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Predicting wavelength dependency of optical modulation of twisted nematic liquid crystal display in the visible range
Optik 126 (2015) 917–922
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Predicting wavelength dependency of optical modulation of twisted nematic liquid crystal display in the visible range Ghaith Makey, Moustafa Sayem El-Daher ∗ , Kanj Al-Shufi Higher Institute for Laser Research and Applications, Damascus University, Airport Highway, Damascus, Syria
a r t i c l e
i n f o
Article history: Received 2 February 2014 Accepted 12 February 2015 Keywords: Spatial light modulators Optoelectronics Simulation Holography
a b s t r a c t One of the main problems in liquid crystal spatial light modulators (LC-SLM) is the prediction of their optical modulation behavior for different wavelengths; this coupled phase/amplitude modulation can be hard to measure for specific wavelengths and setups, yet it is required to be included in the study of a number of SLM’s applications. This work proposes a simulation model to predict the phase/amplitude modulation for different twisted nematic liquid crystal displays (TN-LCD) optical setups, this model incorporates the effects of wavelength in addition to other parameters which already have been studied before like twist angle as internal parameter and polarizers angles, we verified the accuracy of our model compared to results of previously published work. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Extensive work have been dedicated to the topic of twist nematic liquid crystal displays (TN-LCD) modeling and simulation [1,2], yet the individual papers in this field could not answer clearly a key question required for many users of TN-LCD when used as SLM, namely at an arbitrary wavelength what is the coupled amplitude-phase modulation for each grayscale value in a pixel? This question is quiet important to be answered by a simulation model specifically when the measurements of the associated modulation is hard to carry out (e.g. when working on holography’s experiment with amplitude mostly setup and need to predict the values of coupled phase modulation). By keeping the previous question in mind, let us review quickly related work. In 2000 Marquez et al. [3] presented distinctive characterization of edge effects in TN-LCDs; their analysis was supported by measurements taken at 4 wavelengths. In 2001 Marques et al. [4] used the results of [3] to extend their work and gave a quantitative prediction of the modulation behavior of TN-LCD, again the measurements were limited to 4 wavelengths and their analysis requires specific measurement for each wavelength at which they want to predict TN-LCD modulation behavior.
∗ Corresponding author. Tel.: +963 1133920683. E-mail addresses: [email protected] (G. Makey), [email protected] (M. Sayem El-Daher), kalshoufi@gmail.com (K. Al-Shufi). http://dx.doi.org/10.1016/j.ijleo.2015.02.072 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
In 2005 Kim and Lee [5] measured the parameters of TN-LCD with no ambiguity by fitting the theoretical predictions to the intensity of transmittance, then they predicted the phase modulation and verified the results experimentally; their work was limited to the study of one wavelength. In the same year, Wan and He [6] proposed new model for the director distribution of TN-LC cell but again the work did not include a study of wavelength dependency. Results which describe the optical modulation properties of TNLCD without the need to know LCD’s internal structure parameters do exist [7–9]; however these studies did not include a study of wavelength dependency. In this paper we will use the equations derived and listed very neatly in Refs. [3,5], and we will incorporate extended Cauchy equations [10] to propose one unified mathematical model that can be used to predicted phase and amplitude modulation of TN-LCD at any visible wavelength and without the need for any measurement at this wavelength. We will also use the experimental results of [3,4,11–13] to verify our simulation results.
2. Mathematical model In this section we will discuss the mathematical models for two main TN-LCD SLM setups: the first one is a system composed of SLM sandwiched between two polarizers (System A) and the second is a system composed of polarizer-SLM-quarter wave platepolarizer (System B). Fig. 1 shows the differences between two setups.
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G. Makey et al. / Optik 126 (2015) 917–922
g = cos(2ˇedge ) + ×
ˇcenter sin() cos(E + S )
(5)
h = − 1− ×
ˇcenter
cot() sin(2ˇedge )
ˇcenter tan(˛) sin(2ˇedge ) ˛
˛ sin() cos(˛) + cos(2ˇedge ) cos() sin(˛)
j = cos(2ˇedge ) +
ˇcenter
(6)
cot() sin(2ˇedge )
ˇcenter × sin() sin(E + S ) f 2 + g 2 + h2 + j2 = 1
(7)
(8)
S , E are angles of the extraordinary axis of the liquid-crystal molecules at the entrance and exit surfaces of TN-LC cell, in respect. ˛ is the twist angle: ˛ = E − S
(9)
is given by: =
Fig. 1. Different optical setups for TN-LCD as SLM, (a) System A, (b) System B [5].
