COMPLEX OPTICAL FIELDS GENERATION USING A VECTORIAL OPTICAL FIELD GENERATOR

Transcription

1 COMPLEX OPTICAL FIELDS GENERATION USING A VECTORIAL OPTICAL FIELD GENERATOR Thesis Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree of Master of Science in Electro-Optics By Sichao Zhou UNIVERSITY OF DAYTON Dayton, Ohio May, 2016

2 COMPLEX OPTICAL FIELDS GENERATION USING A VECTORIAL OPTICAL FIELD GENERATOR Name: Zhou, Sichao APPROVED BY: Qiwen Zhan, Ph.D. Advisory Committee Chairman Professor Electro-Optics Program Partha Banerjee, Ph.D. Committee Member Director and Professor Electro-Optics Program Joseph Haus, Ph.D. Committee Member Professor Electro-Optics Program John G. Weber, Ph.D. Associate Dean School of Engineering Eddy M. Rojas, Ph.D., M.A., P.E. Dean, School of Engineering ii

3 Copyright by Sichao Zhou All rights reserved 2016 iii

4 ABSTRACT COMPLEX OPTICAL FIELDS GENERATION USING A VECTORIAL OPTICAL FIELD GENERATOR Name: Zhou, Sichao University of Dayton Advisor: Dr. Qiwen Zhan Owing to their unique properties, optical fields with complex spatial distribution in the cross section have attracted great attention. These complex optical vector fields were discovered to have many interesting applications in particle manipulation and acceleration, nanoscale optical imaging and so on. This thesis is mainly composed of two parts. In the first part, the theory and experimental setup for a Vector Optical Field Generator (VOF-Gen) that is capable of creating an arbitrary beam with independent controls of phase, amplitude and polarization on the pixel level utilizing high resolution reflective phase-only liquid crystal (LC) spatial light modulator (SLM) will be reviewed. Experimental results will be presented in the second part, where various optical fields containing phase, amplitude, polarization and retardation modulations are successfully demonstrated. iv

5 The capability to modulate each of the individual degree of freedoms will be verified by our experimental results. These demonstrated capabilities lead to a five ring structured complex vectorial optical field with multiple controllable parameters with unique high-numericalaperture focusing properties that may find important applications in super-resolution imaging, optical nanofabrication, and optical trapping and manipulation. v

6 Dedicated to my parents and my old friends vi

7 ACKNOWLEDGEMENTS First of all, I am deeply indebted to my parents and other family members, for their priceless love and great confidence in me all these years. I gratefully acknowledge my advisor, Dr. Qiwen Zhan, for his tremendous amount of help and support through my master period. Under his guidance, I gained a lot of knowledge and self-confidence. He is a knowledgeable and diligent professor and a leading scientist in the field of space variant polarization engineering and I am especially grateful for his experienced guidance and vision in my research of complex optical fields engineering. I would also like to thank my committee members, Dr. Partha Banerjee and Dr. Joseph Haus. I am grateful for their valuable revision advice on my thesis manuscript. Dr. Haus is a very knowledgeable and versatile. Dr. Banerjee is the Director of the Electro-Optics Program and a very nice and knowledgeable professor with strong disciplines. Finally I sincerely thank the former members of Dr. Zhan s group, including Dr. Wei Han and Dr. Shiyi Wang. I also would like to thank the other students from the Electro-optics Program including Yu Bai, Zhenyu Yang, Hongwei Chen and Zhijun Yang for their help and support. vii

8 TABLE OF CONTENTS ABSTRACT... iv DEDICATION... vi ACKNOWLEDGEMENTS... vii LIST OF FIGURES... xi LIST OF TABLES... xiv LIST OF ABBREVIATIONS AND NOTATIONS...xv CHAPTER 1 INTRODUCTION...1 CHAPTER 2 THEORETICAL BACKGROUND Liquid crystal and spatial light modulators State of polarization, Stokes parameters and Poincaré sphere F imaging system...13 CHAPTER 3 VECTOR OPTICAL FIELD GENERATOR Introduction Principles...18 viii

9 2.1. Spatial light modulator Spatially variant polarization rotator Modulation of light Amplitude modulation (SLM Section 2) Polarization ratio modulation (SLM Section 3) Phase retardation modulation (SLM Section 4) Phase modulation (SLM Section 1) Experimental setup Experimental setup of VOF-Gen f imaging system and its alignment procedure Summary...34 CHAPTER 4 EXPERIMENTAL RESULTS Introduction Experimental results Spatially variant amplitude modulation: Flat-top beam generation Spatially variant polarization modulation: radially polarized beam generation Spatially variant phase retardation Complex vectorial optical fields generation with multiple parameters Complex optical fields for ONF generation Comparison of experimental and theoretical results...48 ix

10 3. Summary...55 CHAPTER 5 CONCLUSIONS AND FUTURE WORK...56 BIBLIOGRAPHY...57 x

11 LIST OF FIGURES Fig (a) Rod-like molecules, and (b) disk-like molecules... 5 Fig Schematic representations of phases of thermotropic liquid crystals... 5 Fig Twisted nematic LCD... 7 Fig Uniaxial molecule in the laboratory coordinate system... 7 Fig Parameters of a state of elliptical polarization Fig Poincaré s spherical representation of the state of polarization Fig Schematic diagram of 4-f imaging system Fig Driver circuits for the color channels of the VOF-Gen system. The upper one is the red channel responsible for the control of SLM1 and the lower one is the green channel for SLM Fig Illustration of the Polarization Rotator setup. (a) The Polarization Rotator comprised of QWP and reflective SLM; (b) Illustration of the effective rotation of the QWP fast axis for the incident (upper) and the reflected (lower) beams due to the mirror Fig Schematic diagram of gamma curve calibration setup Fig Flow chart of the VOF-Gen System Modulation part Fig Experiment setup of the VOF-Gen Fig Schematic diagram of the VOF-Gen xi

12 Fig Pattern with fine features for 4-f imaging system alignment. Each line is 100 μm wide. (a) Pattern with polarized in RCP. (b) Polarized in LP. (c) Pattern with unit amplitude Fig Flow chart of the VOF-Gen system and 4-f imaging system alignment procedure Fig Comparison of the same patterns generated by VOF-Gen system without and with well aligned 4-f imaging systems. (a) Retardation: the intensity is captured after a circular analyzer; (b) polarization rotation: the intensity is captured after a linear polarizer; (c) Retardation: the intensity is captured after a circular analyzer Fig Schematic diagram of the beam shaping setup Fig The circular inverse Gaussian beam phase pattern coded in SLM section Fig Intensity profiles of the (a) incident collimated Gaussian beam (b) FTB of 1.5 mm diameter Fig Phase patterns for radially polarization beams (a) with pre-compensation phase (b) without pre-compensation phase Fig The intensity profile of radially polarization beam with flat-top amplitude distribution. (a)radially polarization beam without pre-compensation phase; (b) radially polarization beam with pre-compensation phase Fig Intensity profiles of radially polarized beam after a polarizer which the fast axis along the black arrows at 0, 45, 90 and 135, respectively Fig Phase information f or UD logo pattern Fig "UD" logo coded in circular polarization. The total field (left), the "UD" logo (upper right) in RCP and the complimentary UD logo (lower right) in LCP xii

