Optical birefringence

Figure 6.1 and 6.2 show the correlation between light polarization and effective refraction index. If the polarization is parallel to the long axis of LC molecule, the extraordinary refraction index is available. If the polarization is perpendicular to the long axis of LC molecule, it would be ordinary refraction index. The polarization dependent effective refraction index leads a potential modulation on the optical constant of material.

Figure 6.1: Optical birefringence property of liquid crystal is dependent on the light polarization

Figure 6.2: Different angle between polarization and LC molecule lead to a variation of effective refraction index.

6.1.2 Diffraction property of cholesteric liquid crystal grating

Figure 6.3: With an application of a proper voltage, an ideal cholesteric liquid crystal grating is constructed in the CLC cell.

As a result of the helical twisting of the liquid crystal director, the dielectric property of the intrinsic CLC phase grating can be written [14]

⎟⎟ note the dielectric tensor is a periodic function of y. The helical twisting leads to a periodic variation of the dielectric tensor. The dielectric tensor in Eq. (1) can be conveniently written term representing an index grating which is responsible for the diffraction. Using Eqs.

(1, 2) we obtain

We now consider the incidence of a beam of polarized light along the z-axis.

The electric field of the incoming beam at the surface of the LC cell (z=0) can be written

where Eo is the amplitude, θ is the angle of polarization measured from the x-axis.

Inside the CLC medium, the incoming beam generates two modes of propagation at different speed. The electric field in the CLC medium can thus be written

o

where the first term represents an extraordinary wave Ee and the second term

represents an ordinary wave Eo.

To understand the diffraction property of the CLC grating, we consider the polarization (ΔP) produced by the electric field in the LC cell due to the presence of the CLC grating

Since ΔεEo=0, the ordinary wave is not interacting with the CLC index grating. The ordinary wave is polarized along the y-axis. The electric field Eo is perpendicular to the director of the liquid crystal, regardless of the twisting of the director. As a result, there's no index grating for the ordinary wave. The ordinary wave will propagate through the CLC medium unaffected by the CLC grating.

For the extraordinary wave, the diffracted wave (1st order) can thus be written (in Raman-Nath regime)

modulation of the index grating.

The diffraction efficiency (1st order) can thus be written

θ

It is noted that the diffraction efficiency proportional to the square of Bessel function of the depth of phase modulation results in that proportions to the square of the cosine of the orientation angle of the polarization state. The diffraction efficiency is predicted to decrease as θ increases. In the Raman-Nath regime, the diffraction efficiency of high orders can be written

θ

6.1.3 Diffraction pattern and diffraction efficiency of CLC grating

The CLC grating was probed by a Nd-YAG laser of 532 nm. The diffraction pattern in Figure 6.4 shows that the intensity of even diffraction orders is generally stronger than that of odd diffraction orders (up to m=4). The measured diffraction efficiencies are in good agreement with the results of our vector analysis. The value of δ was estimated as 9.91 with Δn = 0.1087, λ=532 nm, and d=7.7 μm. The Raman-Nath diffraction behavior was also verified due to the Q factor (^Q=²πλd nΛ²) of ~ 0.196, where the grating spacing Λ= 9 μm and the mean refractive index n = 1.627 [17, 18].

Figure 6.4: CLC grating was probed by a Nd-YAG laser of 532 nm, the calculated efficiency of ±4 diffraction order was shown in below.

In addition, we also investigate the diffraction efficiency of the CLC grating by using a polarized monochromatic beam in spectral range from 450 to 750 nm. In our experimental measurements, we employed a tungsten lamp and a monochromator.

The bandwidth Δλ of probe monochromatic beam was about 2.5 nm and the scanning rate was set 5 nm per step in the measurement. The diffraction signal collected in the optical detector was enhanced by using a locking amplifier.

Figure 6.5 shows the experimentally measured first-order (m=1) diffraction efficiency as a function of wavelength at the polarization angles of 0, 30, 60, and 90 degree (measured from the x axis). At the polarization angle of 0⁰, there are two peak diffractions at around 675 and 530 nm. The wavelength dependence and the oscillating behavior is a result of the Bessel function dependence on the modulation index

δ = 2 π Δ n

₁

d λ

.

Figure 6.5: Diffraction efficiency of the CLC grating probed by a polarized beam in spectral range from 450 to 750 nm.

The diffraction efficiency drops as θ increases. At the polarization angle of 90⁰, the diffraction efficiency should drop down to zero due to the vanishing of the

interaction with the birefringent index grating (according to Eq. (10)). However, in our measurement, a weak diffraction efficiency was still detected. This measured weak diffraction efficiency may be caused by a distortion of the CLC index grating at the boundaries where the liquid crystal molecules are usually strongly anchored by the alignment layer (SiO film).

