Ultrafast Microscopy of Plasmonic Vortex Fields

List of Figures

Figure 1.3 Dispersion relation of SPPs at Ag/Vacuum interface with E E 1 AEE i E ................ Figure 1.4 Schematic of Archimedean Spiral coupling structure. ......................................... Figure 1.5 The basic in-plane SAM distribution of the Skyrmion. ........................................ ........................................................................................................................................... FIB. ................................................................................................................................... Figure 2.13 Principle of Yee's algorithm. ................................................................................. xi Figure 3.2 Time evolution of the plasmonic vortex within one optical cycle. ........................ 50 Figure 3.3 Two-pulse autocorrelation trace and Fourier components of ITR-PEEM experiments. ..................................................................................................................... 51 Figure 3.4 Experimental (a) and simulated (b) x-component of the SPP vortex field in one optical cycle for the SPP coupling structure in Fig. 3.1. .............................................. 52 Figure 3.5 Derived and Simulated Electric field componets. .................................................. 54 Figure 3.6 Three dimensional representation of Poincaré polarization sphere. ................... 55 Figure 3.7 The Stokes parameter maps of the plasmonic vortex In Fig.3.2 .......................... 56 Figure 3.8 Line profile along y-axis at center of Simulated stokes map in Figure 3.7 ......... 57 Figure 3.9 L-line sigularities of plasmonic vortex. ................................................................... 58 Figure 3.10 Line profile pass through the L-line in Figure 3.9 ............................................... 59 Figure 4.1 Schematic of the SPP coupling structure with E ~J L E ¯E E E and defination of the coordinates. ...................................................................................................................... 63 Figure 4.2 Vectorial Plasmonic field ......................................................................................... 66 Figure 4.3 Evolution of the vectorial SPP field components. .................................................. 67 Figure 4.4 The distribution of magnetoelectric coupling pseudoscalar fi . .................. 68 Figure 4.5 Plasmonic spin Skyrmion texture. .......................................................................... 70 Figure 4.6 The helicity angle of in-plane SAM. ........................................................................ 72 Figure 4.7 Skymion boundary and topological charge. ........................................................... 73 Figure 4.8 (a) schematic of the ITR-PEEM experiment. (b)(c) Static PEEM images of plasmonic vortices wit (b) E x J L J F and (c) E x J L J E . ..................................................... 75 Figure 4.9 ħ E E x Fourier filtered ITR-P E E M i m a g e a t 2 ~ 2 3 . 1 fs. ......................................... 76 Figure 4.10 Experimental vectorial field components of plasmonic vortex. ......................... 77 xii Figure 4.11 Experimental SAM spatial distribution at the SPP vortex with OAM E x J L J F . The dashed lines indicate circle with radius E . .......................................................... Figure 5.1 Schematic design of the Archimedean spiral coupling structure. ........................ Figure 5.2 Time dependent " field from analytical model of the dual vortices................... Figure 5.3 FDTD simulated field of dual vortices. ................................................................... Figure 5.4 T h e a n a l y t i c a l m o d e l t o d e s c r i b e t h e m o m e n t u m t r a n s p o r t b e t w e e n t h e v o r t i c e s . ............................................................................................................................ Figure 5.5 Defining the Poynting vector-defined momentum flow in dual vortices system. Figure 5.7 Scanning electron micrograph of the dual Archimedean vortex generator........ Figure 5.8 Schematic of the ITR-2P-PEEM experiment. ........................................................ Figure 6.1 Static PEEM image of SPPs generated by double line CS. ................................. Figure 6.2 Different frequency components of the 2PP signal in Figure 6.1(b) after Fourier filtering. .......................................................................................................................... Figure 6.3 Simulated instantaneous amplitudes of different field components at the illuminated side of the sample with different time delay........................................... Figure 6.4 Simulated Time average profiles of quartic ......................................................... xiii Figure 6.8 The line profile along the propagating direction of experimental static PEEM image at center of parallel slits structure with different slits separation ................ Figure 6.9 The line profile of Fourier filterd static PEEM images in Figure 6.7. ............... Figure 6.10 Calulated line profile at center of parallel lines structure with different separations using Equation 6.1 with FDTD simulated field components . .............. xiv List of Tables Table 5.1 The separation in units of i E E E =530 nm between two vortex CSs to generate the With the understanding of excitation mechanics and topological properties of these plasmonic vortices, we gain the fundamental knowledge of light-material interaction on the nano-femto scales.