13 Fig Focal field intensity along focus (upper) and its line scan (lower) [32]. The longitudinal coordinate is normalized to the wavelength Fig The phase pattern for VOF-Gen Fig Five ring structured polarized in radial polarization. The upper set of graphs show the linear polarization components along 0, 135, 90 and 45 and the lower graph shows the intensity profile of total field Fig Interference setup consists of a beam splitter (BS) and two mirrors Fig The Interference fringes patterns of desired complex field. The right graph shows the total interference pattern and the left graph shows zoom in upper left corner image of the total interference pattern Fig A simple phase pattern for VOF-Gen Fig Interference fringes patterns of π phase shift Fig The Compare of Amplitude modulation Fig. 4-17Intensity distribution and the electric field distribution in the transverse plane (the polarization direction for this ONF is represented by arrows) (a) Overall picture calculated by theory amplitude value. (b) Overall picture calculated by experimentally obtained amplitude value. (c) Zoomed-in picture for the area enclosed with red dashed line in (a). (d) Zoomed-in picture for the area enclosed with red dashed line in (b) xiii

14 LIST OF TABLES Table 1: Parameters of desired field along radial direction xiv

15 LIST OF ABBREVIATIONS AND NOTATIONS VOF- Gen Vector Optical Field Generator CCD Charge coupled device LC Liquid crystal SF Spatial filters SLM Spatial light modulator LP Linear polarization SOP State of polarization NA Numerical aperture LCP Left-hand circular polarization NPBS Non-polarizing beam splitter RCP Right-hand circular polarization ONF Optical needle field DOF Depth of focus F Focal length LCD Liquid crystal display TN Twisted nematic PR Polarization Rotator QWP Quarter wave plate BS Beam splitter HWP Half wave plate xv

16 CHAPTER 1 INTRODUCTION Engineering of optical vector fields with inhomogeneous polarization has drawn a lot of interest owing to their potential applications in various areas [1-5]. Optical traps and tweezers have become powerful tools for the trapping and manipulations of atoms, molecules, nano particles, living biological cells and organelles within cells [6]. Recently applications of engineered vectorial optical fields in optical tweezers was suggested [2]. In addition surface plasmon sensing using radially polarized beams has been studied to achieve higher spatial resolution [7]. Vortex beams, also known as twisted light, has also attracted increasing interest due to its spiral phase wavefront carrying orbital angular momentums [8, 9]. Besides the manipulation of intensity and phase, controlling the vector nature of electromagnetic wave, the state of polarization (SOP) distributions, also plays an important role in beam shaping for flattop generation [10], focus shaping [11], surface plasmon sensing using cylindrical vectorial beams [12], and ellipsometry [13]. With the recent advances in the spatially variant polarization and particularly in the cylindrical vector beams [1], vectorial distribution of the focused electromagnetic fields has drawn a significant amount of attention. 1

17 One specific example that attracted recent interest is the so-called optical needle field (ONF) that is substantially polarized in the longitudinal direction with long depth of focus (DOF), created through the focusing of the filtered radial polarization [14-20]. All of the above applications require the generation of controllable complex vector optical fields.there have been tremendous amount of works that dealt with the generation, focusing and propagation of different types of vector fields. Arbitrary space-variant vector beams with structured polarization and phase distributions using a spatial light modulator (SLM) were presented. Both the phase and the state of polarization of vector beams can be tailored independently and dynamically by a SLM [21]. Using an interferometer constructed from two identical diffractive optical elements to generate optical vector beams was proposed by Kimani C. Toussaint [22]. Another versatile and stable generation of optical vector fields using a noninterferometric approach which can produce all the SOP represented on a higher-order Poincaré sphere was also proposed [23]. A technique for generating arbitrary intensity and polarization distributions was reported using transmissive SLMs and Mach- Zehnder interferometry setup [24]. However, the absolute phase of each electric field component does not cover an entire 2π range. As a result, complete phase control cannot be fully realized. The limitations of this technique also include relatively low transmittance and poor spatial resolution. Very recently a novel method was proposed to fully control the phase and polarization of vector beam at the nanoscale through the use of compact metasurfaces with rectangular nano apertures [25]. However, this approach comes with the limit of amplitude modulation. Besides, each specific vector beams requires an individual filter design, which increases the cost and leads to limited efficiency for practical applications. All above of the existing techniques have their nonigorable limitations and cannot be used to generate a spatially-invariant arbitrary vectorial fields with high spatial resolution on a pixel level. 2

18 In this work, both the theory and experimental setup of a Vector Optical Field Generator (VOF- Gen) will be reviewed in Chapter 2 and Chapter 3. In Chapter 2, the theory of liquid crystal and spatial light modulators, state of polarization, Stokes parameters and Poincaré sphere and 4-F imaging system will be introduced. In Chapter 3, we will describe the principle of how to modulate each individual degree of freedom of an optical field using the VOF-Gen. The experimental setup will also be illustrated. In Chapter 4, various vector fields such as flat top beam, radially polarization beam and a UD logo pattern is polarized in right-hand circular polarization (RCP) immersed within a background is polarized in left-hand circular polarization (LCP), and this has been successfully demonstrated. The capability to modulate each of the individual degree of freedom is verified by these experimental results. Finally, in order to demonstrate the application value of VOF-Gen, a five-ring structured complex vectorial optical field with multiple controllable parameters that could be further focused to produce an optical needle field is successfully generated. 3

19 CHAPTER 2 THEORETICAL BACKGROUND 1. Liquid crystal and spatial light modulators As the key part of our VOF-Gen, the phase-only LC-SLM is used as a variable and addressable retarder. The LC molecule typically has an elongated rod-like or disk-like shape, as shown in Fig The size of the LC molecules is a few nanometers. Liquid crystals materials possess physical properties that are intermediate between conventional fluids and solids. They are fluid like, yet the arrangement of molecules within them exhibit structural orders [26].There are three types of liquid crystals that have been discovered: (1) thermotropic, (2) lyotropic, (3) polymeric. Among those three types of liquid crystals, the thermotropic liquid crystals have been studied extensively and their applications have reached a mature stage [27]. 4

20 Fig (a) Rod-like molecules, and (b) disk-like molecules Thermotropic liquid crystal phases are those that occur in a certain temperature range. If the temperature rise is too high, thermal motion will destroy the delicate cooperative ordering of the LC phase, pushing the material into a conventional isotropic liquid phase. At too low of a temperature, most LC materials will form a conventional crystal [28, 29]. As the temperature increases, thermotropic liquid crystals exhibit three phases: (1) smectic (2) nematic (3) cholesteric. Schematic representations of the phases of thermotropic liquid crystals are illustrated in Fig Fig Schematic representations of phases of thermotropic liquid crystals Fig. 2-2(a), (b) illustrates the smectic phases with one dimensional translation order and orientation order [30]. Fig. 2-2(c) shows the nematic phase exhibits long-range orientational 5