Figure 6.6 shows the first order (m=1) diffraction efficiency as a function of wavelength at various polarization angles according to Eq. (10). The diffraction efficiency is a Bessel function of the modulation index δ which has a (1/λ) wavelength dependence. Peak diffraction occurs at 672 nm and 538 nm. The simple result of Eq. (9) assumes a uniform lossless CLC grating without considering the boundary anchoring and birefringence dispersion. The dispersion of ne and no as well as the distortion of CLC index grating at the boundaries are possible causes of the discrepancy. Further analysis is needed to include the effect of the molecular anchoring at the boundaries as well as absorption.

Figure 6.6: First order diffraction efficiency was calculated with respect to the polarization of 0, 30, 60, and 90 degree.

In summary, we have investigated both theoretically and experimentally the diffraction property of cholesteric liquid crystal gratings. Using a vector analysis, an

approximate calculation of the diffraction efficiencies for a polarized monochromatic beam can be obtained. The diffraction shows a strong wavelength dependence and a strong polarization dependence. A monochromatic light may be manipulated by light polarization and electro-optical effect of CLC material for optical communication field.

6.2 Surface plasmons effect induce by CLC grating environment

Collective oscillation of charge density in metallic nanoparticles known as localized surface plasmon resonance has been studied for nano-optics application. The development of surface plasmon-based nano-scale devices, such as mirror, beamsplitter, coupler and condenser, played an important role in nano-optical field.

Surface plasmons were coupled between metallic nanodots and that increased the conversion of the optical behavior in nano-scale systems [4-8]. However, the fabricating process of these nano-devices was complex and the devices could not be modulated externally. The function of external modulation on the nano-devices is required to be developed for flexible operation.

Surface plasmons are limited at the interface between metal and the dielectric material, when excited by the optical field. The surrounding dielectric property of the metallic material strongly affects the characteristics of the surface plasmons[9-11.

Liquid crystal materials, with the outstanding optical birefringence property, have been employed to cover metallic material as an electrically-switching modulating environment. Applying an electrical field on the cell, one can modulate the reorientation of LC molecule. Then the dielectric property between the metallic material and the environment could be modified. The surface plasmon resonance emerged on the metallic materials could be flexibly modulated by the applying voltage.

The enhanced diffraction in the cholesteric liquid crystal grating was fabricated and demonstrated. It was found that the first-order diffraction efficiency of the CLC grating was enhanced by the localized surface plasmon emerged on the silver nanopariticles. The enhanced diffraction efficiency leads to potential applications for optical switch devices.

6.2.1 Fabrication of CLC grating with Ag nanoparticles

The nematic liquid crystal E7 was doped with the chiral agent CB 15. Both E7 and CB15 were purchased from E. Merck. The nematic phase of E7 ranged from the temperature of -10⁰C to 60.5⁰C. The extraordinary and ordinary refractive indices (ne

and no) were 1.7354 and 1.5175, respectively. They were measured at the wavelength of λ=644 nm and the temperature of T=20⁰C. The helical twist power (HTP) of CB15 was +7.3 m^-1 at 20⁰C for E7. The cell gap was 6 μm. In this study, the CLC cell was constructed with d/p ~ 1 for good grating structure [12, 13], where d and p were the cell gap and the helix pitch of CLC material, respectively. In order to control the helix pitch of cholesteric liquid crystal, the weight percentage of CB15 was 2.28 %.

The CLC-grating device was assembled with two ITO glass plates. One was spin-coated with a polyimide layer and rubbed unidirectionaly for homogeneous alignment. The other was thermally deposited with a 40Å-thicked silver film. The silver film was nucleated by annealing the film at 250⁰C in 1 min to form the silver nanoparticles on the plate surface. For comparison, another CLC planar cell was assembled with two ITO plates. Instead of the deposition of the silver film on one plate, these two plates were spin-coated with a polyimide layer and rubbed in opposite directions for homogeneous alignment.

Figure 6.7: (a) and (b) show the SEM pictures of silver thin films before and after annealing. (c) The AFM picture of the annealed film shows the size of silver nano-particle is ~50 nm.

Figure 6.7 (a) and (b) show the scanning electronic microscope (SEM) pictures of the silver thin films on the glass plates before and after annealing, respectively.

With the application of the atomic-force-microscope (AFM) to probe the silver nano-particles on the substrate, the silver nanoparticles are approximately 50 nm in size. Figure 6.7 (c) shows the three-dimensional morphology of the silver film. The metal nano-particles in this scale can exhibit high localized plasma resonance.

Figure 6.8: (a) Cholesteric liquid crystal cell with silver nanoparticles distributed on one plate of the cell, (b) the CLC grating structure well-formed with the application of a proper voltage to the cell.