Introduction of plasma model

The interaction of metal and electromagnetic wave can be well defined in a classic frame based on macroscopic Maxwell ' s equations 6 . When considering the interaction with matter, four macroscopic fields the dielectric displacement D, the electric field E, the magnetic field H and the magnetic induction B are linked via the polarization P and magnetization M by

where 4 and 4 are the electric permittivity and magnetic permeability of vacuum, respectively.

In linear, isotropic and nonmagnetic media, we can define the following relations:

where represents the dielectric constant (electric permittivity) and is the relative permeability of the material.

In this thesis, I only consider the optical properties of a noble metal, specifically Ag, which can be modeled as a free-electron gas using the Drude model 7 . To describe plasma properties, details of the lattice potential, electron-electron interaction, electron-ion interaction and electron-phonon interaction are not taken into consideration. Instead, some of the aspects of the band structure will be incorporated to define the effective optical mass of electrons. When a time-periodic external electromagnetic field is applied, the electrons oscillate in response against the fixed positive ion cores; their motion is damped with a characteristic collision frequency ª J L s º , where º is the relaxation time of the free electron gas. For typical metal, º is on the order of s r f s at room temperature. However, for noble metals like gold, this simple approach is limited due to inter-band transitions occurring in the visible range near the plasma frequency 8 . This does not happen for silver 9 .

When a time dependent electric field : ; is applied to a metal, the motion for an electron of the plasma sea subjected to an external electronic field is defined by: where the applied field is assumed to have a simple harmonic time dependence, : ; J L A ? where I is the mass of a free electron. The solution of equation 1.5 can be written as which describes the periodic electron displacement with respect to the fixed ion background. The displaced electron will contribute to the macroscopic polarization, J L J F J A : ; , where n is the electron density, as shown in Figure .1.1, where the plasmon is excited by external electron bombardment. Therefore, based on equation (1.5), we can rewrite the equation 1.1 by where ae ª 6 J L AE .

, -

is the plasma frequency of the free electron gas. According to the Drude model, the dielectric function of free electron gas is given by:

The complex dielectric function : ae ; is essential for describing electronic response to optical excitation via reflection, absorption and refraction 10,11 . Among all the noble metals, silver is very unique, because its inter-band absorption only occurs in the UV range, which means the real part of its complex dielectric function, 4 A > : ae ; ?, will pass through the 4 A > : ae ; ? J L r condition in the interband spectral region 9 . According to equation (1.8), the epsilon near zero (ENZ) condition defines the bulk plasma resonance frequency, in which region the free electron can responded collectively to external electromagnetic fields 9 .

: ; J L A I J k ae 6 J E E ª aeJ o : ; AE Equation 1.6 J L 4 : s J F ae ª 6 ae 6 J E E ª ae ; : ; AE Equation 1.7

: ae ; J L s J F ae ª 6 ae 6 J E E ª ae Equation 1.8

Surface Plasma Polaritons

When the collective oscillating modes are excited locally at the metal surface, they can propagate along a metal/dielectric interface. Such propagating modes are named as surface plasmon polaritons (SPPs), which take form of electromagnetic wave on the dielectric side and charge density wave in the metal side 12 .

To describe the SPPs propagating mode, we solve M a x w e l l ' s equations at metal/dielectric interface with the appropriate boundary condition. By assuming a harmonic time dependence of the electric field, the wave equation which is known as the Helmholtz equation can be written as, distance away from the surface or interface, solution for the wave equation (1.9) in the dielectric part ( VJ P r ) is as follow, : V ; J

L # 6 A A ? AE Equation 1.10 " : " ; J L J F # 6 ae 4 6 A A ? AE Equation 1.11 ! : " ; J L E # 6 ae 4 6 A A ? AE Equation 1.12 And the solution in the metal part ( VJ O r ) is, : " ; J L # 5 A A ? AE Equation 1.13 " : " ; J L J F # 5 ae 4 5 A A ? AE Equation 1.14 ! : " ; J L E # 5 ae 4 5 A A ? AE Equation 1.15

where E o E E E is the complex wave vector along the interface, : i J L s AE t ; represent the evanescent z-component of the wave vector perpendicular to the interface in the two media, and 5 and 6 are the dielectric constants for the metal and dielectric media, respectively. As shown in Figure 1.2, the SPPs wave carry a spin angular momentum (SAM), whose direction is locked transversely to their in-plane k-vector.