21 ordering but no positional ordering. The cholesteric liquid crystals phase is typically composed of nematic mesogenic molecules containing a chiral center. Cholesteric liquid crystals are also known as chiral nematic liquid crystals. As shown in Fig. 2-2(d), notably, each plane is perpendicular to the helical axis. This causes the sematic liquid crystals to have the higher positional ordering, and thus is closer to the solid crystals. However the nematic and cholesteric have many electro-optical applications. The aligned nematic liquid crystals molecules, on average, are characterized by one symmetry axis called the director [28]. The most common liquid crystal display (LCD) that is used for everyday items such as watches and calculators is called the twisted nematic (TN) display. In general, LCD consists of a nematic liquid crystal sandwiched between two plates of glass as illustrated by the Fig. 2-3.The director of nematic liquid crystals molecules can be manipulated by an external field, such as an electric field, magnetic field, or optical field. Since liquid crystal is an optical anisotropic substance, the index of refraction depends on the polarization and propagation direction of the incident light. Due to the birefringence of liquid crystals, the director axis reorientation induced by external fields will introduce a large phase change during the optical field traversing the LCD film. 6

22 Fig Twisted nematic LCD Usually, the optical axis of uniaxial liquid crystals molecules is in parallel with the longitudinal orientation of LC molecules (director). The incident light wave vector k on the x-z plane has an angle θ with the directors as shown in Fig Fig Uniaxial molecule in the laboratory coordinate system The effective index of refraction,n eff (θ) can be written as, 7

23 cos 2 ( ) sin 2 ( ) 1, (2-1) n n n ( ) o e eff where n o is the ordinary refractive index and n e is the off-state extraordinary refractive index. θ is the tilt angle dependent on the plied voltage. SLM is made of twisted-namatic LCD usually, consist of a series layers of LC molecules, and each of them can be considered as a thin wave plate. Therefore the Jones matrix used to describe the effect of the twisted-nematic LCD can be shownas [30], M R j cos j sin sin. sin cos j sin LCD ( )exp( ( 0)) (2-2) Here β is the birefringence, α is the twist angle, and γ 2 = α 2 + β 2. The birefringence is defined as β = πdδn λ, where d is the thickness of the displays, Δn is the difference between the ordinary and the extraordinary refractive index of the LC molecules. R(α) is the rotation matrix, Φ 0 is the common phase. For the phase-only LC-SLM, the twist angle α is set to 0 so the Jones matrix can be simplified as [32], M Phaseonly SLM exp( 2 j ) exp( j0) (2-3) The birefringence β varies as a function of the voltage applied across the LC molecules. Therefore the LCD serve as a variable retarder controlled by the applied voltage. From the Jones matrix one can see that the reflective-type phase only LC-SLM directly loads the phase on the horizontal component. Thus, this phase change can be a pure modulation (without changing the polarization state) or a phase retardation depending on the alignment on the input polarization direction. For example, consider the case of a linearly polarized light incident 8

24 parallel to the x axis, the reflected light remains horizontal linearly polarized and only experiences a phase change. But if the incident light has an arbitrary polarization, SLM will load a phase retardation on the horizontal component, which means the polarization of the reflected light will be modulated by the reflective-type phase only LC-SLM. 2. State of polarization, Stokes parameters and Poincaré sphere Polarization is a fundamental property of light that describes the orientation of the lines of electric flux. The polarization of a light is determined by the behavior of the amplitude of E- vector [32]. First let us consider a monochromatic plane wave propagation in the z-direction, due to the transverse nature of the plane wave the real Cartesian components of electric field can be write as: Ex Ax cos( kz t x). (2-4) E A cos( kz t ) y y y By removing the temporal term from the equation above, we obtain the following: 2 2 E Ey ExE x y 2 cos, (2-5) 2 2 A A A A x y x y where δ = δ y δ x ( π δ π). Eq. (2-5).represents the general state of polarization, which we refer to as being elliptically polarized. It is possible to find a new reference coordinate system to reducing the cross-term in Eq. (2-6). In the new reference system the ellipse equation is given by: 2 2 E x E y 1 a, (2-6) b 9

25 where E x and E y are the components of the electric field along the new coordinate. The elliptical polarization as illustrated in Fig. 2-5, the new coordinate system is rotated by an angle α base on original system. The rotation angle is given by tan 2 2AA A x y 2 2 x Ay cos. (2-7) Fig Parameters of a state of elliptical polarization. The values a and b, correspond to the half-length of the major axis and of the minor axis of the ellipse, is given by a Ax cos Ay sin 2Ax Ay cos sin cos b Ax sin Ay cos 2Ax Ay cos sin cos. ab AxAysin (2-8) A widely used simple mathematical tool to represent the vectorical nature of the state of polarization is the so-called Jones vector, which expresses the relative amplitude and phase of the two orthogonal components of the E-vector: 10

26 V Ae Ae i x x i y y. (2-9) After normalization the Jones vector that depends on two parameters, the χ represents the angle between the oscillation direction of electromagnetic wave and x-axis, and δ is the phase retardation between its Cartesian components, that is: V cos i sin e. (2-10) In previous representations, the states of polarization were characterized by complex numbers, Stokes parameters representations only referred to the intensity of the field. Stokes parameters are given as 2 2 S0 I x I y I0 Ax Ay 2 2 S1 I x I y Ax Ay, (2-11) S2 I I 2A cos xay S3 I L I R 2Ax Ay sin From Fig. 2-5 we can easily get A tan A b tan a y x, (2-12) Using these relations into Eq. (2-10, 11), it can be shown as: 2AA 2ab tan tan 2 A A a b x y x y 2AA 2ab sin 2 sin 2 A A a b x y x y. (2-13) Taking the relations (2-13) as well as the relations (2-7) and (2-8), we get: 11

27 cos2 cos2 cos2 cos2 sin 2 sin 2 cos sin 2 sin 2 sin tan 2 sin 2 tan (2-14) Using the definition of the standard Stokes parameters and the relations (2-13) and (2-14), it may be easily shown that: S1 cos 2 cos cos 2 S2 cos2 cos2 sin 2 cos, (2-15) S3 sin 2 sin 2 sin with S S S 3 2 = S 0 2 = 1. The standard Stokes parameters allows a graphical description of SOP using a unit sphere called Poincaré sphere. On Poincaré sphere different states of polarization can be represented. Representing this sphere as well as the different parameters is shown in Fig Fig Poincaré s spherical representation of the state of polarization From the Eq. (2-15) 2α and 2ε are considered as the polar angles of Poincaré s spherical representation, the SOP characterized by the α inclination of the major axis of the ellipse and ε is ellipticity angle. 12

28 3. 4-F imaging system In order to generate arbitrary beams, diffraction effects have to be taken into consideration. The diffraction needs to be minimized so that sharp edges or high frequency information in phase, amplitude and polarization can survive. In the experiment setup the 4-f imaging system is used to relay the optical fields between the object and image planes to minimize diffraction effect and eliminate the quadratic wavefronts. This device - 4 focal lengths long is made up by two identical lens and a transparency plate actually serves a wide variety of holography reconstruction operations that go well beyond what its name implies. A conventional coherent 4-f imaging system is shown in Fig Fig Schematic diagram of 4-f imaging system Assuming lens L1 and lens L2 have the same focal length f, we have an input of the form of a transparency U i (x, y ) located in the object plane which is illuminated by a plane wave. Employ the Fresnel diffraction formula to calculate the field immediately in front of the lens L1 and we have 13