The prepared CLC material was injected into the empty cell to form a planar CLC device. The helix axis of the initial structure of CLC planar cell was parallel along the direction of z-axis as shown in Figure 6.8 (a). After applying a voltage of 2.7 V (frequency = 1 kHz) to the cell, the helix axis was reoriented to y-axis and the grating structure was well-formed in the cell as shown in the Figure 6.8(b). The periodic stripe pattern parallel to the x-axis was observed with an optical microscope.

The grating spacing inspected with the microscope was about 8 μm as shown in Figure 6.9. The grating was probed by a He-Ne laser of 632.8 nm, and the diffraction pattern was significant as shown in the insert of Figure 6.9.

Figure 6.9: Cholesteric liquid crystal grating structure was observed with an optical microscope and the diffraction pattern shown in the top (probed by the He-Ne laser of 632.8 nm).

6.2.2 Extra diffraction band induced by surface plasmon effect

The CLC grating was probed by a monochromatic beam, which was emerged from a monochromator. The white light source of the monochromator was a tungsten lamp. Lens and polarizer were used to collimate the light beam and to polarize the monochromatic light, respectively. The polarization of incident beam was set parallel to the rubbing direction. The first-order diffraction efficiencies were measured by an optical detector. The detector was set close to the test sample, thus the entire first-order diffraction could be covered by the large detector head (opening area:

1cm²). An iris was put in the front of the optical detector to exclude the other-order diffractions.

Figure 6.10: Experimental setup for measuring the first-order diffraction of the CLC grating.

The CLC grating cells were probed by the polarized-monochromatic beam in this study. The range of the probing wavelength was from 450 nm to 700 nm. The first-order diffraction efficiencies of the CLC gratings with respective to the wavelength were shown in the Figure 6.10. In addition, both samples presented oscillatory diffraction efficiencies, but apparent different diffraction efficiencies exhibited on the different treated CLC grating cells with or without silver nanoparticles distributed on the glass plate surface. Without the silver nanoparticles on the glass plate for the CLC grating cell, the distribution of high diffraction efficiencies was observed around 681 nm. The diffraction efficiencies attributed from the intrinsic CLC grating.

However, the distribution of the diffraction efficiencies of the CLC grating with the silver nanoparticles on the glass plate was distinct from that without silver nanoparticles. The distribution spectrum shows two peaks around λ = 654 nm and λ = 505 nm, respectively. At the long wavelength side around λ = 654 nm, the diffraction efficiencies in the spectrum behaved similarly to that without silver nanoparticles. The diffraction arose from the intrinsic CLC grating. The extra band of diffraction efficiencies around λ = 507 nm would contributed by the modulation of surface plasmons emerged from the silver nanoparticles.

Figure 6.11: The first-order diffraction efficiency versus wavelength of the CLC grating with and without the silver nanoparticles distributed on one plate of the cell, and the absorbance of the silver nanoparticles covered with E7 liquid crystal is shown in the insert illustration.

6.3 Theoretical model of periodic surrounding environment

It is noted that based on the theory of Raman-Nath’s diffraction, the first-order diffraction efficiency (η ) can be expressed as follows₁ ^{14, 15},

2 1 1

1 1

1 exp( 2 α d_s)J (2π n d λ i α d_s)

η = − Δ Δ − Δ (1) where the J1 is the first kind Bessel function, the ds is the thickness of the silver nanoparticle layer, the d is the thickness of the grating, and the λ is the wavelength.

The Δn₁ and Δα₁ are the modulation of refractive-index and the modulation of absorption of the grating, respectively. The diffraction efficiency is dependent on the modulation of refractive-index (Δn₁) and the modulation of absorption (Δα₁). In Eq.1, the diffraction efficiency is an oscillatory function of ^d λ. Oscillatory diffraction efficiency thus appeared in the wavelength range from 450 to 700 nm.

As for the extra enhanced diffraction efficiencies appeared around λ = 507 nm, we ascertained that the absorption characteristic due to the surface plasmons emerged from the silver nanoparticles on the glass plate of the CLC grating was determined the diffraction behavior of the grating. The absorbance of the silver nanoparticles in the

CLC grating cell was measured to estimate the modulation of absorption (Δα₁). The absorbance is defined as A = - log (T/T0), where the T and T0 were the transmittances of the LC (E7) cell with or without the silver nanoparticles on the glass plate. The cells were treated in the mode of homogeneous alignment. It is noted that there exists a broad absorbance bump around 500 nm as shown in the inserted illustration in Figure 6.11. The localized surface plasmon was excited on the sliver nanoparticles, and then caused the strong light scattering around 500 nm. These surface plasmons of silver nanoparticles led to the attenuation of the incident beam in the wavelength range around λ = 500 nm. The range of high absorbance corresponded to the enhanced diffraction band in the spectrum. The localized surface plasmon effect was considered to result in the extra diffraction efficiency.