The boundary conditions of Maxwell ' s equations require the and ! fields at the metal/dielectric (vacuum) interface to be continuous, such that # 5 J L # 6 and 5 J L J F 6 . Thus, a propagating mode only exists when the real part of the dielectric constant is negative for the metal ( 4 A : 5 ; J O r ) and positive for the dielectric ( 4 A : 6 ; J P r ). Furthermore, based on continuity condition at the interface, we obtain the dispersion relation of the propagating mode at the metal/dielectric interface, as, axis is normalized by the bulk plasmon frequency ae ª . The blue line represents the light line in the free space, which is defined by the relation ae J L ? G. The dotted line shows the experimental dispersion function of the SPPs mode at Ag/vacuum interface, calculated based on the refraction index data of polycrystalline Ag from Yang 8 . The upper branches represent the dispersion of light in the solid, which approaches the bulk plasmon frequency ae ª 1 u z e V . And the lower branches represent SPPs, which approaches asymptotically the nonretarded surface-plasmon frequency ae ª 1 u y e V for Ag 13,14 . J L J ¤ 5 6 5 J E 6 AE Equation 1.16 As the SPP wavevector increases, the dispersion curve flattens out moving away from the light line as its frequency approaches its asymptotic the surface plasmon (SP) frequency limit ae ª .

According to the classical Drude mode, the SP frequency of a free electron metal/vacuum interface is ae ª J L ae ª ħ t ⁄ . The SP frequency, ae ª , ideally is determined by the properties of the free electron gas and its dielectric environment, according to ae ª J L ae ª J ¥ s J E 6 ⁄ , where 6 is the dielectric constant of dielectric 7 .

Because the wave vector of SPPs is complex, their propagation is damped with an energy attenuation length given by . J L > t + I J k G ª ª J o ? ? 5 . For perfect Ag/vacuum surface, the propagation length falls in the range from 1 ª ª ª when the energy of the excitation light approaches ae ª , to 1 s r r I in the long wavelength limit 7,16 .

Excitation and Imaging Surface Plasmon Polaritons

As I introduced in the last section, because there is a momentum mismatch between excitation light and the SPPs it generates on a flat surface, we need a practical method to couple them. A traditional way to perform this is with a Kretschmann geometry, where the excitation light enters at an angle from a high-refractive-index medium through a thin metallic film 17 . This is inconvenient for many reasons, and is incompatible with PEEM experiments, where the light generates photoelectrons by illumination from the low index, vacuum. A common method to launch SPPs at a flat metal/dielectric interface is to use a variety of surface asperities such as gratings 18 or slits 19 . Slits have many applications that are derived from their design. For example, the radial SPP phase structure can be defined by a slit in form of an Archimedean spiral 20 , which will be discussed in chapters 3-6.

Next, I discuss the SPP field imaging. Surface Plasmons were first detected as electromagnetic energy loss features of metal/dielectric interfaces by electron energy loss spectroscopy (EELS) 7 . Subsequently, several other techniques have been developed to image SPPs in real space with special resolution down to a few nanometers. A practical technique to image SPPs on the nanometer scale is near-field scanning optical microscopy (NSOM) [21][22][23][24][25][26][27][28] . To overcome the diffraction resolution limit of light NSOM uses scattering of the evanescent SPP near-field by a sharp probe tip to achieve sub-nanometer spatial. In this technique, the sharp tip scans cross the sample surface to map out the local SPP field strength. Other techniques, such as leakage radiation microscopy (LRM) 29,30 , fluorescence microscopy 31,32 , are also able to image SPPs fields with subwavelength spatial resolution. Combining with two-pulse interferometry, the mentioned methods can be used of imaging of temporal evolution of plasmonic fields 33 . Such typical microscopy, however, requires spatial raster scanning across the sample surface, which severely limits the capability of recording both the space and time movie of SPP fields evolving at the speed of light.

Such limitations are solved by the interferometric time-resolved multiphoton photoemission electron microscopy (ITR-mP-PEEM) [34][35][36][37] , which is a cutting-edge technology for recording nano-femto movies of coherent field dynamics. This method was developed by Atsushi Kubo in our group at the University of Pittsburgh. The details and principle of PEEM will be introduced in chapter 2.

SPP Spin Angular Momentum (SAM)

The Spin pseudovector of an electromagnetic wave is given by, where, the case of SPP fields, E and H are its electric and magnetic components. A key feature of surface evanescent wave such as SPPs is that they carry non-zero transverse spin-angular momentum (SAM) 29,38,39 that is orthogonal to their propagation vector independent in direction but not the amplitude of the SAM of light that generates them. This is an inherent property of evanescent waves, referred to as the quantum spin-Hall effect of light; I will discuss this further in Chapter 4. Next, to quantize the spin of SPPs, based on equation 1.12, if we consider a general case of SPPs wave, where polarization vector contains all three Cartesian components, propagating in y direction and having evanescent penetration in the VJ P r half plane, which is the dielectric side, then, the electric filed E should be represented as follows, where G ª ª J L G !J J E E G E J , # 4 is the wave amplitude, and the complex number n defines the polarization state with helicity, which is given by the ellipticity of polarization, OEJ L

