From the theoretical results of the governing diffusion-like equation and the threshold intensity for the patterns to be formed we can know that when nematic liquid crystals are used as the nonlinear material to perform the optical pattern formation experiment the intrinsic physical properties, such as the optical birefringence, the unequal property of Franck elastic constants, and the dielectric anisotropy, all play important roles in the pattern formation phenomena. In this chapter, we investigate the influence of these anisotropic properties on the pattern formation phenomena and for convenience the parameters used in our numerical analysis are on the basis of the homogeneous E7 cell with its cell gap of 68 µm though the theoretical results hold for other nematic materials.
3-1 Anisotropic threshold intensity distribution results from the elastic anisotropy of nematic liquid crystals
From Eq. (9) and Eq. (17) we can easily see that the anisotropic nonlinear response of NLC films indeed induces the anisotropic distribution of the needed threshold intensity for the pattern to be formed and from Eq. (5) we can see that the anisotropy of the diffusion lengths comes from the anisotropy of Franck elastic constants. Fig. 3.1(a) shows the theoretical calculated threshold power in various θa when the azimuthal angle is ϕ = 900, 600 and 00. The threshold power is calculated by the product of the threshold intensity and the beam area. Actually, the key physical
parameter for the optical pattern formation is the intensity and we use the threshold power for the consideration of the practical-experimental reality.
From Fig. 3.1(a) we can see that for each fixed ϕ the minimum of the threshold power locates at about θa
= 0.9 radian. For each fixed θ
a the smallest threshold power is at ϕ = 900 and the largest threshold one is at ϕ= 00 and 1800. For convenience in Fig. 3.1(b), we plot the threshold power in various ϕ when θa is kept at 0.787 radian. From Fig. 3.1(b), one can see that the anisotropic property of the threshold power in various azimuthal angles. It’s more clear that the minimum threshold power locates at ϕ = 900 and the maximum one locates at ϕ = 00 and 1800.
0 .6 5 0 .7 0 .7 5 0 .8 0 .8 5 0 .9 0 .9 5 1 1 .0 5 1 .1
0 .5 0 .7 0 .9 1 .1 1 .3
average tilt angle q
a(rad. )
th re shol d pow er ( W )
j=0j=p/3 j=p/2
(a)
0. 73 0. 74 0. 75 0. 76 0. 77 0. 78 0. 79 0. 8 0. 81
0 0. 5 1 1. 5 2 2. 5 3
azim uthal angle q
a(rad.)
th re sh ol d p o w er (W )
(b)
Fig. 3.1 (a) The calculated curves of the threshold power versus the average tilt angle θa when the azimuthal angle ϕ is fixed;
from the top to the bottom are azimuthal angle ϕ = 900, 600 and 00, respectively (b) the calculated curve of the threshold power versus the azimuthal angle ϕ when the average tilt angle θa is fixed as 0.787 rad.
On the basis of this anisotropic distribution of the threshold power, we can propose a method to obtain different optical patterns. The formation of different patterns may be associated with the possibility of the appearance of the modes with different azimuthal angles allowed to exist under the applied input light power and other experimental conditions.
When the input light power is above and close to the minimum threshold power, the modes with azimuthal angles close to ϕ = 900 are allowed to appear. However, these modes will not oscillate to form other patterns since they fulfill no momentum conservation conditions. Therefore, only the mode with the lowest threshold power (in our case the mode with ϕ =
900) will be enhanced and the roll pattern should be observed.
On the other hand, when the input light power is above but not far from the maximum threshold power, all the modes with azimuthal angle from ϕ = 00 to ϕ = 3600 are allowed to appear. This is similar to that in the isotropic case and we can expect the oscillation of three ripple patterns leading to hexagons formation. This is because the hexagons satisfy the momentum conservation condition and the beams can enhance with each other.
3-2 Influence of Frank elastic constant anisotropy on optical pattern formation phenomena
The intrinsic Frank elastic constant anisotropy induces the anisotropy of the diffusion length, which results in the anisotropic distribution of the threshold intensity for optical pattern formation. Therefore, the effects of the elastic constant anisotropy on optical pattern formation are studied in this subsection and the obtained numerical results can reasonably explain the optical patterns that can be formed.