2 ˛2 + ˇcenter
ˇ, ˇcenter , and ˇedge are very important parameters which include wavelength and voltage dependency (and even temperature dependency which is out of our focus in this paper); ˇ is the birefringence of the TN-LCD and it reaches its maximum value in the case of voltage absence: ˇmax =
2.1. Mathematical model for System A We will start directly with the following equations [5] for System A: T˜ = f cos( +
A
−
P ) + h sin(
P ) − ij sin(
A+
A
−
P ) − ig cos(
A
P)
(1)
ı = ˇ − arg(T˜ )
(2)
t = |c T˜ |2
(3)
where T˜ is the normalized transmission coefficient, ı is the phase delay which represents phase modulation, and t can be used to predict analytically intensity transmittance which represents amplitude modulation. P are the angles of the analyzer and the polarizer, respecA, tively, (refer to Fig. 1a). c is a constant represents the loss factor of TN-LC cell (which can be considered unity by normalizing the transmission or can be measured in similar manner to [5]). f, g, h, and j are Jones parameters for TN-LC cell with edge effect taken in account (this effect arises when applying voltage to the cell):
f =
1−
˛ ˇcenter cot(˛) sin(2ˇedge ) sin() sin(˛) ˛
+ cos(2ˇedge ) cos() cos(˛)
(4)
(10)
d(ne − no ) dnmax =
(11)
where d is the thickness of the cell (the liquid crystal part only). is the wavelength of the incident light in vacuum. ne , no are extraordinary and normal refractive indices, respectively, and nmax is refractive indices difference (“max” subscript stands for n value when no voltage is present). From the first look at Eq. (11) one can deduce that ˇmax is changing linearly with 1/ but that is not right experimentally [3,13] because nmax is changing with as well. We will return to this point after completing the definitions of ˇcenter , ˇedge . When applying voltage greater than the threshold of TN-LC cell, edges effect appears [3] and the extraordinary refractive index changes [14]; in this case ˇ will equal to: ˇ = ˇcenter + 2ˇedge
(12)
where ˇedge is the birefringence at each edge area (“edge” is also referred as “surface”). ˇcenter is the birefringence of the center area (“center” is also referred as “bulk”) whose molecule’s rotation is not constrained by edge effect; ˇedge =
dedge nmax
(13)
where dedge is the depth of the area affected by each edge. ˇcenter =
(d − 2dedge )n(V )
(14)
where n(V) emphasizes the dependency of n on the applied voltage to each TN-LC cell. Now let us take a look at the equations from a practical point of view, in the case of the absence of applied voltage ˇedge will equal to zero. And if ˛ and s are known, we only need to know the value of ˇmax to predict amplitude and phase modulation for each
G. Makey et al. / Optik 126 (2015) 917–922
( A , P ) setup. ˛ and s are constant for each TN-LCD model and they can be obtained from references like [3,6,11], or by measurements using the same methods used on those papers. Normally, ˇ can be taken from reported results in papers, calculated theoretically or from measurements, calculating ˇmax analytically from t or ı is very hard, so in order to solve this we wrote a function in Matlab that contains all the required equations to calculate t and ı, the input of this function includes all the required parameters which includes ˇmax as an array whose elements are ranging from a minimum to maximum possible values for ˇmax with a step of 0.01 rad, then if t and ı vs. ˇmax curves are plotted one can work inversely using MATLAB data cursor we can determine and verify ˇmax value for each t and ı measurements. However our intention in the paper is to predict ˇmax value at any visible wavelength without the need to make any measurements at this wavelength. To calculate ˇmax using our proposed scheme we need the following values. (in vacuum) which is certainly known, d which is constant but generally unknown, and nmax which change with itself. For nmax , which is more accurately written as nmax (), we used Extended Cauchy equations in the calculation [10]: Be,o Ce,o ne,o ∼ = Ae,o + 2 + 4
(15)
where Ae,o , Be,o , and ce,o are the three Cauchy coefficients. As we are not interested in the specific values of ne and no we will rewrite (15) as: n ∼ = A +
B C + 4 2
(16)
Since d is most likely to be unknown we will insert the constant d to (16) so we will have: Bd Cd dn ∼ = Ad + 2 + 4
(17)
Ad , Bd , and cd are three coefficients that are constant for a specific TN-LCD model when no voltage is applied. If we have three values for ˇmax at three different wavelengths one can predict the values of ˇmax at any other wavelength in visible range (we will not extend this statement to IR or UV range because we do not have the required measurements to justify this extension, nor we are sure of the behavior of different optical elements in the optical setup when operating in UV–visible or visible-IR ranges). But if we take a closer look at Eq. (17) we will notice that it represent quadratic equation with respect to (1/2 ), and if the three values of , which can be used to calculate the three coefficients, are close to each other they may positioned on the linear part of the quadratic curve and the determined values of the coefficients may not lead to correct predication of n value for a wavelength that is far from the range of the original wavelengths. This is an inherent problem with Cauchy equations and can be solved either by taking the minimum and maximum values of the three wavelengths to be far from each other (e.g. one at red and the second at blue wavelength), or one can use the result which we found by analyzing previous measurements [3,11–13]: we found that dn change linearly with (1/4 ) which means that Eq. (17) can be approximated to: dn ∼ = Ad +
Cd 4
(18)
Only two measurements of ˇmax at two different wavelengths are needed to predict any value of ˇmax at any other wavelength, this assumption is purely empirical (however, even the physical origin of the well-used extended Cauchy equation is not clear [10]). When using Eq. (18) it is important to pay attention that the associated ˇmax of the two wavelengths which are chosen to be used to determine the coefficients in Eq. (18) should be calculated based on accurate measurements (as in Ref. [3]).