29 U U x y 1 i, h, ; f x y 2 2 Ui x, yexp jk dxdy 2 f x y x y exp jk Ui x, yexp jk expjk dxdy 2 f 2 f 2 f (2-16), where, jk jk 2 2 hx, y; z exp jkz exp x y 2 z 2z. (2-17) h(x, y; z) is called the coherent point spread function (CPSF) in Fourier optics. The field distribution U 2 immediately after Lens1 can be calculated by multiplying the phase function as follows: 2 2 U2, U1, expjk, (2-18) 2 f where exp ( jk ξ2 +η 2 ) is the phase function of an ideal thin lens, which means the lens will 2f only affect the phase of the incident field. The field distribution U 3 in the back focal plane of the lens L1 (the Fourier plane) can be found as: x y 2 2 U3 x, y U 2, exp jk dd 2 f 2 2 x y x y exp jk U, exp jk d d 2 f f kx k 2 2 x y f exp jk U 1, 2 f ky k f 1, (2-19) where, 14

30 U 1, U i x, y* h, ; f U ix, y x, y U ix, yexp j f k k 2 2 x y 2k. (2-20) If we let those spatial frequencies k x and k ξ, k y and k η equal to each other. U 3 (x, y) can be simplified as: kx 2 2 k 2 2 x x y f kx k y f U3 x, y exp jk U i x, yexp j 2f 2k ky k y f, kx kx f U i x, y ky k y f (2-21) We can see the lens phase function cancels out the quadratic phase term introduced by the Fresnel diffraction and proportional to the Fourier transform of the incident transparencyu i (x, y ). According to the Eq. (2-20) the field on the image plane will be the Fourier transform of U 3 (x, y).thus U out at Image plane can be written as: out 1, 1 3, U x y U x y kx kx k f U i x, y ky k y k f f U i x 1 y 1 k k k x y kx f 1 ky f 1 x y kx1 f. ky1 f (2-22) 15

31 So far we derived a plane wave from a laser to illuminate U i (x, y ) in the 4-f optical processing system. It shows that the original input on the object plane has been flipped and reimaged on the image plane without any quadratic phase term introduced by the Fresnel diffraction. In other words, the object (transparency) can be reimaged at the image plane in terms of both amplitude and phase. 16

32 CHAPTER 3 VECTOR OPTICAL FIELD GENERATOR 1. Introduction There have been tremendous amounts of works dealing with the generation, propagation and focusing of different types of vector fields. Arbitrary space-variant vector beam with structured polarization and phase distributions using a SLM was presented. Both the phase and the state of polarization of vector beams can be tailored independently and dynamically by a SLM [21]. UItilizing an interferometer constructed from two identical diffractive optical elements to generate optical vector beams was proposed by Kimani C. Toussaint [22]. Another noninterferometric method for vector field generation was proposed which can generate any polarization distribution that corresponds to a coordinate on the higher-order Poincaré sphere, with basis functions of radial and azimuthal polarization [23]. A technique for generating arbitrary intensity and polarization was reported with interesting results using transmissive SLMs and Mach- Zehnder interferometry setup [24]. However, the absolute phase of each electric field component does not cover an entire 2π range. As a result, complete phase control cannot be fully realized. The limitations of this technique also include relatively low transmittance and poor spatial resolution. 17

33 Very recently a novel method is proposed to fully control the phase and polarization of vector beam at the nanoscale through the use of compact plasman metasurfaces with rectangular nano apertures [22].However, this approach comes with the limit of amplitude modulation. In addition each specific vector beam requires an individual filter design, which increases the cost and leads to experiments with limited efficiency for practical applications. All the existing techniques have limitations and cannot be used to generate a spatially-invariant arbitrary vectorial field with high spatial resolution on a pixel level. In this chapter, both the principle and experimental setup of the Vectorial Optical field Generator (VOF-Gen) that is capable of generating arbitrary optical fields through spatially modulating all aspects of optical fields (including phase, amplitude and polarization) on a pixel-by-pixel basis was been reviewed. 2. Principles The goal of a VOF-Gen is to be able to generate an arbitrarily complex desired vectorial optical field.the desired optical vectorial optical field can be represented as a superposition of two orthogonal polarization components. For example, if the complex E-vector is written as: With, (3-1) E x, y xe ye d x y i x x and y y E A e E A e i x y. The corresponding Jones vector for the desired optical field can be expressed as: E d x, y Ae Ae After normalization, we would write i x x i y y. (3-2) id Ed x, y Ad x, ye i x, y. (3-3) yd E,, xd x y x y d E x, ye 18

34 where A d (x, y)represents the amplitude distribution, φ d (x, y) is the common phase for both x and y components and the Jones vector contains the polarization information where E xd (x, y)and E yd (x, y) are both real and normalized (E 2 xd + E 2 yd = 1). δ d (x, y)is the desired phase retardation between the y and x components. Clearly four degrees of freedom, namely the phase, amplitude, polarization ratio and retardation between the x and y components are necessary in order to fully characterize a vectorial optical field. Thus a true vectorial optical field generator needs to be able to control all of these four parameters on a pixel-by-pixel basis for the generation of arbitrarily complex vectorial optical field. The principle of such a VOF- Gen will be discussed in details in the following Spatial light modulator As a key component for the VOF-Gen, the SLM, Holoeye HEO 1080P, is used as a variable and addressable retarder. The SLM is a phase-only, reflective liquid crystal (LC) device featuring a HDTV resolution of 1920 x 1080 with pixel pitch of 8 μm and fill factor of 87%. The VOF-Gen consists of two SLM panels as shown in Fig

35 Fig Driver circuits for the color channels of the VOF-Gen system. The upper one is the red channel responsible for the control of SLM1 and the lower one is the green channel for SLM 2 The retardation for each pixel on the SLM can be described as a function of the voltage (V) applied:δ(v) = 2π/λ(n e (V) n o )d, where d is the thickness of the LC layer, n e and n o are the extraordinary and ordinary refractive indices of the LC retarder, respectively. Due to the birefringent nature, the SLM in our system only responds to the horizontal polarization parallel to the LC directors, meaning that the horizontal component of the reflected beam will carry the phase information specified by the SLM while the vertical one will be reflected unaffected. Since four degrees of freedom in Eq. (3-3) need to be independently controlled in the system, four reflections are required where each SLM section is loaded with one of the phase patterns for the modulations of phase, amplitude, polarization rotation and retardation Spatially variant polarization rotator Before we discuss the details of how to modulate each individual degree of freedom of an optical field, I d like to introduce another key component of the VOF-Gen system called 20

36 Polarization Rotator (PR) based on the concept of a pure polarization rotator in order to realize the linear polarization rotation and amplitude modulation [25].The Polarization Rotator consists of a variable and addressable retarder, a quarter-wave plate (QWP) with a fast axis at 45 respect to the horizontal axis. The schematic diagram of PR is illustrated in Fig Fig Illustration of the Polarization Rotator setup. (a) The Polarization Rotator comprised of QWP and reflective SLM; (b) Illustration of the effective rotation of the QWP fast axis for the incident (upper) and the reflected (lower) beams due to the mirror In the setup, the variable and addressable retarder is realized by the reflective phase-only LC- SLM to achieve spatially variant polarization rotation function on a pixel-by-pixel basis as shown in Fig. 3-2(a). The incident light passes the QWP with fast axis oriented at 45 (upper part in Fig. 3-2(b)) in the laboratory coordinate (x, y), then gets reflected off the SLM surface. The reflected light goes through the same QWP for the second time (lower part in Fig. 3-2(b)). However, due to the reversed propagation direction, the new laboratory coordinate (x, y ) is a mirror image of coordinate (x, y) about the y axis. Therefore, the fast axis of the QWP has been 21