From Eq. (9) one can see that the key factor is the anisotropy of the diffusion lengths. Therefore, in order to clearly analyze the effects of the Franck elastic constants on the pattern formation phenomena we rewrite the parameters in Eq. (5) as
) ,
In order to determine the anisotropic property of the threshold intensity distribution, one should calculate the anisotropy of the diffusion lengths first. From Eq. (17), the anisotropy of the diffusion lengths can be expressed as:
From Eq. (5), the parameter σ can be considered as the effective diffusion
anisotropy parameter. For the vertical-aligned NLC films, the average tilt angle θa, the coefficient G and the elastic anisotropies ka and kb in the above equations have to be replaced by the average polar angle βa,
)]]}, k anisotropy between ka and kb affects the effective diffusion anisotropy parameter σ. The value of the effective diffusion anisotropy σ plays an important role in the formation of optical patterns. Considering both the intrinsically anisotropic distribution of the threshold intensity resulted from the elastic constant anisotropy of the NLC materials and the externally electric-tunable effective diffusion anisotropy, since it is related with the average tilt angle θa which is easily seen in Eqs. (19) and (20), one may employ different methods to obtain different optical patterns.
Generally speaking, if the value of σ is positive, the value of lx is larger than that of ly. This property makes the threshold intensity distribution has its minimum at ϕ = π/2 and maximum at ϕ = 0, π. Therefore once the input light intensity is above and near the minimum threshold, the vertical roll can be obtained since it will experience the maximum gain. On the other hand, when the input light intensity is above and near the maximum threshold, the hexagon is expected to be formed based on its compact structure. If the value of σ is negative, the value of lx is smaller than that of ly. This property makes the threshold intensity distribution has its minimum at ϕ = 0, π and maximum at ϕ = π/2. Therefore, once the input
light intensity is above and near the minimum threshold, the horizontal roll can be obtained and once the input light intensity is above and near the maximum threshold, the hexagon is expected to be formed. The special case for σ = 0 represents the isotropic distribution of the threshold intensity and the hexagon can be obtained when the input light intensity is above and near the threshold.
On the basis of the above description, in order to verify the influence of the elastic constant anisotropy on optical pattern formation, it’s straightforward to study the influence of the elastic constant anisotropy by considering the value of the effective diffusion anisotropy σ. On the basis of Eq. (20), in Table 3.1, we list the possible values of σ for different regions of the elastic constant anisotropy. The material and system parameters used in the calculation are summarized in Table 3.2.
Table 3.1 Classification of regions according to the Frank elastic constant anisotropy.
Table 3.2 The liquid crystal parameters, cell properties, and optical system parameters used in the calculations.
The value of m versus the average orientational angle is plotted in Fig.
3.2. From Fig. 3.2 the value of m is decreasing as the orientational angle increasing. Between the possible range of the average orientational angle θa from 0 to π/2, the value of m is positive and has its maximum as 1 at θa
= 0 and minimum as 0.11 at θa =π/2. This means that we can modulate the value of m from 0.11 to 1 by controlling the average orientational angle θa electrically. For the region A in Table 3.1, the values of both ka and kb are positive and the value of σ can be easily estimated from Eq. (21) and Fig. 3.2.
0 0.2 0.4 0.6 0.8 1 1.2
0 0.25 0.5 0.75 1 1.25 1.5
average orientational angle q
a(rad.)
m
Fig. 3.2 The calculated value of m versus the average orientational angle θa.
To more clearly illustrate the value of σ in this region (region A) we plot an example in Fig. 3.3(a). From Fig. 3.3(a), one can see that the value of σ is positive if kb/ka < 0.11, is negative if kb/ka > 1, and can be modulated to be zero by changing the value of m electrically in the range that 0.11 <
k
b/ka < 1. For example when kb/ka = 0.6, one can have σ = 0 by tuning θa to have m = 0.61962. The possibility of modulating σ to zero by changing the value of m can be clearly seen when we plot the diffusion lengths in Eq. (19) and Eq. (20) in Fig. 3.4(a). From Fig. 3.4(a), one can see that the diffusion anisotropy can be canceled in the range 0.11 < kb/ka < 1.effect ive di ffu si on anisot ropy s ( cm
-2)
m=0.11
effect iv e di ffusi on ani sot ropy s ( cm
-2)
m=0.11m=0.61962m=1 kb/ka=0.11 kb/ka=1
(b)
Fig. 3.3 The calculated effective diffusion anisotropy σ versus the elastic
anisotropies. (a) for region A; with k
a= 1.5 (b) for region B; with k
a= −0.5.