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In short, in the absence of voltage and for any TN-LCD model we need only the values of ˛, s and three or two values of ˇmax at three or two different wavelengths to predict the values of t and ı for any wavelength in the visible range at each ( A , P ) setup ( A and P combinations determine the working mode of TN-LCD as an SLM; it can be set to phase mostly modulation or amplitude mostly modulation for example); verification of our results will be presented in Section 3. If an effective voltage is applied to TN-LC cell, the study can be more complicated. But thanks to Marquez et al. work [3], this study can be much easier by the use of the following relation: ˇedge (V, ) ˇmax () ˇcenter (V, ) = = ˇmax (0 ) ˇcenter (V, 0 ) ˇedge (V, 0 )
(19)
where is an arbitrary wavelength, 0 is a single wavelength at which ˇcenter (V) and ˇedge (V) are calculated based on measurements taken for each voltage applied on TN-LC cell (the voltage is quantized in levels which are referred as grayscale levels of a TNLC cell). According to our assumption, ˇmax () do not need to be measured but can be calculated using (17) or (18). So to fully predict phase and amplitude modulation for TNLCD under voltage operation, and in addition to the requirements needed for no-voltage case, we need to know ˇcenter and ˇedge (or just the total ˇ) values for each grayscale level at one wavelength. This can be measured following the procedures in [3] for example. The verification of this method when the voltage is present is included in Section 3. 2.2. Mathematical model for System B For System B, only (1) and (3) will be changed [5]: T˜ = (f − g cos(2
Q ) − j sin(2
+ (h − g sin(2
Q ) + j cos(2
− i[(g + f cos(2 + (j + f sin(2
Q )) cos(
−
Q )) × sin(
Q ) + h sin(2
Q ) − hcos(2
A
Q )) cos(
Q )) sin(
A
P) A
−
P)
A
+
P)
+
P )]
(20)
c 2 t = √ T˜ 2
(21)
where Q is the angle of slow axis of quarter-wave plate (please refer to Fig. 1b). Beside these modifications all the discussions for System A can be used for this setup. System B is useful to improve phase modulation of TN-LCD. One last note before moving on to the next section; n is also dependent on temperature. We do not have the motivation nor the tools to measure this dependence, which is more likely to be negligible as TN-LCD laser modulation applications are mostly performed in stable temperature and using low power lasers. However in the case that temperature is not stable enough, one can refer to [10,15] to include a study of temperature dependency. 3. Results, comparison and discussion We have carried out simulation using Eqs. (17) and (18) to calculate ˇmax () for a number of TN-LCDs and compared results with experimental results of [3,11–13] to verify the proposed model. In Table 1 we have included the values of s and the twist angle for the studied TN-LCD models. We will start with SONY LCX012BL, measurements on this TN-LCD were carried out by [3,4] for 4 different wavelengths. We will verify Eq. (17) for each value of ˇmax at each wavelength (which are included in [3]) based on the other three values of ˇmax at all other wavelengths, and then we will verify Eq. (18) for each value of ˇmax at a specific wavelength based
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G. Makey et al. / Optik 126 (2015) 917–922
Table 1 The values of s and ˛ for two TN-LCD models. TN-LCD model
s (rad)
˛ (rad)
By reference
SONY LCX016AL SONY LCX012BL
0.792 0.803
−1.594 −1.606
[11] [3]
on the other two values of ˇmax at two other wavelengths (the choice is intended to give maximum error) the results are listed in Table 2. Where we used Matlab’s curve fitting toolbox to determine the coefficients of Eqs. (17) and (18). As one can see from Table 2; the maximum error for ˇ obtained from Eq. (17) is 0.08 rad while the maximum error by the used of (18) is 0.03 rad. In Eqs. (17) or (18) the error gets less if the fitting wavelengths include terminal values in visible range. The sets of Eq. (17) coefficients depends in great manner on the choice of the fitting wavelengths, however the values of Eq. (18) coefficients for each set calculated on two arbitrary wavelengths are relatively close. For the use of SONY LCX012BL, the best values of Eqs. (17) and (18) coefficients based on fitting all the wavelengths of Table 2. For (17), Ad = 466.8 nm, Bd = 3.082e + 6 nm3 , Cd = 3.432e + 12 nm5 , which yield the value of RMSE = 1.023 nm (root mean squared errors) for dn. For (18), Ad = 472.1 nm, Cd = 3.859e + 12 nm5 , which yield the value of RMSE = 0.799 nm for dn. Now we will turn our attention toward SONY LCX016AL which is widely used in holographic applications [16,17]. We could not get the values of ˇmax for multiple wavelengths from a single reference, so we collected data from number of references: In [12] ˇ(V) was measured for full range of grayscales; 406 nm diode laser was used. In [11] dnmax was measured at 632.8 nm. Now we have two values of ˇmax at two well-spaced wavelengths and ˇ(V) for a wavelength so we can predict all other ˇ(V,) in visible range based on both Eqs. (18) and (19) (the case of Eq. (19) is already proven by [3]). But we need at least one value for ˇ to validate Eq. (18) or to calculate Eq. (17) coefficients. The only other reference that provides this value is [13] (Conference paper), this work is great and full of unique ideas and hard measurements, however, even though the reference include a study at 6 different wavelengths (3 in visible), the results of transmittance and the calculated values of ˇ at those wavelengths are mostly inaccurate and need to be compared with [12,11] (the difference is large). Maybe the difference is due to the way the transmittance is measured or in the optical setup, and certainly they should not calculate the values of ˇ by considering the twist angle as variable. However there is one value for the transmittance that we can safely use from this reference, this is 0 at 532 nm in a setup contains TN-LCD between two parallel polarizers; that means that the TN-LCD is
Fig. 2. ˇmax dependency on wavelength for SONY LCX012BL (the upper curve) and SONY LCX016AL.
Fig. 3. Off-state normalized intensity (a), and phase shift (b) vs. wavelength for Sony LCX012BL & Sony LCX016AL, the setup is System A with polarizer angle at 90◦ and analyzer angle at 0◦ both with respect to the entrance director angle s of the corresponding TN-LCD.
almost working as half-wavelength plate at this wavelength and the zero value of the transmittance at this wavelength can be used with very small error regardless to its way of measuring. By using the whole simulation model given in Section 2 (for System A) we were able to calculate the value of ˇ at this wavelength.
