I h k were recorded for 250 points spanned uniformly in the wavelength range from 1200 to 1700 nm. Prior to calculating γk, the spectra were digitally filtered by 64th order finite impulse response filter with the cut-off frequency equal to 0.1 of the sampling frequency and a sinx∕x type kernel. The first and the last 50 samples of the spectrum were discarded due to low intensity of the signal and effects of the nonstationary state in the digital filter output. Each spectrum was processed for different dispersion models: zero dispersion (an bn 0), linear dispersion (an > 0, bn 0), and quadratic dispersion (an > 0, bn > 0). Since birefringence in 1 January 2015 / Vol. 54, No. 1 / APPLIED OPTICS
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liquid crystal materials drops with increasing the wavelength [28,29], only positive values of coefficients an and bn were assumed. Comparison of the results of the reference measurements at 1290 and 1560 nm confirms this assumption (Figs. 9 and 10). Values of constants A1 , A2 , and A3 were experimentally selected as 2.5, 2.5, and 1.25, respectively. For each dispersion model, the optimization algorithm was run 150 times in order to calculate statistical parameters of the proposed method. The following parameters were calculated: 1. Relative standard deviation σ of the measurement at each voltage. 2. Relative bias error for the entire voltage characteristic:
b
M X N Δnm;n − Δnm;ref 1 X ; Δnm;ref MN m1 n1
(21)
where m is the index of the voltage, n is the index of a single run of the PSO algorithm, M is the number of voltage points, N is the number of runs of the PSO algorithm, Δnm;n is the result of nth run of the PSO algorithm for mth voltage, and Δnm;ref is the result of reference measurement for mth voltage.
Fig. 10. Birefringence of the liquid crystal cell as a function of driving voltage. The solid line corresponds with the reference measurement by polarimeter. Points and bars indicate mean value and standard deviation of PSO optimization.
Table 2.
an 0 ≤0.015 ≤0.015
ra
bn
rb
Np
K max
b (%)
0 0.06 0.06
0 0 ≤0.008
0 0 0.03
20 50 50
20 20 50
4.2 −4.1 −6.2
Table 3.
an 0 ≤0.015 ≤0.015
Results of PSO Algorithm for 1290 nm
Results of PSO Algorithm for 1560 nm
ra
bn
rb
Np
K max
b (%)
0 0.06 0.06
0 0 ≤0.008
0 0 0.03
20 50 50
20 20 50
7.3 −3.1 −6.7
The measurement results are presented in Figs. 9 and 10. Statistical parameters of the results are presented in Tables 2 and 3. It is important to stress that the reference measurements were performed by changing the cell driving voltage in small steps; thus, the retardation order was known for each measurement point. As an opposite, values of the cell effective birefringence using our new method were obtained without any a priori information on the retardation order. Fig. 9. Birefringence of the liquid crystal cell as a function of driving voltage. The solid line corresponds to the reference measurement by polarimeter. Points and bars indicate mean value and a standard deviation of PSO optimization. 82
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5. Discussion and Conclusion
The presented results (Tables 2 and 3) indicate that the proposed method allows us to measure values of
birefringence with 7% accuracy. Birefringence dispersion appears to be the main factor limiting the performance of the presented method. Proper modeling of dispersion may reduce the bias error of the measurement and, thus, improve its accuracy. However, the application of such models requires performing multidimensional optimization. This leads to greater values of standard deviation of optimization algorithm results. Moreover, results presented in Tables 2 and 3 show that a too complicated model of dispersion may actually increase the bias error. Plotting the values of MSE for the linear dispersion model (Fig. 8) and comparing fitted spectra for different dispersion models (Fig. 7) reveal another important problem. Very similar spectra can be obtained by different sets of parameters Δn0, an , and bn . This ambiguity cannot be resolved by any modification of the optimization algorithm. Without additional information about the dispersion of birefringence, it is unlikely to obtain measurement accuracy better than a few percent. Despite described difficulties, the proposed method allows us to determine the birefringence of objects, introducing retardation with orders higher than zero. Moreover, the approximate value of birefringence does not have to be known prior to the measurement. The presented research concerns only uniaxial birefringent media. However, one of possible future research topics is the extension of the proposed method on other types of optical anisotropy; e.g., the extension of an optical setup by achromatic phase retarders may allow characterizing the optical activity of the examined sample. Another important question is the alignment of sample main optical axes. The misalignment between sample and polarizers axes results in reduced amplitude of the spectrum oscillations. Perhaps this effect could be used to determine the orientation of the sample axis. Our future work will focus on the application of the described method in polarization-sensitive optical coherence tomography systems (PS-OCT). PS-OCT is an imaging technique that allows us to obtain spectrally resolved polarization images of scattering objects. The spectral analysis of the polarization state of light can greatly increase the number of PS-OCT applications. Another issue is increasing computation speed, which is crucial in the development of real-time imaging techniques. The applications of graphical processing units are promising for computational speedup. Such applications also will be a topic of further research. In conclusion, we have described the method for determination of birefringence of uniaxial birefringent media. The proposed method is based on spectral measurement of the phase retardation between ordinary and extraordinary light rays. The main advantage of this new method is its ability to determine the value of birefringence without knowing the phase
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