Linear and Nonlinear Optical Studies of Liquid Crystalline Materials
By
Ali S. Alshomrany
B.S., Umm Al-qura University, 2000
M.S., Ohio University 2006
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirement for the degree of Doctor of Philosophy
Department of Physics
2013
This thesis entitled: Linear and Nonlinear Optical Studies of Liquid Crystalline Materials written by Ali S. Alshomrany has been approved for the Department of Physics
Noel A. Clark
Joseph E. Maclennan
Date
The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.
Alshomrany, Ali S. (Ph.D., Physics)
Linear and Nonlinear Optical Studies of Liquid Crystalline Materials
Thesis directed by Prof. Noel A. Clark
The electro optical response of liquid crystalline materials while displaying anisotropic properties, has led to numerous optical applications.
A retro-reflecting modulator for optical communication systems using a high numerical aperture objective lens and a planar aligned nematic liquid crystal electrooptic cell that tunes the optical path length in the liquid crystal layer as a function of applied voltage has been developed.
The field-induced phase shift varies the dependence of liquid crystal layer reflectivity in angle of incidence . This dependence can be studied in-situ by performing positional scan, z, along the axis of symmetry of the lens scan, focusing the light with an aberrative lens such that depends on z and measuring the reflectivity in the far field. These data show that a low reflected state can be achieved when liquid crystal thickness such that destructive interference with 휃 ≈ 0 and high field “hometropic” orientation.
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Acknowledgement
First, I thank God for giving me the power and the health to complete the work of this thesis.
I would like to thank my advisor, Noel Clark, for his outstanding support and guidance over the last six years. His enthusiasm and creativity has motivated all the work in this dissertation. I would also like to thank my committee members, who in addition to Noel include
David Walba, Chuck Rogers, Joe Maclennan, and Matt Glaser for their beneficial discussions.
I am thankful for my interaction with the entire liquid crystal materials research center members: Ivan Smalyukh, Robert Blackwell, Dong Chen, Art Klittnick, Zach Kost-Smith,
Taewoo Lee, Cheol Park, Zhiyuan Qi, Renfan Shao, Yongqiang Shen, Yue Shi, Greg Smith, Eva
Korblova, Chenhui Zhu, Zoom Nguyen, Jacquie Richardson, and Maria Kolber.
Additional thanks to the liquid crystal office staff for their supporting cast of assisting especially Annett Baumgartner.
I am extremely grateful to my family for their emotional and immense support.
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CONTENTS
Chapter
1. Introduction 1
1.1. Liquid crystal phases……………………………………………………………………..1
2. Liquid Crystal Electro-Optic Retroreflection…………………………………………………3
2.1. Introduction………………………………………………………………………………3
2.2. Experimental setup and results…………………………………………………………..3
3. Advance Conoscopy Using Fisheye lens…………………………………………………….20
3.1. Introduction……………………………………………………………………………..20
3.2. Conoscopy with Fisheye lens……………………………….…………………….…….20
3.3. Lens characteristic………………………………………………………………………25
3.3.1. Viewing angle…………………………………………………………………....24
3.3.2. Intensity calibration……………………………………………………………...27
4. Second Harmonic Generation Imaging Microscopy of B4-8CB Mixture……...……………33
4.1 Introduction………………………………………………………………………………33
4.2 The SHG confocal laser scanning microscope………………………………….……….33
4.3 B4-8CB Mixture…………………………………………………………………………35
Bibliography…………………………………………………………………………………40
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Chapter 1
Introduction
1.1 Liquid crystal phases
The liquid crystalline phase is a distinct mesomorphic phase of matter that exhibits certain degrees of molecular ordering in the range between that possessed by a crystalline solid and an isotropic liquid. In this mesomorphic phase, the molecules act like a viscous fluid but maintain long range orintational order which allows them to show properties associated with crystalline solid.
Certain chemical structure units (mesogens) are required in order for substances or compounds to form mesophases. The most suitable geometry is a long, narrow, rod-shaped molecule as shown in Figure 1.1. Disc-like and bow-like geometries also tend to form liquid crystal mesomorphic phases. Different thermodynamic mesomorphic phases of these mesogens may coexist at different (phase transitions) heating or cooling temperatures of the same mesogen.
The classification of these phases depends on the macroscopic symmetry as well as the orintaional and positional order of the molecules. G. Friedel [1] proposed the two basic types of mesophase, nematic and smectic.
In Nematic liquid crystals the molecules, N, possess a quasi-long range uniaxial orientational order of the long geometrical axis of the molecules with respect to (a local axis of symmetry) the average director, n, of the individual long axis of the mesogens, but no lack any positional order as illustrated in Figure 1.1a [2, 3].
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Smectic liquid crystalline phases, Sm, exhibit a higher degree of order similar to that of a crystal. In addition to quasi-long range uniaxial orientational order, there is long-range positional
(translation) order of the long geometrical axis of the molecules [2, 3]. Thus, the predominant structure of semcitcs is consistent with stacks of layers, each of which is a two dimensional liquid. These phases are classified in accordance with the molecular arrangement within each layer. Some of the most common phases are Smectic A, and Smectic C phases. In Smectic A, Sm
A, phase, the average director, n, of the individual long axis of the mesogens are orthogonal to the layer plane as illustrated in Figure 1.1b, whereas in the Smectic C, Sm C, phase, the average director, n, are tilted by an angle, , with respect to the layer normal as shown in Figure 1.1c.
Figure 1.1: an illustarion of the common liquid crystal phases (a) Nematic, (b) Smectic A, and (c) Smectic C.