Table 2 The coefficients of Eqs. (17) and (18) with computed ˇmax for each listed wavelength in comparison with ˇmax taken from [3]. Studied values/wavelength
633 nm
514 nm
488 nm
458 nm
ˇmax taken from [3] (rad) dnmax calculated based on [3] (nm) Calculated coefficients of (17) taken by solving it for the other three wavelengths: Ad (nm), Bd (nm2 ), Cd (nm5 ) Calculated ˇmax for each wavelength by using (17) Calculated coefficients of (18) taken by solving it for other two wavelengths (taken to give maximum error): Ad (nm), Cd (nm5 ) Calculated ˇmax for each wavelength by using (18)
2.46 495.8 Ad = 563.9 Bd = −4.233e + 7 Cd = 8.699e + 12 2.54 Ad = 472.1 Cd = 3.859e + 12
3.23 528.3 Bd = 478.2 Bd = −3.965e + 6 Cd = 4.421e + 12 3.22 Ad = 469.4 Cd = 3.979e + 12
3.47 539.5 Ad = 461.9 Bd = 6.014e + 6 Cd = 3.047e + 12 3.48 Ad = 472.1 Cd = 3.859e + 12
3.84 559.8 Ad = 444.1 Bd = 1.785e + 7 Cd = 1.162e + 12 3.81 Ad = 472.1 Cd = 3.859e + 12
2.49
3.22
3.49
3.82
G. Makey et al. / Optik 126 (2015) 917–922
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The values of maximum birefringence from the studied references are: ˇmax = 3.75 rad at 406 nm, based on [12]. ˇmax = 2.25 rad at 632.8 nm, based on [11]. ˇmax = 2.72 rad at 532 nm, based on [13]. We calculated the coefficients of Eq. (18) by taking the first and second values of ˇmax as they are the most far and accurate, the results are: Ad = 446.8 nm, Cd = 1.028e + 12 nm We used ˇmax at 532 nm to verify Eq. (18) and we got small error which is 0.01 rad. By calculating the coefficients of Eq. (18) by taking all the wavelengths in account we get: Ad = 446.8 nm, Cd = 1.0165e + 12 nm5 with RMSE = 0.7544 nm for dnmax . We calculated the coefficients of Eq. (17) by taking all the wavelengths, and we got: Ad = 441.2 nm, Bd = 3.186e + 6 nm3 , Cd = 6.558e + 11 nm5 Before listing our conclusion we will point out some remarks on the data, which – we hope – will help readers to be able to reproduce our results:
Fig. 5. Intensity & phase modulation by Sony LCX016AL at different wavelengths, the setup is System A with (a) polarizer angle at 90◦ and analyzer angle at 0◦ (b) polarizer angle at 77◦ and analyzer angle at 167◦ , all with respect to the entrance director angle s . The phase shift is taken with respect to gray level 0 in (a) and to gray level 255 in (b).
(1) Fig. 2 shows ˇmax relation vs. wavelength for both studied TNLCDs, the curves calculated by using (18) where we used the final calculated coefficients for each TN-LCD. (2) In Fig. 3 and depending on the previously calculated ˇmax (using (18)), we showed the off-state normalized intensity and phase shift vs. wavelength for SONY LCX012BL and SONY LCX016AL, Eqs. (1)–(11) were used. (3) Finally, we used Eqs. (12)–(14) to show the ability of the analysis done by both [3,4] to predict optical modulation. Where we started by using: - The analysis done by [3,4] which we applied on Sony LCX016AL (those references already proved the analysis on Sony LCX012BL). - Phase and amplitude modulation curves from [16] to calculate both ˇcenter & ˇedge at 632.8 nm. Normally, calculating ˇcenter & ˇedge requires measurement of intensity modulation at three specific polarizer–analyzer setups (like the work in [4] for example), that was not present in the references we used for Sony LCX016AL. However, we noticed that with the setup used in [16] and with good accuracy, each value of normalized intensity corresponds to a specific value of ˇcenter regardless to the value of ˇedge (see Fig. 4a), so we calculated ˇcenter based on amplitude modulation curve then we calculated ˇedge based on phase modulation curve of [16]. Fig. 4b shows why we cannot use this method with any polarizer–analyzer setup, where Fig. 4b is based on the same polarizer–analyzer setup as in [12].
Fig. 4. Normalized intensity modulation with respect to both ˇcenter & ˇedge or Sony LCX016AL at 632.8 nm in System A with (a) polarizer angle at 90◦ and analyzer angle at 0◦ (b) polarizer angle at 77◦ and analyzer angle at 167◦ , all with respect to the entrance director angle s .
Then we compared the results of using those values of birefringence to predict intensity and phase modulation at different wavelength and different polarizer–analyzer angles which correspond to the work and results of [12]. The results are shown in Fig. 5 where intensity and phase modulation were calculated at different polarizer–analyzer angles, and at different wavelengths (at
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G. Makey et al. / Optik 126 (2015) 917–922
632.8 nm which corresponds to [16], at 406 nm which corresponds to [12], and 532 nm which been chosen arbitrary), the intensity in Fig. 5 was normalized for each wavelength. The maximum error of normalized intensity modulation was about 0.05, and the maximum error in phase modulation was about 0.05. 4. Conclusions In this work we presented a method to include wavelength dependency in simulating models of TN-LCD optical modulation, this addition is based on extended Cauchy equations. We verified our results by comparing simulation result to experimental results of a number of published results which is highly regarded and accepted in the field. We used well spread TN-LCD models in our study so that the results can easily be implemented in wide range of future work. And for this very purpose, we included in this paper all the equations needed for the full modeling of typical SLM setups.