6 ´ < AE = 5 > AE . B ->J F s AE s ?. The energy flow of the SPP fields is defined by the Poynting vector, J L 4 A : B J H ; , J L s v ae : 4 B J H J E 4 B J H ; AE Equation 1.17

which can be decomposed into two terms as J L J E : here represents the observable, canonical or orbital angular momentum (OAM) density, which represents the energy transport, while represents the SAM density, which does not contribute to the energy transport. is defined by the spin angular momentum density in Equation 1.17 via the relation, Furthermore, by substituting the electric field from Equation 1.14 into Equation 1.13 and 1.15, we obtain expressions for and as follows, where S J L ? $ 6 A ? 6 º donates the spatially inhomogeneous energy density of the SPP wave. As is evident, both momentum and spin of SPPs wave in Equation 1.13 contain a transverse component in x direction. Unlike the transverse component of momentum ! , however, which dependents on helicity OE, the transverse component of the spin ! is independent of helicity OE. It signifies that an evanescent wave such as SPP always carries a non-zero transverse SAM independent of the helicity of the incident light employed to launch it. Additionally, the transverse component of the spin ! depends on wave vector ! along the propagating direction, which demands that counter-propagating SPPs must have the opposite transverse SAM direction. This explains the momentum-spin locking of SPPs wave, which is an inherent property of SPP waves that can be experimentally observed 34,40 .

J L s t : ‚ J H ; Equation 1.19

Topological Spin texture

In the previous section, I presented the SAM of SPPs field and how it generates plasmonic vortices. In this section, I will introduce the principle of topological spin textures, which include the type and key factors of the topological spin textures of plasmonic vortices.

The topological properties of the SAM spin textures of SPP fields can be characterized by the topological charge 50 , Q, defined as where, & J L 5 8 S fi :

; is defined as the charge density, S : T AE U ; J L S : N c o s AE N s i n ;

is the normalized SAM of the SPPs field defined by Equation 1.17, where N and are the radial and azimuthal co-ordinates, and the integrating region is the domain confining the topological quasiparticle. The topological charge of a quasiparticle quantifies how many times the normalized SAM vector wraps around a unit polarization sphere, which is known as the Poincaré sphere 51,52 .

For example, for a meron quasiparticle, the vector starts from north or south pole and terminates at the equator, whereas the Skyrmion wraps from one pole to the other. Next, we consider the consider the normalized SAM vector in spherical coordinates with the longitude and latitude angles and , which can be expressed as

S J L : c o s : ; s i n : N ; AE s i n : ; s i n : N ; AE c o s : N ; ; . Then, the topological charge, Q, can be expressed as in terms of two additional topological numbers. The first factor named the polarity, L , whose sign, ±1, defines whether the spin at the center of the quasiparticle points up or down. And the second factor is the vorticity, R AE which describes the in-plane components of S . Q J L J & J L s v Ł J S fi : S T J H S U ; AE Equation 1.24 L J L s t J -s i n : N ; @ : N ; J L J G s Equation 1.26 R J L s t Ł J -: ; @ AE Equation 1.27

Furthermore, if we consider diverse helical distributions, an initial phase V should be added: : ; J L R J E V, as shown in Figure 1.5. With different R and initial phase V, we can get Q J L s v Ł J -@ N J -@ @ : N ; @ N @ : ; @ s i n : N ; J L s t J -s i n : N ; @ : N ; fi s t Ł J -: ; @ J L L fi R Equation 1.25 several types of Skyrmion SAM spin textures, generally referred to as the Néel-type, Bloch-type and anti-Skyrmion.

Overview of LEEM/PEEM Instrument

By separating the roles of electrons and photons in achieving the space and time resolution,