37 effectively rotated to 135 in the coordinate (x, y ). Thus, the Jones matrix representation of the PR can be calculated as [32]: M PR R JQWPR M SLM R JQWPR x, y 1 1 i 1 i 1 0e 01 i 1 i 4 1 i 1 i i 1 i i e e xy, 2 i x, y x, y sin cos 2 2 x, y x, y cos sin 2 2 xy, 3 xy, i 2 R 2 2 (3-4) The Jones matrix of PR consists of a rotation matrix R ( 3π δ(x,y) ) and an extra phase term. 2 The rotation matrix R indicates an effective polarization rotation of ( 3π δ(x,y) )at each pixel 2 in the clockwise direction, which depends on the phase loaded onto the SLM. By precisely measuring the amount of rotation based on the nulling effect with a linear analyzer for each gray level, we are able to calibrate the gamma curves of the SLM so that the gray level and the actual phase imposed by the SLM are more concisely correlated. The calibration setup consists of a beam splitter (BS), a Polarization Rotator, a polarizer and an Optical power meter are shown in Fig

38 Fig Schematic diagram of gamma curve calibration setup The horizontally polarized incident beam first goes through beam splitter, then the beam reflected by BS goes through a polarizer after passing through the Polarization rotator system and is collected by the power meter. We got the minimized transmission from the power meter after rotate the polarizer, which is so-called nulling effect. Eq. (3-5) indicates that for the beam passing through PR system an effective clockwise polarization rotation of ( 3π δ(x,y) )at each 2 pixel can be achieved. In other words, the counter-clockwise polarization rotation at each pixel is found to be( δ(x,y) 2 + π ).Therefore the relationship between actual phase value δ(x) loaded 2 into each SLM pixel and the amount of rotation θ(x) for the polarizer based on grey level X can be found by 2 X x, y X (3-5) Then a precise calibration gamma curve can be obtained by generating the one to one mapping of the grey level and the actual phase relationship using the same calibration procedure as 23

39 mentioned above. For any phase value to be generated by the SLM, a grey level is found by interpolating the phase value in the nonlinear relationship as opposed to using the linear relation: Grey level = [ mod(φ,2π) 255] [32]. 2π 2.3. Modulation of light Fig Flow chart of the VOF-Gen System Modulation part From the discussions above, in general four SLMs would be needed in order to fully control all of the degree of freedoms to create an arbitrarily complex optical field. In the VOF-Gen setup two SLM panels are used with each of the SLM panel divided into two halves. Each half of the SLM panels is used to realize the control of one degree of freedom as described in Fig. 3-4.The Incident beam that we use in our modulation part is a horizontally polarized collimated Gaussian beam. Then the phase information are loaded onto each of SLM Sections 1 to 4 to realize phase, amplitude, polarization ratio and retardation modulations. At the image plane, the desired complex optical field can be analyzed by a CCD camera. The 4-f imaging systems are used to relay the optical field from one SLM section to the next and to the image plane. These 4-f imaging system is also necessary to minimize the diffraction effect after careful alignment. Details of the modulations and phase pattern calculations are given in the following. 24

40 Amplitude modulation (SLM Section 2) We put a linear polarizer with transmission axis oriented along horizontal direction just after the first polarization rotator system utilizing the second SLM section to achieve the amplitude modulation. The Jones vector for output beam can be given as: 2 xy, i J SLM 2 x, y Einput x, ycos e 2 xy, , 2 xy, 2 xy, i 2 1 Einput x, ysin e 2 0 (3-6) where E input (x, y) is the amplitude of incident beam and δ 2 (x, y) is the phase information loaded in SLM section 2. The Jones vector for the output beam indicates that amplitude modulation is realized by sin ( δ 2 (x,y) ) while the output beam is still horizontally polarized due 2 to the transmission axis orientation of the linear polarizer. Compared with the general expression for desired a vector field shown in Eq. (3-3), the phase information δ 2 (x, y) loaded in SLM section 2 can be given as: -1 2 x, y 2sin Ad x, y. (3-7) Polarization ratio modulation (SLM Section 3) Linear polarization rotation is achieved by another PR system, the spatially variant polarization rotation on a pixel-by-pixel basis can be realized by the phase information loaded onto the SLM section 3.Recall Jones Vector of output field for SLM section2 as shown in Eq.(3-6), the input field of SLM section 3 is horizontally polarized. The resulting output field can be represented in Jones Vector form as: 25

41 3 xy, 2 x, y 3x, y cos 2 xy, i J SLM 3 x, y Einput x, ysin e 2 3 xy, sin 2 2, (3-8) where δ 3 (x, y) is the phase pattern for SLM Section 3. Compared with the general expression for the desired field as shown in Eq. (3-3). δ 3 (x, y) can be found via the following expression: 3 xy, 2tan 1 E E yd xd x, y x, y. (3-9) Phase retardation modulation (SLM Section 4) Due to the birefringence nature of the LC molecules, the phase retardation can be readily introduced by the last SLM section. The Jones Vector after phase retardation modulation can be written as: xy, i x, y 3 4 2x, y 3x, y cos e 2 xy, i J SLM 4 x, y Einput x, ysin e 2 3 xy, sin 2 2 (3-10) where δ 4 (x, y) is a phase pattern for SLM Section 4. Similarly, δ 4 (x, y) can be found from the desired field distribution given by Eq. (3-3): x y x y 4, d,. (3-11) 26

42 Phase modulation (SLM Section 1) Phase modulation is the first step of generating vectorial optical field, which can be realized as the phase information loaded on the SLM section 1 will be directly imposed on the horizon component of the reflected beam. From Eq. (3-10) we can see that, after amplitude and polarization ratio modulations, two additional phase term are introduced into the vector beam due to the geometrical phase effects arising from the two PR system. Therefore, the phase term of final output field is given by: x, y x, y x, y x, y, (3-12) output 1 where δ 1 (x, y) is the phase pattern loaded onto SLM Section 1 of the VOF-Gen. In order to correctly generate the desired phase, pre-compensation phase term and the desired phase must be combined in the phase modulation process. Assuming δ output (x, y) equal to δ d (x, y), we have: where δ c (x, y) = ( δ 2 (x,y) 2 x y x y x y 1, c, d,, (3-13) + δ 3 (x,y) + π) is the pre-compensation phase. As we previously 2 discussed the required phase for each of SLM sections can be calculated based on the desired output field. Also noteworthy is the phase information added to the beam is distributed spatially inhomogeneous. In other words, the vectorial optical field with arbitrarily phase, amplitude, polarization ratio and retardation can be achieved by loading those spatially inhomogeneous distributed phase patterns onto each of the SLM sections. 27