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Chapter 2
Liquid Crystal Electro-Optic Retroreflection: Generating Conical Illumination with High
Angular Resolution Using the Spherical Aberration of a High Numerical Aperture Microscope
Objective
2.1 Introduction
There are possible uses of liquid crystal electro-optics in situations where the dynamic modulation of a retroreflected laser beam is used to transmit information in a remote sensing application. The coincidence of the focal surface of a refractive element, such as ball lens or beads, with a liquid crystal thin layer of thickness 1 m < t < 20 m serves as a retroreflector when the refractive index of the LC layer, nlc, is tuned between the refractive index of the refractive element, nre, and nre < nlc < nre [4].
In order to explore and develop relevant geometries we have investigated the reflection characteristics of an electro-optic geometry having a liquid crystal layer as the thin film dielectric in a transparent capacitor made of closely spaced glass plates coated with transparent electrically conducting indium-tin oxide films.
2.2 Experimental setup and results
The basic test setup, shown in Figure 2.2.1a, has a collimated 632.8 nm laser beam focused by an infinity corrected 100X microscope objective (NA=0.8) onto various optical geometries, with the reflected light collected effectively at infinity by a small area (2 mm2) amplified Si detector, D2.
The objective employed [Olympus LMPlanFlI] is engineered for imaging at the air sample
3 interface, producing a diffraction limited focus of the incident beam on a glass surface, for example Figure 2.2.1b. Measurement of the reflected intensity on D2 during positional scanning of the objective parallel to the beam (z-scanning) provides a measurement of I(z) the depth of focus of the diffraction limited beam, as shown in Figure 2.2.1b. The focal peak is found to be characterized by a full width at a half height of 1.619 µm, and in Figure 2.2.1b is compared to the expectation for the focusing of a Gaussian beam in the 1.4 m [4] approximation thus an assessment of quality of focus yielded by the objective. This difference in the axial optical resolution is based on the assumption that the wavefront of the illumination light is free from any distortion and its intensity is uniform in the lateral direction, x and y.
Figure 2.2.1: Schematic diagram of the optical reflectometer. (a) A monochromatic linearly polarized light from He-Ne laser is spatially filtered by a pinhole and then collimated through a beam splitter which directs the collimated light into two separate beams, one of which passes through a microscope objective designed to focus collimated light to a diffraction limited focal spot in air, and the other which is used as an intensity reference. The sample position along the beam axis (z) can be scanned using a motorized drive with 50 nm repeatability and precision. Light focused by the objective is reflected by the planar interfaces of the sample cell (oriented normal to z), and that part of the reflection that passes back through the objective and is sufficiently parallel to the z axis is reflected by the beamsplitter through a distant (3 m away), 2 mm diameter aperture to be detected by silicon photodiode D2.
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However useful LC electro-optic geometry requires LC to be between solid plates enabling electric field application and alignment of the LC, in which case the probe light must pass through some thickness of glass or other optical material. If an objective corrected to focus in air
Figure 2.2.1b is used then its behavior when focusing through glass will be affected by aberration. This produces classic features of a degraded focus including an asymmetric distribution of on axis intensity, as shown in the calculated I(z) in the inset of Figure 2.2.2c for illuminating the air glass interface of a specimen with refractive index n=1.33 and thickness 50
m sandwiched between cover glass of thickness 170 m and refractive index n=1.54 by objective with numerical aperture NA=0.9 [6].
Figure 2.2.1: Snapshots of the geometry and the notation used to describe the axial z-scan of the focal spot through a slab of dielectric, where z gives the relative separation of the objective and sample. (b) The scan through z=0, where the air-glass interface lies in the plane of the diffraction-limited focus of the unaberrated rays gives a sharp single reflection peak at z=0. (c) For z > 0 spherical aberration causes the focal distance in the glass plate to depend on the angle of incidence a, where the rays with larger a focus further into the glass. (d) As z is scanned
5 through z=350 µm, where the aberrated focal plane of the (peripheral) ray of largest a intersects the planar back face of the glass a secondary reflection peak is detected in D2. (e) As z is further increased the value of a producing a back face reflection decreases, until (f) at z(a=0), on-axis rays are reflected. (h) In this range of a a broad reflection band is observed, having a series of reflection peaks, a maximum near a=0 where spherical aberration is minimum, and an exponential tail toward increasing a. (g) Angular dependence of a on z for nglass=1.52 of maxima and minima that repeatedly appear close to the main peak at z=450 m.
The geometrical optics description of this ascribes the aberration to the nonlinearities inherent in
Snell's law, which will cause the rays perfectly focused in air to intersect the system axis in glass at distances increasingly further from the lens as the angle of incidence in air, a, is increased. If z measures the distance between the first glass interface and the position of the focal spot if the glass were removed, then the crossing point is a distance tg(z) into the glass for an angle of incidence a(z, tg), where
( )
휃 √ (2.2.1)
( ) [ ] is plotted in Figure 2.2.1g. If the glass is of finite thickness, tg, introducing a second reflecting interface a distance tg into the glass, then incident collimated light at the angle of incidence a(z, tg) will produce reflected collimated light, as sketched in Figure 2.2.1e. As z is increased from z=0 the first rays to be so reflected will be those for the largest a, amax, when z=z(amax), as in
Figure 2.2.1d. As z is increased further the a for collimated reflection will decrease until z=z(a=0)=tg/ng, where the on-axis rays focus. Figure 2.2.1g plots a(z) vs. z from equation
(2.2.1), for the range of a corresponding to our objective. Figures 2.2.1h and 2.2.2b shows
Nglass(z), the corresponding collimated reflected intensity from the glass collected on detector D2, normalized such that the front surface reflection has unit peak intensity, i.e., N(z=0)=1. The
6 aberrated reflected intensity is largest near a= 0 where the small angle rays focus and decreases approximately exponentially with decreasing z until a~amax beyond which angle the light is cut off by the objective aperture. Also shown are N(z) for the same glass plate with either indium tin oxide (ITO) or ITO and polyimide (PI) coatings on the back interface. Since these are of thickness much smaller than the wavelength (tITO=44 nm and tPI=17 nm), their N(z)’s differ from that of the glass only by a a -independent factor. For a small and a~amax Airy-like fringes appear in Nglass(z) as a result of refractive effects [10].