[3]
[4]
[5] [6] [7]
[8]
[9]
[10] [11]
Acknowledgments The authors would like to thank both Dr. Imad Assad and Mr. Sinan Al-Jalali of Laser Institute, Damascus University for their help on some technical issues, the authors also extend appreciation to Mr. Mousa Ayoub of Physics Department, Munster University and Stefan Osten of Holoeye, for useful discussion and for providing helpful references. References [1] J.W. Goodman, Introduction to Fourier Optics, 3rd ed., Roberts & Company, 2005. [2] A.S. Ostrovsky, C. Rickenstorff-Parrao, M.Á. Olvera-Santamaría, Using the liquid crystal spatial light modulators for control of coherence and polarization
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Optik journal homepage: www.elsevier.de/ijleo
Predicting wavelength dependency of optical modulation of twisted nematic liquid crystal display in the visible range Ghaith Makey, Moustafa Sayem El-Daher ∗ , Kanj Al-Shufi Higher Institute for Laser Research and Applications, Damascus University, Airport Highway, Damascus, Syria
a r t i c l e
i n f o
Article history: Received 2 February 2014 Accepted 12 February 2015 Keywords: Spatial light modulators Optoelectronics Simulation Holography
a b s t r a c t One of the main problems in liquid crystal spatial light modulators (LC-SLM) is the prediction of their optical modulation behavior for different wavelengths; this coupled phase/amplitude modulation can be hard to measure for specific wavelengths and setups, yet it is required to be included in the study of a number of SLM’s applications. This work proposes a simulation model to predict the phase/amplitude modulation for different twisted nematic liquid crystal displays (TN-LCD) optical setups, this model incorporates the effects of wavelength in addition to other parameters which already have been studied before like twist angle as internal parameter and polarizers angles, we verified the accuracy of our model compared to results of previously published work. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Extensive work have been dedicated to the topic of twist nematic liquid crystal displays (TN-LCD) modeling and simulation [1,2], yet the individual papers in this field could not answer clearly a key question required for many users of TN-LCD when used as SLM, namely at an arbitrary wavelength what is the coupled amplitude-phase modulation for each grayscale value in a pixel? This question is quiet important to be answered by a simulation model specifically when the measurements of the associated modulation is hard to carry out (e.g. when working on holography’s experiment with amplitude mostly setup and need to predict the values of coupled phase modulation). By keeping the previous question in mind, let us review quickly related work. In 2000 Marquez et al. [3] presented distinctive characterization of edge effects in TN-LCDs; their analysis was supported by measurements taken at 4 wavelengths. In 2001 Marques et al. [4] used the results of [3] to extend their work and gave a quantitative prediction of the modulation behavior of TN-LCD, again the measurements were limited to 4 wavelengths and their analysis requires specific measurement for each wavelength at which they want to predict TN-LCD modulation behavior.
∗ Corresponding author. Tel.: +963 1133920683. E-mail addresses: [email protected] (G. Makey), [email protected] (M. Sayem El-Daher), kalshoufi@gmail.com (K. Al-Shufi). http://dx.doi.org/10.1016/j.ijleo.2015.02.072 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
In 2005 Kim and Lee [5] measured the parameters of TN-LCD with no ambiguity by fitting the theoretical predictions to the intensity of transmittance, then they predicted the phase modulation and verified the results experimentally; their work was limited to the study of one wavelength. In the same year, Wan and He [6] proposed new model for the director distribution of TN-LC cell but again the work did not include a study of wavelength dependency. Results which describe the optical modulation properties of TNLCD without the need to know LCD’s internal structure parameters do exist [7–9]; however these studies did not include a study of wavelength dependency. In this paper we will use the equations derived and listed very neatly in Refs. [3,5], and we will incorporate extended Cauchy equations [10] to propose one unified mathematical model that can be used to predicted phase and amplitude modulation of TN-LCD at any visible wavelength and without the need for any measurement at this wavelength. We will also use the experimental results of [3,4,11–13] to verify our simulation results.
2. Mathematical model In this section we will discuss the mathematical models for two main TN-LCD SLM setups: the first one is a system composed of SLM sandwiched between two polarizers (System A) and the second is a system composed of polarizer-SLM-quarter wave platepolarizer (System B). Fig. 1 shows the differences between two setups.
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g = cos(2ˇedge ) + ×
ˇcenter sin() cos(E + S )
(5)
h = − 1− ×
ˇcenter
cot() sin(2ˇedge )
ˇcenter tan(˛) sin(2ˇedge ) ˛
˛ sin() cos(˛) + cos(2ˇedge ) cos() sin(˛)
j = cos(2ˇedge ) +
ˇcenter
(6)
cot() sin(2ˇedge )
ˇcenter × sin() sin(E + S ) f 2 + g 2 + h2 + j2 = 1
(7)
(8)
S , E are angles of the extraordinary axis of the liquid-crystal molecules at the entrance and exit surfaces of TN-LC cell, in respect. ˛ is the twist angle: ˛ = E − S
(9)
is given by: =
Fig. 1. Different optical setups for TN-LCD as SLM, (a) System A, (b) System B [5].