-PEEM is non-perturbative ultrafast microscopy method for capturing the vectorial electromagnetic fields in the form of SPPs with sub-optical cycle temporal and sub-opticaldiffraction limited spatial resolution, which I will discuss in detail in Chapter 3. The PEEM technique was established by Ernst Bauer and his collaborators in 1970s by utilizing spatial imaging of the photoelectric effect to produce high resolution images of structural and electronic properties of surfaces. This method enables experiments to visualize surface structures and electronic properties with exceptional detail, offering insights into surface phenomena with elemental contrast at the nanoscale 53 . For the experiment describe in my thesis, I have imaged the SPPs field with a commercial FE-LEEM P90 instrument from Specs GmbH with specified resolution of ~8 nm in the PEEM mode. This instrument was manufactured based on the design developed by Rudd Tromp of IBM 54,55 . There are two distinct modes of operation in this commercial LEEM/PEEM system, the PEEM and LEEM modes. In the PEEM mode, the PEEM signal is generated by a light source that drives the linear or nonlinear photoelectric effect. In experiments that I will describe, I use a broadly tunable femtosecond pulse laser system, which produces 20~25 fs duration pulses, at a repetition rate of usually 1 MHz in the ~270-900 nm range. As shown in the Figure 2.1, we modify the purchased PEEM instrument by replacing two optical access windows with low dispersion CaF2 lenses that serve for focusing light at two different incident angles, either oblique with y r " , or normal incident angle. The normal incident lens placement is indicated by the black circle in the figure. Besides focusing the excitation light with a variable translation, the lenses maintain ultrahigh vacuum. The electrons photoemitted by a nonlinear process are gathered and deflected by electron optics to be imaged on a 2D electron detector. The photoelectron imaging is performed by amplifying single photoelectron signals with a microchannel plate (MCP) and recording the resulting electron pulse impact on a phosphorescence plate with a CCD camera. This records the photoemission image distribution from the sample surface.

The PEEM imaging can be performed with single pulses or with a time (phase) correlated identical pulse pair. In a time-resolved mode the excitation is performed by excitation with a correlated pump-probe pulse pair. Transmitting the excitation laser pulses through a Mach-~50 as precision, to record the dynamic photoemission electron microscopy images.

In the LEEM mode, a surface is interrogated by recording the reflection of electrons from a cold field emission electron gun. The same electron optics as in the PEEM mode record the static or dynamically changing surface structure as communicated by the reflected electron signal. In my research, I only performed experiments in PEEM mode, which is a cutting-edge method to record the vectorial SPPs field. I will also describe, however, the ultra-high vacuum (UHV) components of LEEM mode in the following section to provide a brief outline of FE-LEEM P90 instrument. The FE-LEEM P90 from Specs is a complex instrument combining a variety of UHV components. In following section, I will introduce the most important sub-parts and working principle of the instrument; the detail of each part of the instrument can be found at the official website of Specs 56 and chapter 2 in the thesis of Yanan Dai 57 , who contributed to setting up the instrument in our laboratory. Figure 2.1 shows an overview photographic image of the FE-LEEM P90 instrument with its essential parts labeled.

Electron Optics

Figure 2.3 shows the schematic of the FE-LEEM P90 system that emphasizes the electron optics system. Here, I will provide a brief introduction about how electrons are formed and processed by optics. As I mention in the last section, in the LEEM mode, electrons are emitted from the cold field emission gun and accelerated to s w k e V . The electrons pass through the magnetic gun lens and condenser lens to adjust the magnification of the illumination system and to focus the electron beam at the entrance plane of the Magnetic prism array (MPA). As shown in field to deflect the electron beam by 90 degrees and refocuses the beam in a symmetrically located exit plane in the objective lens arm. The transfer lens, M1, focuses the electron beam on the backfocus plane of the cathode objective lens. After decelerated by applying high voltage between sample surface and objective lens, a well-defined, collimated low energy electron beam (typically having energy of a few eV) impinges onto the sample. In the crystalline sample, these low energy electrons undergo diffraction, and after reflection and re-acceleration by passing through the electric field between objective lens and sample surface once again, a low energy diffraction (LEED) pattern is formed in the back-focal plane of the objective lens. The objective lens can also form a real-space image of the sample surface at a larger distance of around x r m m . In a PEEM experiment, instead of the backscattered electrons from the sample surface, we use photoemitted electron as the source of sample imaging. The photoelectrons can be linearly excited by either a UV lamp, or nonlinearly by an intense femtosecond laser pulse. As shown in Figure 2.3, the incidence angle of the laser pulse can be set either to y r " from surface normal, or r " for the normal incident excitation but not both simultaneously. The experiments that I report use the normal excitation geometry. When photoemitted electrons are imaged, the back-focal plane has a superposition of the angular distribution of the photoelectrons from the irradiation spot.

The transfer lens M1 refocuses the LEED pattern onto the entrance plane of the MPA1 and forms a further magnified virtual real-space image in the center of MPA1. With the electron velocity reversed, the MPA1 deflects the electron beam by 90° towards the imaging optics column.

The MPA1 transfers the LEED pattern onto its exit plane, towards the electrostatic transfer lens placed midway between MPA1 and MPA2. The transfer lens transfers the real-space image from the diagonal of MPA1 to the diagonal of MPA2.