Figure 2.2.2: (a) Schematic arrangement for retroreflection from an air-gap cell, yielding N(z) in Figure 2.2.2.e below.
The resemblance of the calculated I(z) and Nglass(z) suggests that measuring Nglass(z) is an effective way to probe I(z). In order to test this proposition and as a prelude to the study of liquid crystal cells we introduced a third interface to make an air-filled cell of thickness tcell4.5 µm, as shown in Figure 2.2.2a. This cell becomes a low finesse multiple beam interferometer with a path difference for interference that depends on g=(1/ng)sinathe angle of incidence in the glass
Figure 2.2.1e. Ncell(z) measured for the glass cell, plotted along with Nglass(z) in Figure 2.2.2d, shows distinctive interference minima for z in the range 350 µm < z < 470 µm, corresponding to
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a in the range 50º > a > 0º. The reflected intensity ratio Rcell(z)=Ncell(z)/Nglass(z), plotted in
Figure 2.2.2d, exhibits the following features: (i) constructive interference maxima of R(z)4 for
a < 50º; destructive interference minima of R(z)~0.05 for a < 50º; reflection without interference from two glass surfaces giving R~2 for z > 470 µm, where focus is not possible.
For obliquely incident parallel monochromatic light this air gap cell should have a reflected intensity ratio (휃 ) given by the intensity reflectivity interference
2 2 function of the cell, r cell, divided by the glass/air Fresnel reflectivity, r g/a,
( ) (휃 ) (2.2.2) ⁄ ( ⁄ )
where r(a) is glass/air Fresnel amplitude reflection coefficient, is the optical phase difference and is given by
(2.2.3)
Here m is the interference order which depends on the wavelength of the laser λ, the index of refraction of the gap ncell, the thickness of the gap tcell, and the scanning position z. For the phase change on reflection at oblique incidence through the gap the interference order is obtained from the relation
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Figure 2.2.2: (b) Nglass(z), normalized retro-reflectivity of single uncoated glass (tg=724 µm, ng=1.55), and of similar plates coated on the back side as indicated. The coatings substantially change the magnitude of N(z), but not the shape. (c) I(z), on-axis intensity vs. scan position z, obtained using the diffraction integral representation calculation [6, 7, 8, 9] for an on-axis point 250 µm deep into an ng=1.55 glass plate and a uniformly illuminated objective with numerical aperture NA=0.9. Comparison with (b) shows that N(z) is proportional to on-axis intensity at the reflecting interface. (d) Normalized retro-reflection Ncell(z) for a tcell=4.5 µm glass air-gap cell plotted with Nglass(z), showing the interference effect of multiple reflection between the back and front interfaces of the air gap. At constructive interference Ncell(z)/Nglass(z)4, and for z > z(a=0), where there is no longer interference Ncell(z)/Nglass(z)2, both as expected. (e) Plot of R(z)=Ncell(z)/Nglass(z) fitted to the interference function, Rcell(z). (f) Fit to Rcell(z) convolved with a flat-topped distribution of width a, showing that the angular width of the cone of light selected in the retroreflection geometry is very narrow: a~0.8º.
( ) √ ( ) [ ] (2.2.4)
( )
with the zeros in Rcell occurring for integer values of m.
The black curve in Figure 2.2.2e is a least squares fit of Equation (2.2.2) to the Rcell(a), showing that this basic interference function depending on a single selected angle of incidence fits the data well in the range of z where there is focus of some selected ray at a(z). That is, it appears that a(z), the range of angles of incidence selected in the aberrated focus at a(z) by the detection of collimated reflected light is much narrower that the range corresponding to that of the fringe spacing. The effective range of angles selected am in the vicinity of interference minimum m can be probed quantitatively by comparing the Rcell(z) data of Figures 2.2.2d and e in the vicinity of the minimum to a model in which there is a distribution of incident angles about
a(z). For this purpose we performed convolutions of the calculated Rcell(a) in Equation (2.2.2) with the flat-topped distribution: D(a)=1/a for -a/2 < a < a/2, D(a)=0 for a <-a/2 and a > a/2. Sample convolutions R(a) D(a) in the vicinity of the m=10 minimum are 10 shown in Figure 2.2.2f for several a values along with the measured extinction. Least squares fit of the convolution to the m=10 data (dashed curve) gives a10 =0.8º. This value shows that the angular selection of cone angle of illumination in the retroreflection of aberrated light using the Olympus 100X objective is quite narrow.
The reflectivity data of Figure 2.2.2e and the fit Rcell(a) are shown again in Figure 2.2.3b, along with a plot of the dependence of the interference order m on z, ncell, and tcell, which determines m(0)m(a=0). For the air gap cell tcell4.5 µm and ncell=1 give m(0)=14.6, and minima in the a in the range 50º > a > 0º for m =10, 11, 12, 13, and 14, as indicated by the construction of violet lines in Figures 2.2.3 a and b.
Figure 2.2.3: (a) Liquid crystal electro-optic retro-reflector geometry, along with the structure of 5CB, the room-temperature nematic liquid crystal used.