2 ˛2 + ˇcenter
ˇ, ˇcenter , and ˇedge are very important parameters which include wavelength and voltage dependency (and even temperature dependency which is out of our focus in this paper); ˇ is the birefringence of the TN-LCD and it reaches its maximum value in the case of voltage absence: ˇmax =
2.1. Mathematical model for System A We will start directly with the following equations [5] for System A: T˜ = f cos( +
A
−
P ) + h sin(
P ) − ij sin(
A+
A
−
P ) − ig cos(
A
P)
(1)
ı = ˇ − arg(T˜ )
(2)
t = |c T˜ |2
(3)
where T˜ is the normalized transmission coefficient, ı is the phase delay which represents phase modulation, and t can be used to predict analytically intensity transmittance which represents amplitude modulation. P are the angles of the analyzer and the polarizer, respecA, tively, (refer to Fig. 1a). c is a constant represents the loss factor of TN-LC cell (which can be considered unity by normalizing the transmission or can be measured in similar manner to [5]). f, g, h, and j are Jones parameters for TN-LC cell with edge effect taken in account (this effect arises when applying voltage to the cell):
f =
1−
˛ ˇcenter cot(˛) sin(2ˇedge ) sin() sin(˛) ˛
+ cos(2ˇedge ) cos() cos(˛)
(4)
(10)
d(ne − no ) dnmax =
(11)
where d is the thickness of the cell (the liquid crystal part only). is the wavelength of the incident light in vacuum. ne , no are extraordinary and normal refractive indices, respectively, and nmax is refractive indices difference (“max” subscript stands for n value when no voltage is present). From the first look at Eq. (11) one can deduce that ˇmax is changing linearly with 1/ but that is not right experimentally [3,13] because nmax is changing with as well. We will return to this point after completing the definitions of ˇcenter , ˇedge . When applying voltage greater than the threshold of TN-LC cell, edges effect appears [3] and the extraordinary refractive index changes [14]; in this case ˇ will equal to: ˇ = ˇcenter + 2ˇedge
(12)
where ˇedge is the birefringence at each edge area (“edge” is also referred as “surface”). ˇcenter is the birefringence of the center area (“center” is also referred as “bulk”) whose molecule’s rotation is not constrained by edge effect; ˇedge =
dedge nmax
(13)
where dedge is the depth of the area affected by each edge. ˇcenter =
(d − 2dedge )n(V )
(14)
where n(V) emphasizes the dependency of n on the applied voltage to each TN-LC cell. Now let us take a look at the equations from a practical point of view, in the case of the absence of applied voltage ˇedge will equal to zero. And if ˛ and s are known, we only need to know the value of ˇmax to predict amplitude and phase modulation for each
G. Makey et al. / Optik 126 (2015) 917–922
( A , P ) setup. ˛ and s are constant for each TN-LCD model and they can be obtained from references like [3,6,11], or by measurements using the same methods used on those papers. Normally, ˇ can be taken from reported results in papers, calculated theoretically or from measurements, calculating ˇmax analytically from t or ı is very hard, so in order to solve this we wrote a function in Matlab that contains all the required equations to calculate t and ı, the input of this function includes all the required parameters which includes ˇmax as an array whose elements are ranging from a minimum to maximum possible values for ˇmax with a step of 0.01 rad, then if t and ı vs. ˇmax curves are plotted one can work inversely using MATLAB data cursor we can determine and verify ˇmax value for each t and ı measurements. However our intention in the paper is to predict ˇmax value at any visible wavelength without the need to make any measurements at this wavelength. To calculate ˇmax using our proposed scheme we need the following values. (in vacuum) which is certainly known, d which is constant but generally unknown, and nmax which change with itself. For nmax , which is more accurately written as nmax (), we used Extended Cauchy equations in the calculation [10]: Be,o Ce,o ne,o ∼ = Ae,o + 2 + 4
(15)
where Ae,o , Be,o , and ce,o are the three Cauchy coefficients. As we are not interested in the specific values of ne and no we will rewrite (15) as: n ∼ = A +
B C + 4 2
(16)
Since d is most likely to be unknown we will insert the constant d to (16) so we will have: Bd Cd dn ∼ = Ad + 2 + 4
(17)
Ad , Bd , and cd are three coefficients that are constant for a specific TN-LCD model when no voltage is applied. If we have three values for ˇmax at three different wavelengths one can predict the values of ˇmax at any other wavelength in visible range (we will not extend this statement to IR or UV range because we do not have the required measurements to justify this extension, nor we are sure of the behavior of different optical elements in the optical setup when operating in UV–visible or visible-IR ranges). But if we take a closer look at Eq. (17) we will notice that it represent quadratic equation with respect to (1/2 ), and if the three values of , which can be used to calculate the three coefficients, are close to each other they may positioned on the linear part of the quadratic curve and the determined values of the coefficients may not lead to correct predication of n value for a wavelength that is far from the range of the original wavelengths. This is an inherent problem with Cauchy equations and can be solved either by taking the minimum and maximum values of the three wavelengths to be far from each other (e.g. one at red and the second at blue wavelength), or one can use the result which we found by analyzing previous measurements [3,11–13]: we found that dn change linearly with (1/4 ) which means that Eq. (17) can be approximated to: dn ∼ = Ad +
Cd 4
(18)
Only two measurements of ˇmax at two different wavelengths are needed to predict any value of ˇmax at any other wavelength, this assumption is purely empirical (however, even the physical origin of the well-used extended Cauchy equation is not clear [10]). When using Eq. (18) it is important to pay attention that the associated ˇmax of the two wavelengths which are chosen to be used to determine the coefficients in Eq. (18) should be calculated based on accurate measurements (as in Ref. [3]).