0.001
average tilt angle q
a(rad.)
di ffusi on l engt h (cm)
Fig. 3.4 The calculated diffusion lengths versus the average tilt angle θ
a. (a) for region
A; with k
a= 1.5 (b) for region B; with k
a= −0.5. Please note that l
xis the same for
different k
b/k
aif k
ais not changed.
This means that the vertical roll, the horizontal roll and the hexagon can be expected to be obtained in region A by considering suitable elastic constant anisotropy range and employing suitable methods. For the region B in Table 3.1, the value of both ka and kb are negative. From Eq. (19) and Fig. 3.2, the value of σ can also be easily estimated. We plot an example in Fig. 3.3(b) to see the value of σ. From Fig. 3.3(b), one can see that the value of σ is positive for kb/ka > 1, is negative for kb/ka < 0.11, and can be modulated to be zero by changing the value of m electrically in the range that 0.11 < kb/ka < 1. The possibility of modulating σ to zero by changing the value of m can also be seen from the diffusion lengths plotted in Fig.
3.4(b). This means that the vertical roll, the horizontal roll and the hexagon can also be obtained by considering suitable elastic anisotropy range and employing suitable methods in this region.
A simple method to obtain the vertical roll, the hexagon and the horizontal roll can be suggested if the effective diffusion anisotropy σ can be canceled electrically like the case in region A and B. For example, Figs. 3.5(a)-(c) show the results for ka = 0.625 and kb/ka = 0.512 in region A. From Fig. 3.5(a) we find that lx and ly are equal at θa = 0.905 radian and the threshold intensity distributions for all the azimuthal modes are the same at θa = 0.905 radian as shown in Fig. 3.5(b). This property makes the formation of the stable vertical roll, the stable horizontal roll and the stable hexagon possible by simply modulating the average tilt angle electrically. The threshold intensity versus the azimuthal angles for several fixed tilt angles is shown in Fig. 3.5(c). If the input light intensity is about 55.6 W/cm2 and θa is 0.85 radian, then the modes which are allowed to appear is closed to the minimum at ϕ = π/2 and thus the stable
0. 0012
diffusion length (cm) l x
l y
Fig. 3.5 (a) The calculated diffusion lengths versus the average tilt angle θa (b) The calculated threshold intensity versus the average tilt angle θa when the azimuthal angle ϕ is fixed (c) The calculated threshold intensity versus the azimuthal angle ϕ when the average tilt angle θa is fixed With ka = 0.625, kb/ka = 0.512.
vertical roll is expected to be obtained. If θa is 0.905 radian, all the modes in different azimuthal ϕ are allowed to appear and then the stable hexagon is expected to form due to its compact structure. If θa is 0.95 radian, the modes that are allowed to appear are closed to ϕ = 0 or ϕ = π.
Therefore the stable horizontal roll is supposed to appear. This method to obtain different patterns is similar to the method used by Santamato by rotating the NLC sample in Ref. 11.
Now we continue to discuss the cases in regions C and D. One can easily realize from Eq. (21) and Fig. 3.2 that the value of σ in region C is always positive and the value of σ in region D is always negative. In region C, the value of ka is positive and the value of kb is negative. From Fig. 3.2, one can see that the value of m is always positive and then the resulted value of σ in region C is always positive. Similarly, in region D the value of ka is negative, the value of m is positive and the value of kb is positive. Then the resulted value of σ is always negative in region D. The examples of the threshold intensity distribution in regions C and D are plotted in Fig. 3.6(a) and (b), respectively. The possible optical patterns can be obtained in region C are the vertical roll and the hexagon, and in region D they are the horizontal roll and the hexagon.
As for the vertical-aligned case, the analysis can be easily done by following the similar procedure for the parallelly planar-aligned case.
One just needs to replace θa, G, ka, and kb by βa, G*, ka* and kb*, respectively and the results are similar.
From the above description, one can see that the Frank elastic constant anisotropy indeed plays an important role for optical pattern formation in the studied system. It is not easy to illustrate all the above cases
experimentally since the commercially available rod-like liquid crystal materials generally posses the property that k33 > k11 > k22. Therefore, only the cases in region C for with parallelly planar-aligned LC films and the cases in region B with the vertical-aligned LC films may be obtainable now.