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Figure 2.2.3: (b) Sketch of the planar-aligned LC cell having the average molecular long axis, n, parallel to the plates and horizontal. Also shown are z-scan plots of normalized retro-reflected intensity for incident polarization parallel to n, N0V(z) and N6V(z) for v=0V and the field aligned cell with v=6V, respectively. These N(z) curves show multiple beam interference fringes in the scan range 0º < a < 50º. Maxima and minima of N(z) are observed at scan positions separated by z~50 µm for which the path difference between light reflecting from the front and rear LC/cell interfaces satisfies constructive and destructive interference conditions, respectively. Cell conditions here are chosen such that a scanning position exists, in this case at z=465 µm, where N0V(z) has a maximum reflection peak and N6V(z) has a destructive interference minimum, leading to the possibility of high contrast voltage-modulation of the intensity of the retro-reflected light for this z.
This aberrated illumination geometry was applied to the investigation of the retrorefletion characteristics of planar aligned nematic liquid crystal electro-optic cells, in the geometry shown in Figures 2.2.3a and 2.2.3b. The nematic LC 5CB was introduced into a gap between glass plates coated on the gap faces with the ITO electrodes mentioned above to make a transparent capacitor with the LC as the dielectric. The electrodes were also coated with the rubbed PI layer
12 to make a planar-aligned cell, having the nematic director n(r), the local average mean molecular long axis orientation nearly parallel to the plane of the gap surfaces and parallel to the incident optical polarization, as shown in Figure 2.2.3b. Rubbing of the PI films orients n at the surface along the rubbing direction with a pretilt angle of 3º relative to the surface. Antiparallel rubbing on the two cell plates then gives a uniform orientation of n(r) with (r)=3º, where (r) is the angle between n(r) and the x-y plane. Application of an AC voltage (peak-to-peak amplitude, v) across the ITO electrodes produces a torque on n due to the dielectric anisotropy of the LC, inducing a further increase of (r) throughout the cell. Since the cell is homogeneous in the x-y plane, with a field applied (r) can be written as (s), depending only on s, the distance into the
LC from the front cell surface. At high voltage the orientation of n is nearly along the applied electric field, normal to the cell plates ((s)90º @ v=6V). The orientation in the cell center,
c=(s=tcell/2) is shown vs. applied voltage in Figure 2.2.3e.
Figure 2.2.3b presents the normalized cell reflectivity z-scans N(z) obtained at the limits of planar orientation (v=0V), and field-induced homeotropic orientation (v=6V), showing that
N(z) depends strongly on applied voltage. At v =0V the maximum of the reflectivity peak is
N(z=465 µm)~0.1, which is much smaller than that which would be obtained from an empty glass/ITO/PI cell (N(z=465 µm) ~1 from Figure 2.2.2d, a result of the much smaller reflectivity at the glass/ITO/PI – LC interface than the glass/ITO/PI – air interface. The incident light is polarized parallel to the x-z plane which, with n also parallel to x, is a plane of mirror symmetry of the cell. As a result the light is polarized on average parallel to n and is only weakly depolarized by the LC birefringence, and the electro-optic effect is due principally to varying n and multiple beam interference between reflections at the cell surfaces as in the air gap cell. The field dependence of the refractive index in the cell substantially alters the variation of N(z) with
13 field, and, in particular produces strong variation of N(z) at certain values of z. For example, setting z465µm, near the reflectivity maximum at a=0, yields high contrast variation of N(z) as interference minima in the vicinity are tuned through by varying v, as shown in Figure 2.2.5e.
The field dependence further explored by tracking the evolution of N(z) vs. applied voltage in the full range from planar to saturation as shown in Figure 2.2.3. In the low voltage regime the
N(z) possesses a maximum reflection at peak position z=465 m correspond to v=0 V with the nematic LC director n is parallel to the x-y plane. As the applied field increased in the vertical direction, the Nematic LC director n reorients in the field direction, resulting in a dispersion of
n, the optical birefringence of the uniaxial nematic LC. Maximum contrast ratio of N(z) was obtained at the peak position, z=465 m, when the applied voltage tunes the optical birefringence to a value that satisfies a distractive interference condition at v=0.83 V. The high voltage regime, in the range 2.15 V – 6 V, results in optical response identical to what obtained when the voltage scanned in the low voltage regime.
The spatial distribution of the multiple beam interference fringes order of the experimental data is predicted using equation (2.2.4) as plotted in Figure 2.2.4a. The state corresponds to optical birefringence of the uniaxial nematic LC, n=1.54, possess three minima at m=22, 23, and
24 as shown by the black curve in Figure 2.2.4c and the spatial distribution of the minima is predicted theoretically by the black curve in Figure 2.2.4a. When the optical birefringence of the uniaxial nematic LC tunes to, n=1.7, at saturation voltage (blue curve of Figure 2.2.4c), v=6 V, the three minima shift to a new interference order m=20, 21, and 22 and this shift the spatial distribution on the z scan axis as plotted in Figure 2.2.4a (blue curve) to satisfy the condition of multiple beam interface.
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In a summary, a low reflected state can be achieved when liquid crystal thickness such that destructive interference with 휃 ≈ 0 and high field “hometropic” orientation.