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In short, in the absence of voltage and for any TN-LCD model we need only the values of ˛, s and three or two values of ˇmax at three or two different wavelengths to predict the values of t and ı for any wavelength in the visible range at each ( A , P ) setup ( A and P combinations determine the working mode of TN-LCD as an SLM; it can be set to phase mostly modulation or amplitude mostly modulation for example); verification of our results will be presented in Section 3. If an effective voltage is applied to TN-LC cell, the study can be more complicated. But thanks to Marquez et al. work [3], this study can be much easier by the use of the following relation: ˇedge (V, ) ˇmax () ˇcenter (V, ) = = ˇmax (0 ) ˇcenter (V, 0 ) ˇedge (V, 0 )
(19)
where is an arbitrary wavelength, 0 is a single wavelength at which ˇcenter (V) and ˇedge (V) are calculated based on measurements taken for each voltage applied on TN-LC cell (the voltage is quantized in levels which are referred as grayscale levels of a TNLC cell). According to our assumption, ˇmax () do not need to be measured but can be calculated using (17) or (18). So to fully predict phase and amplitude modulation for TNLCD under voltage operation, and in addition to the requirements needed for no-voltage case, we need to know ˇcenter and ˇedge (or just the total ˇ) values for each grayscale level at one wavelength. This can be measured following the procedures in [3] for example. The verification of this method when the voltage is present is included in Section 3. 2.2. Mathematical model for System B For System B, only (1) and (3) will be changed [5]: T˜ = (f − g cos(2
Q ) − j sin(2
+ (h − g sin(2
Q ) + j cos(2
− i[(g + f cos(2 + (j + f sin(2
Q )) cos(
−
Q )) × sin(
Q ) + h sin(2
Q ) − hcos(2
A
Q )) cos(
Q )) sin(
A
P) A
−
P)
A
+
P)
+
P )]
(20)
c 2 t = √ T˜ 2
(21)
where Q is the angle of slow axis of quarter-wave plate (please refer to Fig. 1b). Beside these modifications all the discussions for System A can be used for this setup. System B is useful to improve phase modulation of TN-LCD. One last note before moving on to the next section; n is also dependent on temperature. We do not have the motivation nor the tools to measure this dependence, which is more likely to be negligible as TN-LCD laser modulation applications are mostly performed in stable temperature and using low power lasers. However in the case that temperature is not stable enough, one can refer to [10,15] to include a study of temperature dependency. 3. Results, comparison and discussion We have carried out simulation using Eqs. (17) and (18) to calculate ˇmax () for a number of TN-LCDs and compared results with experimental results of [3,11–13] to verify the proposed model. In Table 1 we have included the values of s and the twist angle for the studied TN-LCD models. We will start with SONY LCX012BL, measurements on this TN-LCD were carried out by [3,4] for 4 different wavelengths. We will verify Eq. (17) for each value of ˇmax at each wavelength (which are included in [3]) based on the other three values of ˇmax at all other wavelengths, and then we will verify Eq. (18) for each value of ˇmax at a specific wavelength based
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Table 1 The values of s and ˛ for two TN-LCD models. TN-LCD model
s (rad)
˛ (rad)
By reference
SONY LCX016AL SONY LCX012BL
0.792 0.803
−1.594 −1.606
[11] [3]
on the other two values of ˇmax at two other wavelengths (the choice is intended to give maximum error) the results are listed in Table 2. Where we used Matlab’s curve fitting toolbox to determine the coefficients of Eqs. (17) and (18). As one can see from Table 2; the maximum error for ˇ obtained from Eq. (17) is 0.08 rad while the maximum error by the used of (18) is 0.03 rad. In Eqs. (17) or (18) the error gets less if the fitting wavelengths include terminal values in visible range. The sets of Eq. (17) coefficients depends in great manner on the choice of the fitting wavelengths, however the values of Eq. (18) coefficients for each set calculated on two arbitrary wavelengths are relatively close. For the use of SONY LCX012BL, the best values of Eqs. (17) and (18) coefficients based on fitting all the wavelengths of Table 2. For (17), Ad = 466.8 nm, Bd = 3.082e + 6 nm3 , Cd = 3.432e + 12 nm5 , which yield the value of RMSE = 1.023 nm (root mean squared errors) for dn. For (18), Ad = 472.1 nm, Cd = 3.859e + 12 nm5 , which yield the value of RMSE = 0.799 nm for dn. Now we will turn our attention toward SONY LCX016AL which is widely used in holographic applications [16,17]. We could not get the values of ˇmax for multiple wavelengths from a single reference, so we collected data from number of references: In [12] ˇ(V) was measured for full range of grayscales; 406 nm diode laser was used. In [11] dnmax was measured at 632.8 nm. Now we have two values of ˇmax at two well-spaced wavelengths and ˇ(V) for a wavelength so we can predict all other ˇ(V,) in visible range based on both Eqs. (18) and (19) (the case of Eq. (19) is already proven by [3]). But we need at least one value for ˇ to validate Eq. (18) or to calculate Eq. (17) coefficients. The only other reference that provides this value is [13] (Conference paper), this work is great and full of unique ideas and hard measurements, however, even though the reference include a study at 6 different wavelengths (3 in visible), the results of transmittance and the calculated values of ˇ at those wavelengths are mostly inaccurate and need to be compared with [12,11] (the difference is large). Maybe the difference is due to the way the transmittance is measured or in the optical setup, and certainly they should not calculate the values of ˇ by considering the twist angle as variable. However there is one value for the transmittance that we can safely use from this reference, this is 0 at 532 nm in a setup contains TN-LCD between two parallel polarizers; that means that the TN-LCD is
Fig. 2. ˇmax dependency on wavelength for SONY LCX012BL (the upper curve) and SONY LCX016AL.
Fig. 3. Off-state normalized intensity (a), and phase shift (b) vs. wavelength for Sony LCX012BL & Sony LCX016AL, the setup is System A with polarizer angle at 90◦ and analyzer angle at 0◦ both with respect to the entrance director angle s of the corresponding TN-LCD.
almost working as half-wavelength plate at this wavelength and the zero value of the transmittance at this wavelength can be used with very small error regardless to its way of measuring. By using the whole simulation model given in Section 2 (for System A) we were able to calculate the value of ˇ at this wavelength.