In general, electron optics, such as MPA and objective lens have chromatic and spherical aberrations, which limit the resolution of electron microscopes. By symmetry, the chromatic aberration from MPA can be canceled as the electrons are deflected over a { r " angle by MPA2, which places the diffraction pattern at the center of the magnetic lens M2. The transfer lens M3

then transfers the LEED pattern to the reflection plane of electron mirror optics, which performs the aberration correction to cancel the chromatic and spherical aberrations % and % of the objective lens. The aberration correction system consists of a set of electron mirrors (not shown) including a front lens at the ground potential, and two ring-shaped elements at intermediate potentials V3 and V2, followed by a mirror electrode at potential V1. By adjusting the combination of these three potentials while observing a PEEM image, we strive to obtain the best combination, to compensate for the chromatic and spherical aberrations of the objective lens. After correction, the electron beam is reflected back into the MPA2, where it is further deflected by 90° into the projector column.

Projector Column

The projector column is a system of four lenses that project the sample image onto the 2D imaging detector; it is followed by three sets beam deflector coils to align the beam. The LEED pattern located at the diagonal plane of the MPA2 is placed in the center of the first electrostatic projector lens, P1, to reimage it in front of P3 for further magnification. Then the second electrostatic projector lens, P2, is used to switch imaging between the real and momentum space, by applying a DC current to activate the lens. If the current is zero, P1 will transfer the real image from the prism diagonal to the object plane of P3. If P2 is activated, it will transfer the diffraction from the contrast aperture plane to the P3 object plane. Next, P3 magnifies the image of diffraction pattern present in its object plane into the object plane of the final projector lens P4. The final projector lens, P4, consists of two identical, closely coupled lenses (P4A and P4B). At lower excitation, when the focal length of each is equal to the lens separation, the pair forms an upright image with low magnification. At high activation, when each lens is set to its shortest focal length, the pair forms an inverted image with significantly larger magnification. Additionally, the whole projector column works at high current, causing the electrostatic lens heating with a consequent thermal drafting of imaging. Therefore, the water-cooling system is installed to maintain the system operating at a manageable temperature and avoid heating damage. Finally, P4 projects the image onto the detector, where it is recorded, digitized and sent to computer for further analysis.

Non-colinear Optical Parametric Amplifier Excitation Source (NOPA)

For ultrafast PEEM experiments, the sample surface is illuminated by an external optical system to launch SPPs waves and to induce nonlinear multiphoton photoemission. The optical system, I employed is based on a non-colinear optical parametric amplifier (NOPA) 59 , as shown schematically in Figure 2.8. The excitation laser source for exciting NOPA, is an Yb-doped fiber oscillator-amplifier laser system (Clark-MXR Impulse). This pump laser has a center wavelength at ae 4 J L 1035 nm and pulse duration of 250 fs 35,60,61 . Typically, the fiber laser is operated at 1MHz reputation rate with an output energy of s J J per pulse. On entering into NOPA, the output from the fiber laser is split by a polarization beam splitter (BS); 80% of its power passes through two different >-BaBO3 (BBO) crystals to produce the second ( t ae 4 ) and third harmonics ( u ae 4 ) of the fundamental beam. The BBO crystals arrangement uses type-p h a s e m a t c h i n g f o r t ae 4 second harmonic generation, and a type-f o r ae 4 J E t ae 4 = u ae 4 sum frequency mixing, which are consistent with the incident light polarization. The remaining t r ¤ of ae 4 light is focused into a sapphire plate to produce a white light continuum by self-phase modulation 62,63 (indicated as white light generation WLG) . The generated white light is further split by a dichroic beam splitter that reflects above and transmits below 650 nm. The longer wavelength part of the white light (J P

x w r n m ) is combined with the t ae 4 pump beam at another BBO crystal, where a part of white light, which acts as a seed light is selectively amplified by the t ae 4 pulse in a parametric pulse. By adjustment of angle of the BBO crystal and overlap of the white light and pump beam the amplified output can be tuned in a range between 650 nm (1.91 eV) to 900 nm (1.38 eV) 57,59 . The wavelength of the final output is tuned by adjusting the spatial overlap and relative temporal delay between the seed light and pump light in the BBO crystal. Such amplification process is called non-colinear optical parametric amplification (NOPA) process, and the criterion for amplification is known as a phase matching condition, where the momenta of the pump, seed and final output are set to conserve. Similarly, the shorter wavelength region of the white light (J O x w r n m ) meets the u ae 4 beam to produce the final laser output in the wavelength range of 500 nm (2.48 eV) to 650 nm

(1.91 eV) 57,59 . In addition, both output pulses can be frequency doubled using a self-build Second Harmonic Generation optical system to extend the wavelength range down to around 270 nm (4.59 eV). Through the whole wavelength the, the pulse duration of the final output is well compressed to be around 30 fs or less, as shown in Figure 2.9.