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Figure 2.2.3: (c,d) Multi-z-scan plot showing the interference patterns and their voltage dependence, NV(z), for a planar aligned nematic LC 5CB cell with voltage in the ranges 0 V < v < 0.83 V and 2.15 V < v < 6.0 V. Also plotted is the coated plate reflectivity Ncoated(z) for 16 reference. The ratio R(z)=NV(z)/Ncoated(z) ~4 for constructive interference, as was the case for the air-gap cell. (e) Inset showing numerical calculation of , the field-induced rotation of n from planar alignment in the midplane of the cell. (c) The electro optic response of NV(z) for increasing the applied voltage amplitude from v=0 V to v=0.83 V, the regime where is small. Increasing voltage produces a continuous shift in the interference minima toward smaller a. (d) The electro optic response of NV(z) for increasing the applied voltage amplitude from v=2.15V to v=6.0 V, the regime where ~90º. Increasing voltage produces a continuous shift in the interference minima toward smaller a. The dependence of NV(z) on V is very similar in (c) and (d) because a principal axis of the optical dielectric tensor is nearly along the normal to the cell plane, z, in both cases (see text).
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Figure 2.2.4: (a) Graphical representation of the dependence of a and of interference order, m(a(z)), on scan position, z. Here m is defined such that integral m(a) corresponds to destructive interference and m(a=0)=(2ncelltcell)/(a) Calculated a(z) and of m(a) for the three 18 relevant cases of ncell, tcell: air gap (purple); 5CB LC gap @ v=6.0 V (blue); 5CB LC gap @ v=0 V (black). (a,b) The extinctions (white circles) for the air gap case are projected up onto (b), which match accurately with air N(z) data. (c) Black curve: R0V(z)=N0V(z)/Ncoated(z); blue curve: R6V(z); white curve: Rcell(a(z)) calculated from Eq. 2.2.2. The extinctions (white circles) for the 5CB LC gap cases are projected up onto (c), giving the minima for the white curves which, for ncell=ne =1.70 @ v=0 V and ncell=no=1.55 @ v =6.0 V, respectively match the R0V(z) and R6V(z) data well. (d) Comparison of the v=0 V (black curve) and v=2.15 V (gray curve) cases. These both have half-integral order m at a=0, and a principal optic axis nearly along z (~0º and m =24.5 for v=0 V and ~90º and m=22.5 for v=2.15 V). Also shown is a comparison of the v=0.83 V (black curve) and v=6.0 V (gray curve) cases. These both have integral order m at a0, and a principal optic axis nearly along z (~0º and m =24 for v=0.83 V and ~90º and m =22 for v=6.0 V), providing a way to adjust ncell and tcell for maximum electro-optic contrast. (e) On-axis (a=0) reflectivity vs. applied AC voltage, showing that the full voltage range scans m through 2.5 interference orders multiple beam interference fringes order.
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Chapter 3
Advanced Conoscopy Using a Fisheye Lens
3.1 Introduction
Conoscopy [11, 12] is a long established method for observing the conoscopic image, interference patterns, produced by anisotropic crystals under the polarizing light microscope. It is based on the observation of the anisotropic crystals between crossed polarizer and analyzer with large converging light. The resulting interference patterns in this method of observation are used to determine the orientation as well as the optical properties of the optical anisotropy which is very useful in the field of liquid crystals. For example, the tilt angle of the liquid crystal director of a nematic liquid crystal may be determined from the observation of the conscopic interference pattern [13].
3. 2 Conoscopy with fisheye lens
In order to view conoscopic image through the polarizing microscope, an Amici-Bertrand lens is inserted into the optical system between the analyzer and the ocular of the microscope. As a result, the focal plane of the ocular and eye lens system is brought to a focus on the interference pattern formed in the back focal plane of the objective and produces a virtual image. An example of interference figures obtained with uniaxial and biaxial crystals placed between crossed polarizer and analyzer under the polarizing microscope is shown in Figure 3.2.1 a-c. The different propagation directions of the light rays through the sample lead to sequence of interference colors in the conoscopic image when a white source of light is used.
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Figure 3.2.1: Conoscopic images under white illumination for uniaxial film (a), biaxial crystal film (b-c) between crossed polarizer and analyzer [14, 15, 16].
Effective conoscopic image can be observed through the polarizing microscope only with a wide angle cone of rays to be passed through the object; hence, a high numerical aperture objective lens and condenser are used.
Fisheye lenses [17, 18] provide a super wide field of view over 180o with large depth of field and accurate angular resolution. Therefore, it can be utilized to observe and examine conoscopic images with high optical image quality. Figure 3.2.3b illustrates the experimental arrangement under which conoscopic image can be observed with this approach. The optically anisotropic crystal plate between crossed polarizer, P, and analyzer, A, is placed between the diffuser and the fisheye lens. The beam from a high intensity source (blue laser) is passed through a crossed polarizer and analyzer after it has been attenuated by a neutral density filter.
An illuminated diaphragm, white sheet of paper, is used to cover a suitably area of a diffuser
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Figure 3.2.3: Experimental setup for fisheye conoscopy in a laboratory setting. (a) Arrangement of the experiment to perform the viewing angle measurement. (c) The inset sketches the essentials of the fisheye lens optics, where is the angle of incidence in air relative to the lens axis, and x() is the corresponding imaging radius from the lens axis. (b)The sample, polarizing and analyzing sheets are in the plane normal to the lens axis. In order to achieve the minimum required sample diameter (~21mm in this case) sample is in contact with the analyzing sheet which is, in turn, in contact the front rim of the lens. The peripheral limit of for the Olloclip lens is max=75º (NA=0.97). plate with scattered light, situated just above the diaphragm. The diffuser prevents any image of the light source iris being formed in the object plane and scatters light in all direction. The optical system of the fisheye lens collects the transmitted light from the analyzer above the sample and brings each light bundle to focus on the upper back focal plane of the fisheye lens,
CCD, as shown in Figure 3.2.3c. Each point in the back focal plane field is associated with a particular propagation direction of the illuminating light rays. The point in the center of the field corresponds to light rays traveling parallel to the axis of symmetry of the lens while the points at
22 the edge of the field corresponds to light rays traveling with wide angles relative to the axis of symmetry of the lens.