Table 2 The coefficients of Eqs. (17) and (18) with computed ˇmax for each listed wavelength in comparison with ˇmax taken from [3]. Studied values/wavelength
633 nm
514 nm
488 nm
458 nm
ˇmax taken from [3] (rad) dnmax calculated based on [3] (nm) Calculated coefficients of (17) taken by solving it for the other three wavelengths: Ad (nm), Bd (nm2 ), Cd (nm5 ) Calculated ˇmax for each wavelength by using (17) Calculated coefficients of (18) taken by solving it for other two wavelengths (taken to give maximum error): Ad (nm), Cd (nm5 ) Calculated ˇmax for each wavelength by using (18)
2.46 495.8 Ad = 563.9 Bd = −4.233e + 7 Cd = 8.699e + 12 2.54 Ad = 472.1 Cd = 3.859e + 12
3.23 528.3 Bd = 478.2 Bd = −3.965e + 6 Cd = 4.421e + 12 3.22 Ad = 469.4 Cd = 3.979e + 12
3.47 539.5 Ad = 461.9 Bd = 6.014e + 6 Cd = 3.047e + 12 3.48 Ad = 472.1 Cd = 3.859e + 12
3.84 559.8 Ad = 444.1 Bd = 1.785e + 7 Cd = 1.162e + 12 3.81 Ad = 472.1 Cd = 3.859e + 12
2.49
3.22
3.49
3.82
G. Makey et al. / Optik 126 (2015) 917–922
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The values of maximum birefringence from the studied references are: ˇmax = 3.75 rad at 406 nm, based on [12]. ˇmax = 2.25 rad at 632.8 nm, based on [11]. ˇmax = 2.72 rad at 532 nm, based on [13]. We calculated the coefficients of Eq. (18) by taking the first and second values of ˇmax as they are the most far and accurate, the results are: Ad = 446.8 nm, Cd = 1.028e + 12 nm We used ˇmax at 532 nm to verify Eq. (18) and we got small error which is 0.01 rad. By calculating the coefficients of Eq. (18) by taking all the wavelengths in account we get: Ad = 446.8 nm, Cd = 1.0165e + 12 nm5 with RMSE = 0.7544 nm for dnmax . We calculated the coefficients of Eq. (17) by taking all the wavelengths, and we got: Ad = 441.2 nm, Bd = 3.186e + 6 nm3 , Cd = 6.558e + 11 nm5 Before listing our conclusion we will point out some remarks on the data, which – we hope – will help readers to be able to reproduce our results:
Fig. 5. Intensity & phase modulation by Sony LCX016AL at different wavelengths, the setup is System A with (a) polarizer angle at 90◦ and analyzer angle at 0◦ (b) polarizer angle at 77◦ and analyzer angle at 167◦ , all with respect to the entrance director angle s . The phase shift is taken with respect to gray level 0 in (a) and to gray level 255 in (b).
(1) Fig. 2 shows ˇmax relation vs. wavelength for both studied TNLCDs, the curves calculated by using (18) where we used the final calculated coefficients for each TN-LCD. (2) In Fig. 3 and depending on the previously calculated ˇmax (using (18)), we showed the off-state normalized intensity and phase shift vs. wavelength for SONY LCX012BL and SONY LCX016AL, Eqs. (1)–(11) were used. (3) Finally, we used Eqs. (12)–(14) to show the ability of the analysis done by both [3,4] to predict optical modulation. Where we started by using: - The analysis done by [3,4] which we applied on Sony LCX016AL (those references already proved the analysis on Sony LCX012BL). - Phase and amplitude modulation curves from [16] to calculate both ˇcenter & ˇedge at 632.8 nm. Normally, calculating ˇcenter & ˇedge requires measurement of intensity modulation at three specific polarizer–analyzer setups (like the work in [4] for example), that was not present in the references we used for Sony LCX016AL. However, we noticed that with the setup used in [16] and with good accuracy, each value of normalized intensity corresponds to a specific value of ˇcenter regardless to the value of ˇedge (see Fig. 4a), so we calculated ˇcenter based on amplitude modulation curve then we calculated ˇedge based on phase modulation curve of [16]. Fig. 4b shows why we cannot use this method with any polarizer–analyzer setup, where Fig. 4b is based on the same polarizer–analyzer setup as in [12].
Fig. 4. Normalized intensity modulation with respect to both ˇcenter & ˇedge or Sony LCX016AL at 632.8 nm in System A with (a) polarizer angle at 90◦ and analyzer angle at 0◦ (b) polarizer angle at 77◦ and analyzer angle at 167◦ , all with respect to the entrance director angle s .
Then we compared the results of using those values of birefringence to predict intensity and phase modulation at different wavelength and different polarizer–analyzer angles which correspond to the work and results of [12]. The results are shown in Fig. 5 where intensity and phase modulation were calculated at different polarizer–analyzer angles, and at different wavelengths (at
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632.8 nm which corresponds to [16], at 406 nm which corresponds to [12], and 532 nm which been chosen arbitrary), the intensity in Fig. 5 was normalized for each wavelength. The maximum error of normalized intensity modulation was about 0.05, and the maximum error in phase modulation was about 0.05. 4. Conclusions In this work we presented a method to include wavelength dependency in simulating models of TN-LCD optical modulation, this addition is based on extended Cauchy equations. We verified our results by comparing simulation result to experimental results of a number of published results which is highly regarded and accepted in the field. We used well spread TN-LCD models in our study so that the results can easily be implemented in wide range of future work. And for this very purpose, we included in this paper all the equations needed for the full modeling of typical SLM setups.
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[10] [11]
Acknowledgments The authors would like to thank both Dr. Imad Assad and Mr. Sinan Al-Jalali of Laser Institute, Damascus University for their help on some technical issues, the authors also extend appreciation to Mr. Mousa Ayoub of Physics Department, Munster University and Stefan Osten of Holoeye, for useful discussion and for providing helpful references. References [1] J.W. Goodman, Introduction to Fourier Optics, 3rd ed., Roberts & Company, 2005. [2] A.S. Ostrovsky, C. Rickenstorff-Parrao, M.Á. Olvera-Santamaría, Using the liquid crystal spatial light modulators for control of coherence and polarization
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