Mach-Zehnder Interferometer for Coherent Pump-Probe Measurements

Mach-Zehnder Interferometer (MZI) is an instrument for performing interferometric measurements, particularly with ultrashort pulses. In my experiment it is used to control and measure the relative phase (delay) between two copropagating light beams by recording their interference. I use it to generate two identical pulses with a known phase pair with known phase delay. The schematic of the MZI is shown in Figure 2.10. It consists of four mirrors, which form two retroreflectors in two separate paths, and two ~50% beam splitters, which send the pulses into each path and ultimately recombine them as they exit to propagate collinearly with a well-defined delay. One of the retroreflecting mirror pairs is translated to vary the delay between the pumpprobe pulses. The beam path length in one of the interferometer arms is scanned by translating the flexure mirror stage with a piezoelectric actuator. The stage translation by amount x generates a time delay between the pulses of J L t T ? ⁄ . The piezoelectric motor is controlled by applying external voltage through a LabView virtual instrument (VI) to the actuator power amplifier. By controlling the parameters in LabView program, we can control the step or motion speed of the piezoelectric motor, which will control the time delay step ¿ of pump-probe pulse in ITR-PEEM experiment. A typical step in my experiment is around ~0.1 fs. Next, the two pulses meet at the second beam splitter, where the pulse pair further split into two pulse pairs. The MZI has two output paths: The path with balanced dispersion excites the sample, while the other output is sent through a monochromator, to record a sinusoidal interference pattern at the central wavelength of the laser pulse. This path is also observed dynamically on an oscilloscope to optimize the MZI alignment. Theoretically, there is no resolution limitation, because an infinitely small change in electric field gives rise to an infinite mechanical displacement. The real world, however, offers some limits in resolution, which are caused by electrical, mechanical acoustic and thermal noise.

Therefore, the MZI requires to be operated in very stable measurement conditions to reach the highest position resolution. In our lab, the MZI is operated in stable measurement conditions as follow to get a better position resolution: (1) a well-grounded environment with stable temperature to avoid the electrical and thermal noise. ( 2) an actively damped table to provide vibrationally isolated condition.

(3) cover the lid during measurement to avoid the effect of air fluctuation.

Photoemission Electron Microscopy (PEEM)

PEEM utilizes spatial imaging of low energy photoemitted electrons, following lightinduced photoemission. If photoemission occurs as a linear response, the microscopic PEEM image will record the spatial distribution of light intensity as well as those of the sample electronic properties and material. It relies on the spatial variation of local electron emission to generate contrast in the captured photoemission images. Our PEEM experimental setup is operated with either a UV discharge lamp or a femtosecond pulsed laser source for exciting the linear or non-linear photoemission. When the incident light field with photon energy that is higher than the work function of the sample, single photon absorption is sufficiently energetic to excite electrons into vacuum. The photoelectron spatial distribution only depends on the material properties or structuring, which provide contrast in a PEEM images with only structural information of the sample. If the work function of the sample is higher than the photon energy of the light source, the electron cannot be excited through linear or single photon emission process. However, if the incident light is a femtosecond laser pulse with a high peak power, a non-linear, two or more photon photoemission process can take place when the total absorbed photon energy is sufficient to excite electrons to energies above the work function to be emitted into vacuum 33,69,70 . In that case, PEEM image is dominated by the local field amplitude to the t J power, where n is the photonic order of photoemission process [33][34][35]37,71 . Therefore, the PEEM image signal can be expressed as,

+ ħ ħ ˘ : T AE U AE P ; 1 J -: : AE ! AE ae ; ; 6 l ae E x ? ¶ AE Equation 2.1

where the 2 is the total polarization field induced at the sample surface; the integration of the signal is over the total interaction time between the excited laser pulse and local surface field because the photoelectron position recording is much slower the duration of the excitation pulses. described in this thesis, where two purple lines show the material absorb two photon energy quanta and two black line show electron is excited into vacuum through non-linear two-photon photoemission (2PPE) process. I have used incident laser pulse train of wavelength ª ¯J L w w r n m ( D R ¯J L t t w e V ) to illuminate polycrystalline silver surfaces with FIB milling generated structures [examples are show in section 2.2.2]. The wavelength of incident laser ª ¯ is chosen to have feature, which can be a coupling structure formed by FIB milling, as shown in section 2.2, is illuminated by external laser pulse to generate a continuous momentum distribution for coupling of the external light into an SPPs field. The illumination of a sharp structure acts as a source of momentum to couple into the SPP field. As shown in Figure 2.12(b), there are two different incident laser pulse geometries that can be used in a PEEM experiment. Conventionally, the laser pulse can incident at an oblique angle of 1 y r " with respect to the surface normal (J E V) direction, which is defined by the light access to the sample surface from the side. Alternatively, the light can enter in the normal direction ( J L r " ) taking advantage of the clear path through the magnetic prism structure in front of the sample. The photoemission signal is excited by a periodic modulation (blue line in Figure 2.12 (b)), which we refer to as a surface plasmon polariton interference beating pattern 33 . This beating pattern is stationary and depends on the time structure of the excitation light; it records the difference between the local speeds of the external light and