43 3. Experimental setup 3.1. Experimental setup of VOF-Gen The experiment setup and schematic diagram of the VOF-Gen are shown in Fig. 3-5 and Fig. 3-6, respectively. A He-Ne laser running at the wavelength of 632.8nm is used as the light source. We use the non-polarizing beam splitters (NPBSs) to properly direct the laser beam. The combination of a half wave plate (HWP) and Polarizer P1 is used to adjust the intensity of the incident beam. As we discussed above, two SLM panels in our VOF-Gen setup are divided into four sections, as shown in Fig Each section of the SLM panels is used to realize the control of one degree of freedom. The laser beam first incidents on the SLM Section 1 where the phase modulation will be realized. The optical field after the phase modulation can be relayed to SLM Section 2 with a 4-f system consisted of the Lens L1 and Mirror M1.The distances from SLM to L1 and from L1 to M1 is 300mm which equal to the focal length of L1. Amplitude modulation is realized in SLM section 2 by with the PR system we mentioned in section 2.2 and a polarizer P2, while the transmission axis of P2 is along the horizontal direction. After the amplitude modulation the optical field is relayed to SLM section 3 by using the second 4-f system comprised of lens L2 and L3.Polarization rotation is achieved in SLM section 3 with another PR system. After that the field will relayed to SLM section 4 using another 4-f system consisted of Lens L4 and Mirror M2, where retardation is added to the optical field. Finally, Lenses L5 and L6 form the last 4-f system to relay the optical field from SLM section 4 to a CCD camera. 28

44 Fig Experiment setup of the VOF-Gen Fig Schematic diagram of the VOF-Gen As we discussed above, all of the degrees of freedom should be control by two SLM panels to create an arbitrarily complex optical field. Therefore independent and simultaneous control of the SLM panels is required. This feature can be realized through coding different color channel to each of SLM panels. Specifically, we only code the green color information to SLM 1 and code the red color information to SLM2. Besides, as previously mentioned, each of the SLM 29

45 panel is divided into two halves. Therefore the overall phase pattern is combined through two color channels and multiplexed spatially into the right and left halves as shown in Fig This architecture utilizes the high resolution of the SLM panel such that the entire VOF-Gen can be controlled by one computer that is capable of outputting resolution color graphics. In order to minimize the diffraction effect and maintain high frequency information, 4-f imaging systems will be used in our setup. Spatial filters SF1 and SF2 located in the Fourier planes of the 4-f systems are used to suppress the interferences caused by bulk cube beam splitters as shown in Fig f imaging system and its alignment procedure In order to relay the optical fields between two SLM sections, diffraction effects have to be taken into consideration. The diffraction needs to be minimized so that sharp edges or high frequency information in phase, amplitude and polarization modulations can survive [32]. In our setup, four 4-f imaging systems [33] are used as discussed previously. In order to align the entire 4-f imaging system, I use a pattern with fine features that were designed by a former group member Dr. Wei Han and used the CCD camera to resolve the pattern as a figure of merit for the alignment. In the experiment, 3 horizontal lines crossed with 3 vertical lines are generated where each of the lines is only 100 μm wide shown in Fig. 3-7[32]. 30

46 (a) (b) Fig Pattern with fine features for 4-f imaging system alignment. Each line is 100 μm wide. (a) Pattern with polarized in RCP. (b) Polarized in LP. (c) Pattern with unit amplitude 31

47 Before proceeding to the complex optical field generation, 4-f systems need to be well aligned to eliminate the diffraction effects. The alignment is performed in a sequential order, as shown by the arrows in Fig The last 4-f system between CCD and SLM Section 4 needs to be aligned first. It is worth noting that the patterns are designed so that the degree of freedom to be realized in the specific SLM section can be revealed. Therefore the pattern in the computer to align the 4-f system with fine features is polarized in RCP and the rest of the background of this pattern polarized in LCP as shown in Fig. 3-7(a). The circular analyzer in front of the CCD camera is used to resolve of the RCP component. The distance between the lens L6 and the CCD of the last 4-f system and the circular analyzer should be adjusted until we got the sharpest edge pattern, so that the last 4-f system is well aligned. Then the third 4-f imaging system consisting of L4 and M2 should be adjusted secondly. The second pattern we used to adjust the third 4-f system shown in Fig. 3-7(b) is polarized in horizontal polarization, the rest of the pattern polarized in vertical polarization. While the alignment needs to be performed for the second 4-f imaging system in a similar way where the third pattern as shown in Fig. 3-7(c) with unit amplitude and the amplitude of rest of pattern is zero. The 4-f imaging system s alignment is determined by adjusting the distances to obtain the maximum sharpness where diffraction is minimized. At this point the entire 4-f imaging system is well aligned. 32

48 Fig Flow chart of the VOF-Gen system and 4-f imaging system alignment procedure To compare, the patterns generated by our VOF-Gen system with and without well aligned 4- f system alignment obtained by the CCD Camera as shown in top and bottom row of the Fig. 3-9, respectively. Fig Comparison of the same patterns generated by VOF-Gen system without and with well aligned 4-f imaging systems. (a) Retardation: the intensity is captured after a circular analyzer; (b) polarization rotation: the intensity is captured after a linear polarizer; (c) Retardation: the intensity is captured after a circular analyzer 33

49 As we can see when the optical field captured by a CCD camera without a well aligned 4-f system alignment is blurred, with unclear edges as shown in bottom row of the Fig The high frequency informations are not survive. However, with the well calibrated 4-f systems the diffraction effects have been minimized and sharp edges are obtained. 4. Summary I review the theory and experiment setup of optical field generator (VOF-Gen) that is capable of generating arbitrary optical fields by spatially modulating all aspects of the optical field (including phase, amplitude and polarization) on a pixel-by-pixel basis. This arbitrary complex optical field generator may find extensive applications in areas where exotic input fields are required, such as particle manipulation and beam shaping. In the next chapter, various complex vector fields are generated and tested to demonstrate the functionality and flexibility of the proposed VOF-Gen. 34

50 CHAPTER 4 EXPERIMENTAL RESULTS 1. Introduction Vector optical fields with spatial variant amplitude, phase and polarization distribution have proved useful for a wide variety of applications. Shaping both the amplitude and the phase of optical field yielded non-diffracting beams such as the Airy beams and Bessel beams [1]. Vortex beams with spiral phases have been the subject of considerable interest in optical tweezers and quantum information processing due to its spiral phase wavefront carrying orbital angular momentums [8, 9]. Besides, tailoring the polarization structure of vector beams has drawn a lot of interest owing to their broad potential applications in various areas, such as optical trapping [6], flattop generation [10], focus shaping [11], surface plasma sensing using cylindrical vectorial beams [12]. Furthermore, controllable polarization characteristics within the focal volume can be realized with spatial variant polarized illuminations. One example that attracted recent interest is the so-called optical needle field that is substantially polarized in the longitudinal direction with long depth of focus (DOF) created through the focusing of filtered radial polarization [17-20]. All the above applications require modulations of certain aspect of optical field. 35