Figure 3.2.4a-d illustrates examples of conoscopic images obtained through the fisheye lens system for three biaxial PET films when the observations were carried out in white light, showing the typical decreased fringe spacing and color contrast with increasing the film thickness. The isogyres form a cross black brushes when the trace of the optic axial plane lies parallel to either the polarizer or the analyzer as shown in Figure 3.2.4c. Upon rotation of the film from this extinction position the cross splits into two hyperbolic brushes which are centered on the melatops as shown in Figure 3.2.4a-b.
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Figure 3.2.4: Example white light conoscopic images, N(x,y), of biaxial PET films of different thickness (a-c), showing the typical decreased fringe spacing and color contrast with increasing t. Angle of incidence =75º at the circular periphery of the patterns. Panels (a) and (b) also show the corresponding N(x,y) for 445 nm laser illumination. (d) Conoscopic image of the lamination of two biaxial films.
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3. 3 Lens characteristic
3. 3. 1. Viewing angle
In order to explore and investigate the angular resolution of the fisheye lens system a monochromatic light illumination source, green laser pointer, with radius of 1 mm is shined parallel to the axis of symmetry of the lens from a distance = 30 m allowing the beam light to diverge and cover the entire field of view of the fisheye lens with a uniform light as shown in
Figure 3.2.3 a. The optical fisheye lens system is then rotated by a rotational stage mount with angle in a step of 10 with respect to the optical axis of the lens. Mapping of pixel position of the focused incident light rays in the x-y image plane coordinate system of the CCD during rotational scanning of the lens perpendicular to the beam provides a measurement of the intensity level of the green laser, N(x). The focal peak is found to be characterized by a full width at a half maximum FWHM of 0.3 and 0.7 for angle of incidence 0 and 65 respectively as shown in
Figure 3.3.1.1a. Angular scan of N() of a white LED point source positioned at the same distance 30m from the lens showing the angular resolution and chromatic aberration of the fisheye/iPhone imaging with the beam incident at =25º and at =65º as shown in Figure
3.3.1.1b. Red and green wavelengths focus at the same pixel coordinate whereas the blue focus is shifted to smaller angle by ~0.5º at =65º.
Mapping of pixel positions, x, of the focused incident light rays in the x-y image plane coordinate system of the CCD during rotational scanning is plotted in Figure 3.3.1.1c for both
523 nm light (solid green circles) and white LED source (open circle). These data strongly overlap in the range -75o < < 75o, along with the fitting function x( solid lines for the RG (red line) and B (blue line) wavelengths of the LED light. The fit shows that the Olloclip
25 fisheye/imager combination is nearly orthographic, i.e. x()sin. The best theoretical fit of the experimental data is giving by the function
Figure 3.3.1.1: (a) Angular scan of N(θ) of a 532 nm beam from a laser positioned 20 m from the lens showing the angular resolution of the fisheye/iPhone imaging with the beam incident on- axis (θ=0) and at θ=65º. The focal spot full width at half maximum (FWHM) is less than 0.5º for most of the angular range. (b) Angular scan of N(θ) of a white LED point source positioned 30 m
26 from the lens showing the angular resolution and chromatic aberration of the fisheye/iPhone imaging with the beam incident at θ =25º and at θ=65º. Red and green wavelengths focus at the same pixel coordinate whereas the blue focus is shifted to smaller angle by δθ~-0.5º at θ=65º.
휃 (휃) [ ( )] (2.3.1.1) 0
where is angle of incidence, β is a fitting parameter, xmid is the mapping of the x coordinate in the x-y image plane coordinate system of the CCD at angle of incidence and is given by
xmid=(xmax+xmin)/2, xmax is the mapping at and xmin is the mapping at and x=( xmax-xmin)/2.
3. 3. 2. Intensity Calibration
The gradual rotation of the analyzer in the experimental setup of Figure 3.2.3 b by angle from the initial crossed position with no sample or polarizers above the diffuser provides measurements of the diffuse illumination response, the pixel blue level N(x,y), for 445 nm blue laser light as shown in Figure 3.3.2.1a. As seen in Figure 3.3.2.1a, the diffuse illumination response, N(x,y)=0 at extinction angle of the polarizer/analyzer, and increases with uncrossing the polarizers angle to a saturation level at . Moreover, the diffuse illumination response can be converted to angle of incidence, N(as shown in Fig. 3.3.2.1 b. by equation
(3.3.1.1)
(휃 ) (휃) 휃 0 (3.3.1.2) where x(pixels.
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Figure 3.3.2.1: Diffuse illumination response function for 445 nm laser light: (a) N(x,y), with polarizer and analyzer crossed at angle ψ (no sample). Data for ψ > 0 and ψ < 0 are shown for each |ψ|. (b) N(x,y) data of (a) converted to N(θ) using x(θ) from Figure 2.3.1.1a-c.
The saturation data of the analyzer uncrossing angle need to be corrected in order to measure the ideal intensity transmitted between crossed polarizer and analyzer, I(. This correction is calculated for each data set by the following expression
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Figure 3.3.2.1: Diffuse illumination response function for 445 nm laser light (c) N(θ) data of (b) converted to I(θ) using I(N). (d) I(θ)/[1-cos(πψ/90)], the I(θ) data of (c) scaled up to remove the polarizer/analyzer extinction (the 10º data is excluded). The colors correspond to those used in (a-c) and the black curve is the average over the curves.