48 53 58 63 68 73 78
0. 6 0. 7 0. 8 0. 9 1 1. 1 1. 2
average tilt angle q
a(rad.)
th re sh o ld in ten sity ( W /cm
2)
j=p/2j=p/4 j=0
(a)
35 40 45 50 55 60
0.66 0.76 0.86 0.96 1.06 1.16
average tilt angle q
a(rad. ) th re shol d in te ns it y ( W /c m
2) j=p/2
j=p/4 j=0
(b)
Fig. 3.6 (a) The calculated threshold intensity versus the average tilt angle θa when the azimuthal angle ϕ is fixed; with ka = 0.625, kb/ka = −0.4. (b)The calculated threshold intensity versus the average tilt angle θa when the azimuthal angle ϕ is fixed with ka = −0.2, kb/ka = −1.
3-3 Analysis of the influence of dielectric anisotropy of nematic liquid crystals on optical pattern formation phenomena
From Eq. (19), we can see that not only the anisotropy of the diffusion lengths is affected by the orientation of the liquid crystal molecules but also the effective nonlinearity is related with the orientation of the liquid crystal molecules. Therefore, once a positive dielectric anisotropy liquid crystal material is used and treated as the parallel-aligned configuration the orientation of the liquid crystal materials can be modulated by the externally applied electric field. Furthermore, the nonlinearity of the system can be modulated. In this subsection, we analyze the effect of the biasing voltage through the ability to change the nonlinearity by modulating the orientation of the NLC molecules electrically and see the influence on the optical pattern formation phenomena.
Since the positive-dielectric-anisotropic NLC films are used the orientation of the liquid crystal directors is changed when the applied voltage exceeds the Freedericksz voltage Vth (in our case Vth = 0.9Vrms).
As an example, from Eq. (11) we plot the changing of θa with respect to the applied biasing voltage and is shown in Fig. 3.7(a). From Fig. 3.7(a), we can see that θa can be modulated by the applied biasing voltage.
Therefore, from Eq. (17) we plot α with respect to θa as shown in Fig.
3.7(b). From Fig. 3.7(b), we see that α initially increases with θa increasing and reaches a maximum when θa is about 0.9 rad.. However, when θa is larger than 0.9 rad. α decreases with θa increasing.
Furthermore, the changing behavior of α with respect to θa directly affects the threshold intensity distribution when θa is changed.
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 biasing voltage (V
rms)
av er ag e tilt an gl e q
a(rad)
(a)
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
0 0.2 0.4 0.6 0.8 1 1.2 1.4
average tilt angle q
a(rad) eff ective nonlinear coef fi cient a (cm
2/W )
(b)
Fig. 3.7 (a) The calculated curve of the average tilt angle θa versus the biasing voltage (b) the effective nonlinear coefficient α
versus the average tilt angle θa; with beam diameter=1.4 mm, d=68 µm, L=1.9 cm, R=0.65, input light power=0.91 W
azim uthal angle j (rad.)
th re sh ol d po w er (W )
qa=0.9 qa=1.05 qa=1.2
(b)
Fig. 3.8 (a) the calculated threshold power distribution for different θa; from the top to the bottom are θa = 0.6, 0.75 and 0.9 rad, respectively (b) the calculated threshold power distribution for different θa; from the bottom to the top are θa = 0.9, 1.05 and 1.2 rad, respectively; with beam diameter=1.4 mm, d=68 µm, L=1.9 cm, R=0.65, input light power=0.91W
Fig. 3.8(a) and 3.8(b) show the theoretical curves of the threshold power at different θa. The threshold power is obtained by the product of Eq. (17) and the beam area. In our calculation the beam diameter is assumed to be 1.4 mm. From Fig. 3.8(a) and 3.8(b), one can see that the threshold power distribution decreases with θa increasing when θa is smaller than 0.9 radian and increases with θa increasing when θa is larger than 0.9 radian.
Therefore, considering both the anisotropic distribution of the threshold intensity and the electric-modulated property of the effective nonlinear coefficient α an electric method to obtain different optical patterns can be expected. When θa is initially biased at a value smaller than 0.9 radian, one can input a light power larger than the maximum threshold (in our case the maximum locates at ϕ=00 and 1800) the hexagon is expected to be formed based on its stable and compact structure. This is similar to that in the isotropic case and we can expect the oscillation of three ripple patterns leading to hexagons formation. Decreasing the biasing voltage, which decreasing θa, results in the increasing of the threshold. Once the minimum threshold (in our case the minimum locates at ϕ=900) is tuned below and near to the input light power, only the mode with the lowest threshold power will be enhanced and the roll pattern should be observed.
On the other hand, when θa is initially biased larger than 0.9 radian one has to increase the biasing voltage to see the roll pattern.