51 In this chapter, to demonstrate the functionality and flexibility of the proposed VOF-Gen, various complex vector fields such as flat top beam, radially polarization beam and a UD logo pattern with unit amplitude polarized in RCP while the rest of the window polarized in LCP, are generated and tested. Especially, to the end of this chapter a more complex engineering of the optical field is proposed, where the modulations of phase, amplitude, polarization and retardation are simultaneously realized using VOF-Gen. The engineered optical field could then be focused by a high numerical aperture (NA) objective to produce high-purity optical needle field that expected with flattop distribution along longitudinal direction and an extended depth of focus. 2. Experimental results 2.1. Spatially variant amplitude modulation: Flat-top beam generation A laser beam with flat-top intensity profile is desirable in many applications [34]. Many techniques exist for the creation of FTB external to the laser cavity [35-37], which can be accomplished with low loss, albeit with some complexity in the optical delivery system (e.g., requiring careful alignment and fixed input beam parameters to the shaping elements)[38]. Deformable mirrors were originally developed to correct for atmospheric disturbance, but have proved useful for beam shaping applications, and have been used for producing circular and rectangular flat-top intensity profiles [39]. They have the drawback however that the number of mirror elements is limited, and so the feature size of the beams produced by deformable mirrors is therefore limited [40, 41]. A more common approach today is to use LCD in the form of SLMs to dynamically mimic both amplitude and phase transformations. The advantage of using an addressable SLM is the ability to manipulate the beam intensity profile so dynamically and programmable. 36

52 In order to fully control amplitude modulation, a beam shaping setup is proposed to transfer Gaussian beam into flat-top beam. The beam shaping setup is part of a VOF-gen system consists of a BS, a PR system we mentioned in section 3.2.2, a linear polarizer with transmission axis in horizontal direction as depicted in Fig Fig Schematic diagram of the beam shaping setup The input is a collimated horizontally polarized (x-polarized) Gaussian beam and its goes through the PR setup, where the Sections 2 of SLM panel 1 will load an appropriate phase on the incident beam to perform the transformation of a Gaussian beam into a circular FTB. In order to realize and demonstrate the amplitude modulation, we designed a circular inverse Gaussian beam amplitude pattern. Only the incident field within this circular pattern can go through the polarizer and the transmission for the incident field outside of pattern is set to be zero. Note that the entire window would have been illuminated by the input Gaussian beam without the amplitude modulation. SLM Section 1 is loaded with a pre-compensation phase and SLM Section 3 and 4 both have a flat phase. The phase pattern loaded onto SLM section 37

53 2 is shown in Fig. 4-2.The intensity profiles of the collimated Gaussian beam and the circular FTB with 1.5mm diameter as shown in Fig Fig The circular inverse Gaussian beam phase pattern coded in SLM section 2 Fig Intensity profiles of the (a) incident collimated Gaussian beam (b) FTB of 1.5 mm diameter The result shows that the collimated Gaussian beam is successfully converted into an FTB with fine features such as sharp edges. However, the diffraction pattern has not been entirely 38

54 removed. This is due to the fact that reducing the pinhole size in the Fourier plane of the 4-f system will remove both the diffraction pattern and high frequency components. In order to maintain a high resolution, I balanced the pinhole size so that high frequency terms are well preserved while only allowing a minimum amount of diffraction pattern Spatially variant polarization modulation: radially polarized beam generation A beam of light has radial polarization if at every position in the beam the polarization vector points towards the centre of the beam. Due to the unique polarization symmetry properties [1], the radially polarization light was discovered to have many interesting applications in Optical Trapping [42], nanoscale optical imaging [7] when focused by a high-na objective lens. Many methods of generating radially polarization beams have been reported [21-25]. We show the capability of generate a radially polarized beam using our VOF-Gen. The polarization rotation function will be realized in SLM section 3. After the amplitude modulation, the input beam with FTB intensity profiles can be rotated pixel-by-pixel as we mention in section In order to generate the radially polarization beam, the phase loaded on SLM Section 3 will with an azimuthal dependence. However, there is an extra phase that will be introduced by the amplitude and polarization modulations. Therefore,in order to generate radially polarization beam with a flat phase, a pre- compensation phase term needs to be loaded on to SLM Section 1. To verify the phase pre-compensation, the phase patterns for radially polarization beam with and without pre-compensation phase are shown in Fig Respectively, the intensity profiles captured by CCD camera as shown in Fig

55 Fig Phase patterns for radially polarization beams (a) with pre-compensation phase (b) without precompensation phase 40

56 Fig The intensity profile of radially polarization beam with flat-top amplitude distribution. (a)radially polarization beam without pre-compensation phase; (b) radially polarization beam with pre-compensation phase From Fig. 4-5(a), we can see that without phase pre-compensation, a FTB is obtained by a CCD camera. This can be understood as the additional spiral phase cancelled the polarization singularity at the center of the radially polarized beam. Once pre-compensation phase is introduced a singularity at the center as expected (shown in Fig. 4-5(b)). This confirms that the geometrical phase generated due to the operation of SLM Section 2 and SLM Section 3 is successfully compensated. In general, the phase pre-compensation scheme can be used to compensate any extra phases that are introduced during the whole modulation progress as shown in Eq. (3-13). Furthermore, this confirmation of the phase pre-compensation also serves as another evidence of the phase modulation capability as we discussed in Section 3.1. The generation of radially polarized beam is then shown in Fig The arrows in the upper set of graphs indicate the directions of the linear analyzer in front of the camera and the intensity of each linear polarization component (0, 45, 90 and 135 ) is shown respectively. As shown in 41

57 the figure, radial polarization in the output field is generated. Thus the polarization rotation capability is demonstrated. Fig Intensity profiles of radially polarized beam after a polarizer which the fast axis along the black arrows at 0, 45, 90 and 135, respectively 2.3. Spatially variant phase retardation Right-hand circular polarization (RCP) and left-hand circular polarization (LCP) have phase retardation of +π/2 and -π/2, respectively. In order to demonstrate the phase retardation modulation capability, we designed an UD logo pattern with zero phase and unit amplitude is polarized in RCP while the rest of the window polarized in LCP. The required phase pattern for VOF-Gen and the total field as well as the RCP and LCP components are shown in and Fig. 4-7 and Fig. 4-8, respectively. 42

58 Fig Phase information f or UD logo pattern Fig "UD" logo coded in circular polarization. The total field (left), the "UD" logo (upper right) in RCP and the complimentary UD logo (lower right) in LCP As shown in the Fig. 4-8, circular polarization in the output field is generated. Thus the phase retardation can be achieved by our VOF- Gen. At this point, the capability of modulate each of 43

59 individual freedom describing in intensity profiles has been successfully demonstrated. In the next part, a complex vector optical pattern with multiple parameters is proposed and experimentally demonstrated. In addition, the experimental and theoretical results will be compared Complex vectorial optical fields generation with multiple parameters Complex optical fields for ONF generation In this part, a more complex vector optical pattern with multiple parameters is proposed and experimentally demonstrated. The engineered optical field is designed such that when focused by a high NA objective, a high-purity optical needle field is expected with flattop distribution along longitudinal direction and an extended depth of focus. The investigation of focused electromagnetic field over a three dimensional volume has always been an interesting and important area of theoretical and applied optics [32]. The focusing properties of linearly polarized (scalar) field have been well established by Richards and Wolf [43]. However, beams that possess polarization axial symmetry, hadn t been thoroughly investigated until recent years. With the recent advances in the spatially variant polarization and particularly in the cylindrical vector beams [1], vectorial distribution of the focused electromagnetic fields has drawn significant amount of attentions. Focused with a high NA objective lens, vector field distributions in the focal volume [5] can be exploited for focal intensity shaping [6, 7]. Furthermore, controllable polarization characteristics within the focal volume can be realized with spatial variant polarized illuminations [8 11]. One example that attracted recent interest is the so-called optical needle field that is substantially polarized in the longitudinal direction with long DOF. The unique electric field distribution opens the door to applications in particle manipulation [4-6] and acceleration [44], nanoscale optical imaging 44