(휃 ) ( ) ( ) (2.3.1.3) where is the observed imager response plotted in Figure 3.3.2.1b. The dependence of the intensity, I(N), of the imager on the angle of incidence, is plotted in Figure 3.3.2.1c. This dependence characterizes the nonlinearity of the imager response for different incidence angles.
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Each data set of Figure 3.3.2.1c scaled up to remove the polarizer/analyzer extinction as plotted in Figure 3.3.2.1d. The degree of overlap indicates the extent to which the nonlinearity of the imager response has been corrected for. When N(and I(plotted with the scaled data averaged over the near overlap of I(and the average shows that the nonlinearity of the imager response has been quantitatively corrected as shown in Figure 3.3.2.1e. The best fit of analytical expression to N(is given by
휃 (휃) [( ) ( )] (3.3.1.4)
as shown in Figure 3.3.4.1f.
Equations (3.3.1.3) and (3.3.1.4) serve as the effective incident intensity function for calculations of conoscopic distributions. For example, the scan of 445 nm conoscopic image along the line passing through the optic axes indicated in Figure 3.3.2.2b converted from N(x) to
N(using equation (3.3.1.2) as shown in Figure 3.3.2.2c-d and then converted to I( using equation (3.3.1.4) as shown in Figure 3.3.2.2e. The fringe extinction is limited by light scattering in the PET film of image 3.3.2.2b.
In s summary, Fisheye lenses enable convenient, flexibly applied, precision conoscopic observation.
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Figure 3.3.2.1: Diffuse illumination response function for 445 nm laser light (e) N(θ,ψ=0) and I(θ,ψ=0) (magenta lines) plotted with the scaled data averaged over ψ (black curve). (f) Fit of analytical functions to N(θ,ψ=0) and I(θ,ψ=0).
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Figure 3.3.4.2: (a) white light conoscopic image, N(x,y), of a thick biaxial PET film. (b) 445 nm laser light conoscopic image, N(x,y), of the same film. (c) Scan, N(x), of the 445 nm conoscopic image along the line passing through the optic axes indicated in (b). (d) N(x) in (c) converted to N(θ) using x(θ). (e) N(θ) in (d) converted to I(θ) using I(N).
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Chapter 4
Second Harmonic Generation Imaging Microscopy of B4-8CB Mixtures
4.1 Introduction
The induced dipole moment, macroscopic polarization, in molecular species when incident electromagnetic radiation field of amplitude E probes this specie can be expressed in the time domain [19]
( ) ≈ (4.1.1)
where ij is the linear susceptibility and ijkis the second order nonlinear susceptibility terms.
The second term in equation (4.1.1) describes the second harmonic generation, SHG, response from non-centrosymmetric molecule.
SHG has been integrated to optical microscopy for visualizing the microscopic polar structure [20] and later the technique was implemented on scanning microscopy by Sheppard
[21] for 3D visualization.
4.2 The SHG confocal laser scanning microscope
In addition to the optical microscope, the SHG laser confocal scanning microscope requires a short pulse laser to generate the SHG signal. Figure 4.2.1 shows a schematic of the
SHG confocal laser scanning microscope. A tunable monochromatic linearly polarized light from a Ti:sapphire laser is spatially filtered by a pinhole and then collimated through a beam splitter or a dichroic mirror which directs the collimated light through a microscopic objective designed to
33 focus collimated light to a diffraction limited focal spot in air or cover slip of thickness 170 m.
The sample position along the beam axis (z) can be scanned using a motorized drive with nm repeatability. A half waveplate is inserted between the dichroic mirror and the microscope objective to control the polarization orientation of the laser beam for sensitive measurement of the SHG signal.
Figure 4.2.1: SHG confocal laser scanning microscope. A scanner unit is attached to the microscope for fast scanning. Two detection units can be employed to generate SHG images of the sample under study in the transmission or reflection modes.
In order to perform the lateral xy scanning, a pair of galvanometer mounted mirrors is attached to the microscope as a scanner unit. The adjustment of the mirrors results in a spatial shift of the focus in the focal plane. SHG signal can be detected either in the transmission or reflection mode, forward or backward direction. In the transmission mode, a high numerical
34 aperture microscope objective is used to capture the highly directional transmitted SHG signal. A focusing lens is inserted to focus the captured light into the photomultiplier after it has been filtered by a bandpass filter.
4.3 B4/8CB mixture
In this chapter the structure of binary mixture of NOBOW and 8CB liquid crystal is studied by SHG laser scanning Microscope. The chemical structure and phase sequences on cooling of NOBOW and 8CB are shown in Fig. 4.3.1.
Figure 4.3.1: Chemical structure and phase sequence of Nobow (a) and 8CB (b) [22].
The B4 phase of the bent core molecule, the achiral molecules organize spontaneously into helical nanofilaments to form macroscopic chiral domains (helical winding smectic layers), appears on cooling from the B2 phase, the achiral molecules organize to form a polar order and tilt in planar layers as shown in Figure 4.3.2.
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Figure 4.3.2: illustration of the B4 phase of Nobow. (a) The orination of the achiral molecule of Nobow in the smectic layer, where the molecular long axis of the molecule tilts relative to the z axis. The net dipole moment of molecules results in a macroscopic polarization normal to n and z. (b) twist of the stacking of the smectic layers of (a) with a cross section (pink plane) of the twist structure of the nanofilament, where the polarization oriented along the helix axis, z. (c) Freeze fracture transmission electron microscopy image of pure Nobow showing the layer edges in the inset [22].