Furthermore, the silver nanoparticles were covered by a CLC grating, the incident polarization light encountered different dielectric properties at the interface between the metal film and the LC media. The characteristics of surface plasmons of the metallic nanoparticles were strongly affected by the surrounding dielectric medium. The localized surface plasmons under the grating stripes would exhibit different spectral intensity distribution. Detailed analysis should consider that the absorbance generated by the CLC grating environment is dependent of wavelength.

A theoretical model was presented to demonstrate the extra enhanced diffraction efficiencies appeared around λ = 507 nm due to the silver nanoparticles in the CLC grating cell. In the model, the modulation of absorption (Δα₁) induced by CLC grating environment was estimated by the theoretical absorbance of silver nanoparticles. In the experiment, the CLC grating with a spatial periodic refraction index distribution covered on the silver nanoparticles. It was assumed that the silver nanoparticles spatially experienced the environments with refraction indices of 1.74 (n) and 1.52 (n ), respectively. TheΔα was the difference between the absorbance

spectra with respect to the n=1.74 and n=1.52. The calculation of absorbance and the diffraction efficiency of CLC grating without and with silver nanoparticloes were discussed as follows.

By heating silver film, the oblate spheroidal shape silver nanoaprticles were formed to lie on ITO substrate [17]. One neglected the interaction among spheroids, the total cross sections of silver nanoparticles in s- and p-polarization are given as follows [17-19] where the^ε

( )

^ω is the complex dielectric response function of material, the a is the focal length of the spheroid, the θ is the angle between the incident light and x-y plane, as shown in the insert of Fig. 5. The parameter of η₀=R (1−R²)^1/2 determines the shape of the spheroid, the R is the ratio of the minor axis to the major axis of the spheroid. The ε_1,0^{, and} ε1,1, are the responses to polarizability component. And the

0 ,

Q , and 1 Q are the associated Legendre function of second kind, respectively. They _1,1 are derived from the solution of Laplace’s equation in oblate spheroid coordinate and

one dipole approximation. The detail derivation is referred to the previous work of T.

L. Ferrell et al. [17-19].

Based on the information of total cross section, the optical absorbance A of silver nanoparticles can be written as the following expression

⎟⎠

where the N is the density of spheroids in a unit of surface area. The value of N can be determined by an analysis of scanning electron micrograph. So far, the absorbance of silver nanoparticles can be determined by the particles size, shape, and density. In addition, the environment effect on surface plasmons was important to the absorbance property of silver nanoparticles.

Since the silver nanoparticles layer was covered by top CLC grating and bottom ITO substrate, the environment effect is significant in the absorbance calculation. We assumed that the ITO substrate and the CLC grating were a thin uniform dielectric film to cover the silver nanoparticles. The following dispersion relation was applied to modify the absorbance property affected by environment effect [20].

( ) ( )

^{( )} 0

( )

₂ 0

( )

1,1 1,0 ⁰ environment, respectively. The optical constant of element was found in Palik’s work and the ε_s was the average dielectric function of nematic liquid crystal and ITO substrate [21]. The substitute values of ε_1,0^andε1,1 were obtained from the solution of this quadratic equation to modify the absorbance characteristics of silver nanoparticles.

Figure 6.12: Theoretical absorbance of silver nanoparticles surrounded by the environment with refractive indices of 1.52 and 1.74. The insert shows the silver nanoparticles layer was posited between the CLC grating and ITO plate. The particles experienced the environments with index of ne and no.

Figure 6.6 shows the theoretical absorbance spectra of silver nanoparticles encounted by the environment with the refractive indices of 1.52 and 1.74. The insert presents the silver nanoparticles layer are posited between the CLC grating and ITO plate. The particles experience the environments with index of ne and no. The incident angle (θ), ratio of the minor axis to the major axis (R), and focus length (a) are 0⁰, 0.6, and 40 nm, respectively. It was noted that the absorbance was a function of wavelength. The maximum absorbance occurred at around 476 and 500 nm due to the refraction index of 1.52 and 1.74, respectively. The difference in the optical absorbance spectra of silver nanoparticles led to an amplitude modulation of ^Δα₁ occurred in the CLC cell. Since Δα₁ is strongly dependent on the wavelength and that is the cause of the extra band around 500 nm in diffraction efficiency.

Figure 6.13: Theoretical diffraction efficiencies show the extra diffraction band of CLC grating with silver nanoparticles.

Figure 6.13: Theoretical diffraction efficiencies show the extra diffraction band of CLC grating with silver nanoparticles.

在文檔中 金屬奈米粒子表面電漿效應之研究 (頁 102-0)