60 and so on [7]. During the past few years, researchers have established various methods to create needle-like fields. For examples, Wang et al. [45] achieved a needle of longitudinally polarized light by focusing a radially polarized Bessel-Gaussian beam with a combination of a binaryphase optical element and a high- NA lens. Yuan et al. [46] presented a needle-like field by tightly focusing an azimuthally polarized beam with a high-na lens and a multibelt spiral phase hologram. Hu et al. [47] generated a super-length optical needle by focusing hybridly polarized vector beams with an annular high-na lens. Kitamura et al. [48] demonstrated the needle-like focus generation by radially polarized halo beams. Guo et al. [49] reported the optical needle by tightly focusing of a higher-order radially-polarized beam transmitting through multi-zone binary phase pupil filters. The above-mentioned needlelike fields are achieved by means of transmitting a vector beam through a multi belt pupil filters and then tightly focused by a high-na lens. As the complexity of the conventional system increases, the fabrication of the individual filter design gets more and more challenging, lead to the need for a new method for engineering the radial-variant polarization on the incident field to achieve a needle of longotudinal polarized field without any pupil filters. In this work, we provide a method to create complex vector optical pattern without any pupil filters by using our VOF- Gen. To obtain the desired complex field for ONF generation, a method based on reversing the radiation pattern of an electric dipole has been developed [4]. Owing to the vectorial characteristics of the illumination, a high purity and ultra-long ONF can be realized. The proposed approach for obtaining longitudinal field possesses uniform axial intensity profile, extended DOF as long as 8λ and longitudinal polarization with high purity. The focal field intensity along focus is shown in Fig. 4-9 (upper) and the line scan of the normalized intensity in Fig. 4-9 (lower) [32]. 45

61 Fig Focal field intensity along focus (upper) and its line scan (lower) [32]. The longitudinal coordinate is normalized to the wavelength The depth of focus is found to be 7.9 λ. In order to generate such optical needle field, the required parameters and required phase pattern of the optical field in the pupil plane are shown in Table 1 and Fig. 4-10, respectively. The desired patterns of the optical field in the pupil plane have five discrete ring polarized in radially polarization and each of ring has different amplitude, the phase takes binary values that alternates between 0 and π from one zone to another [50]. 46

62 Table 1: Parameters of desired field along radial direction. RING# Normalized Amplitude Phase 0 π 0 π π Normalized Ring Widths Fig The phase pattern for VOF-Gen The experimental results is shown in Fig The field distribution after polarizer at 0, 135, 90 and 45 and the total field are shown in the upper and lower portions of Fig. 4-11, respectively. From the experimental results, we can see the diffraction pattern around the ring structure has not been entirely removed. This is due to the fact that reducing the pinhole size in the Fourier plane of the 4-f system will remove both the diffraction pattern and high frequency components. In order to maintain a high resolution, we balanced the pinhole size so that high frequency terms are well preserved while only allowing a minimum amount of diffraction pattern. 47

63 Fig Five ring structured polarized in radial polarization. The upper set of graphs show the linear polarization components along 0, 135, 90 and 45 and the lower graph shows the intensity profile of total field Comparison of experimental and theoretical results In previous part, we successfully demonstrated complex vectorical optical field generation and showed the experimental results. The radially polarization of our desired complex field has been was successfully verified by using a polarizer. However, the phase modulation cannot be presented in intensity profiles directly. In order to verify the π phase shift between adjacent rings, an interference setup is used to generate the Interference fringes. The interference setup 48

64 consists of a BS and two mirrors as depicted in Fig In addition, the experimental results of amplitude modulation will be compared with the theoretical design. Fig Interference setup consists of a beam splitter (BS) and two mirrors The input beam after the fully modulation by VOF-Gen will directly goes through the beam splitter and is collected by the CCD camera. The reference beam carry the same phase information with input beam will be reflected by two mirrors and eventually into the CCD. The Interference fringes patterns of desired complex field as shown in Fig Due to the unique polarization symmetry properties of the radially polarization light, If we want see the Interference fringes, we still need put a polarizer in front of the CCD camera to make sure the input beam and reference beam with same linear polarization. 49

65 Fig The Interference fringes patterns of desired complex field. The right graph shows the total interference pattern and the left graph shows zoom in upper left corner image of the total interference pattern From the Fig. 4-13, we can see the interference fringes located in ring 2 and ring 4, 5 was complementary with the interference fringes located in ring 3, which means there has a π phase different between the ring 2, 4, 5 and ring 3. However, due to the intensity of reference beam is only one quarter of input beam and also because the optical path of reference beam does not obey the 4-F imaging system. The edge of interference fringes is not very sharp. Essentially, we only can observe interference fringes in part of the pattern. In order to improve image quality, we design a simple phase pattern with one half coded in π phase and the other with zero phase. The required phase pattern and the Interference fringes patterns are shown in Fig and Fig. 4-15, respectively. 50

66 Fig A simple phase pattern for VOF-Gen Fig Interference fringes patterns of π phase shift For this time the interference fringes is clearly show in the Fig The π phase different between the left and right part of the pattern was been successfully proved. In the next section, we will compare the experimental and theoretical results of amplitude modulation, the result is shown in Fig From the comparison, the amplitude modulation for ring 4, 5 obtain by the CCD camera is match very well to the theoretical designed values. However, the amplitude 51

67 modulation for the first three ring still have some different from theoretical results. The difference may be introduced by the diffraction effects and other interference pattern. In order to maintain a high resolution, we cannot entirely remove interference pattern. That is the main reason that the amplitude modulation for experimental results in these regions are higher than theoretical design. NORMALIZED AMPLITUDE The Compare of Amplitude modulation #RING 1 1 Theoretical results Experimental results Fig The Compare of Amplitude modulation To explore the focusing properties of such a radial-variant vector fields and the changing of generated optical fields how to affect the ONF generation, we calculate the total intensity pattern in the transverse plane and the electric field distribution along the longitudinal direction with different parameters. In order to create a needle-like field with high-purity longitudinal polarization and uniform axial intensity, it is important to tailor phase or amplitude distribution of the incident radially polarized beam. Therefore, in practical application, the deviation of structure, phase and amplitude distribution in each ring will influence the quality of optical needle field generation. Owing the structure and phase modulation of our result pattern agree 52

68 with theoretical design very well, we only show the total intensity direction fluctuations arousing by amplitude changing. After tightly focused our design pattern, the total intensity distribution in transverse plane and the electric field distribution along the longitudinal direction are illustrated in Fig Fig. 4-17Intensity distribution and the electric field distribution in the transverse plane (the polarization direction for this ONF is represented by arrows) (a) Overall picture calculated by theory amplitude value. (b) Overall picture calculated by experimentally obtained amplitude value. (c) Zoomed-in picture for the area enclosed with red dashed line in (a). (d) Zoomed-in picture for the area enclosed with red dashed line in (b) The parameters of the outermost region (Ring #4 and #5) will strongly influence the longitudinal components in the focal region while axial intensity distribution of the focal field and DOF mainly depends on the characters of the inner zone (the first three rings) [4]. The simulation results based on experimentally obtained amplitude value are shown in Fig. 4-17(b) and 4-17(d) respectively. The out-most annular zone has the largest incident angle and the 53