The helical nanofilaments grow into a homochiral radial network with orientation of the filament alone the growth direction when NOBOW mixes with organic guest molecules like
36
8CB. Figure 4.3.3 shows the optical images of the radial domains formed in the B4 phase of
NOBOW/8CB mixture using polarizing light microscope.
Figure 4.3.3: Polarizing optical microscope images of the binary mixture c=50% of NOBOW and 8CB liquid crystal with (a) 4X microscope objective lens and (b) 20X microscope objective under crossed polarizer and analyzer.
The polarity of the nanofilaments in the macroscopic domains of B4 phase is studied by
SHG laser confocal scanning microscope. In order to investigate the polarity of the nanofilaments radial network in the macroscopic domains of B4 phase a linearly polarized pulse laser of wavelength, =980 nm, was used to probe one of the radial domains in Fig. 4.3.3. Fig.
4.3.4a-i shows the polarization dependence of the SHG signal from the radial domain. The resulting scanning images for one slice of the radial domain at different orientation of the analyzer, situated before the PMT, with respect to the laser excitation polarization are shown in
Fig. 4.3.4. The top three images of Fig. 4.3.4(a-c) shows the SHG signal when the analyzer is rotated by 0o, 45o, and 90o respectively with respect to the laser excitation polarization angle equal to 0o, the vibration direction parallel to the x-axis. As shown in Fig. 4.3.4a and c the two different orientation of the analyzer yield highly different SHG images. Light polarized at angle
37
0o with respect to the x-axis generate the strongest SHG signal from filaments oriented parallel to the x-axis. The SHG signal starts to decreases in amplitude when the net dipole moment of the filaments orient wise or clockwise by angle higher than 0o with respect to the x-axis until the signal reaches 0 at angle 90o and -90o with respect to x-axis as shown in Figure 4.3.4a. The rotation of the analyzer by angle 45o with respect to the x-axis results in transmission of SHG component parallel to the analyzer Figure 4.3.4b. At extension orientation of the polarizer and analyzer, the SHG signal generates the lowest contracts image since the analyzer blocks most of the generated SHG singles Figure 4.3.4c. Furthermore, rotation of the laser polarization by angle
45o results in no SHG signal from filaments oriented perpendicular to the polarization of the excitation light as shown in Figure 4.3.4d and 4.3.4f. Excitation of the filaments with polarized light along the y-axis results in three different SHG images contrast for different orientation of the analyzer as shown in Figure 4.3.4 g-i. Filaments grow along the y-axis generate SHG signal stronger than what is generated by an orthogonal filament as shown in Figure 4.3.4i.
This contrast in the SHG images suggest that in addition to the polarity of the nanofilaments the net dipole moment of the filament is oriented parallel to the long symmetric axis of the filaments, along the filaments growth axis.
In a summary, the B4 phase of Nobow/8CB mixture forms radial networks of helical nanofilments upon cooling from the isotropic phase. These nanofilaments exhibit a net dipole moment along the filaments growth axis.
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Figure 4.3.4: SHG confocal laser scanning microscope images of the B4 nanofilaments. (a-c) SHG Images of laser excitation of polarization oriented at 0o, pink arrow, with respect to x axis, horizontal axis, at different orientation of the analyzer. (d-f) SHG Images of laser excitation of polarization oriented at 45o, pink arrow, with respect to x axis, horizontal axis, at different orientation of the analyzer, red arrow. (g-i) SHG Images of laser excitation of polarization oriented at 90o, pink arrow, with respect to x axis, horizontal axis, at different orientation of the analyzer.
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Bibiliography
[1] G. Friedel , Ann Physique, 18, 273 (1922).
[2] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Oxford Science Publication, 2nd ed. 1993).
[3] S. Chandrasekhar, Liquid Crystals. (Cambridge University Press, 2nd ed., 1992).
[4] M. A. Geday, G. del Campo, Molecular Crystals and Liquid Crystals, 494, 213 (2008).
[5] W. R. Zipfel, R. M. Williams, Nature Biotechnology, 21, 1368 (2003).
[6] P. Torok, P. Varga, J. Opt. Soc. Am. A, 12, 325 (1995).
[7] S. H. Wiersma, P. Torok, J. Opt. Soc. Am, 14,1482 (1997).
[8] P. Torok, P. Varga, Applied Optics, 36, 2305 (1997).
[9] C. J. R. Sheppard, Journal of Microscopy, 185, 366 (1997).
[10] M. Born, Principle of Physics (Cambridge University publication, 7th ed. 1999).
[11] F. D. Bloss, An introduction to the methods of optical crystallography, (Holt Rinehart and Wilson, 2nd ed., 1961).
[12] E. E. Wahlstrom, Optical Crystallography, (John Wiley and Sons, 4th ed., 1969).
[13] L. V. Brett, H. H. Winter, Applied optics, 40, 2083 (2001).
[14] http://jm-derochette.be/Conoscopy/Biaxial_minerals_5.htm
[15] http://www.olympus-ims.com/en/microscope/cx31p
[16] http://jm-derochette.be/Conoscopy/Uniaxial_minerals_5.htm
[17] K. Miyamoto, Fisheye lens, J. Opt. Soc. Am, 54, 1060 (1964).
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[18] R. Hill, Proc. Opt. Conv, 878(1926).
[19] R. W. Boyd, Nonlinear optics, (Academic press, 3rd ed. 2008).
[20] R. Hellwarth, and P. Christensen, Optics Commm., 12, 318 (1974).
[21] C. J. R. Sheppard, R. Kompfiner, J. Gannaway, and D. Walsh, IEEE J. Quantum Electronic 13E,100D (1977).
[22] D. Chen, Nanostructure of liquid crystal phases. (PhD dissertation, University of Colorado at Boulder, 2012).
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