the SPP pulse that it generates [73][74][75] . Since at a given frequency the wave vector of SPPs field, real part of complex wave vector, is always greater than that of incident light, an interference pattern between the two fields forms at the difference between the in-plane component of the excitation field, oe J L oe s i n , and the SPP field, . Such difference defines the periodicity of the beating pattern, ª » , as,

is the in-plane projection of the excitation wavelength for the oblique angle of incidence, , ª ª ª J L 6 is the SPPs wavelength, and is the in-plane angle between oe and . However, in the case of normal incidence, J L r " , oe J L r such that Eq (2.2) gives ª » J L ª ª ª . This implies that under normal incidence illumination the wavelength of the 2PP spatial interference beating pattern is the SPPs wavelength 76 , making possible direct imaging of SPP field movies. Therefore, for all my experiments and simulations in this thesis, I employed normal incident optical excitation to directly perform imaging of SPP dynamics.

Interferometric Time-resolved PEEM (ITR-PEEM)

In the case of a time-resolved PEEM experiment, unlike the static PEEM experiment in the previous section, there are two identical ultrafast laser pulses for pump and probe SPP field. Such pair of phase lock pump-probe pulses with certain temporal separation, , is generated by the MZI, as described in section 2.3.3, and then, illuminate onto the sample surface. Follow the same mechanism in Figure 2.12(b), SPPs launched by pump pulse at the structure edge propagate along the sample surface, while the probe pulse, with certain time delay, , interferes with the pump excited SPPs and liberates photoelectrons in a 2PPE process. For a specific delay time , when the probe pulse arrives at the sample surface, the SPP fields launched by the pump pulse have already propagated for a distance determined by the delay time. Therefore, the probe pulse can create interference patterns with the pump excited SPPs at the regions the pump excited SPP fields have already reached in the time scale . By scanning with a small temporal delay step, one could obtain a dynamic motion of the interference pattern (in femtosecond time scale), which reveals the time evolution of the pump excited SPP fields. Thus, according to equation (2.1), a pump-probe time dependent interference pattern with specific delay time can be obtained by the following equation:

where the photoemission signal is integrated over one cycle of pump-probe excitation. According to the above discussion, if the delay time is sufficiently small, there is a dynamical interaction signal where the pump pulse generates an SPPs field wave packet, and the external probe pulse interferes with it. In that case, ITR-PEEM enables us to record the generation, propagation and decay of SPPs fields, which is the main experiment method I employ in this thesis.

Simulation Framework

Since PEEM experiments obtain information of vectorial electromagnetic fields integrated in the optical cycle of excitation, a numerical method is necessary to reproduce all field components to simulate a PEEM experiment. In this section, I will introduce a vectorial simulation method, which I employ to reproduce the PEEM experiments in this thesis, named finite difference time domain (FDTD) algorithm. The FDTD algorithm is a fully vectorial simulation method, w h i c h s o l v e s M a x w e l l ' s e q u a t i o n s a t e a c h p o i n t i n s p a c e a n d t i m e , c o n s t r a i n e d b relation to evolve the electromagnetic fields, until a steady state solution is obtained 77 . In this thesis, I have performed FDTD simulation by a commercial software, Lumerical FDTD.

The FDTD algorithm is based on the curl equation of the electromagnetic fields as an update method, when there is a change in the electric field, the surrounding magnetic field is updated and then resulting in surrounding electric fields. This update process expends the entire problem space, and over designated time window obtains either a steady state solution, or a timepropagating field solution. In a time-domain method, the update equations are all based on timed e p e n d e n t M a x w e l l ' s e q u a t i o n s , where the validity of the remaining M a x w e l l ' s e q u a t i o n s is ensured by choosing an appropriated initial field configuration 78 , as following:

ˇJ H : ; AE Equation 2.4 : ; P J L J F B ˇJ H : ; AE Equation 2.5 : ; J L : ; AE Equation 2.6 : ; J L : ; Equation 2.7 In typical FDTD simulation which is done by Lumerical FDTD software in my thesis, both the continuous physical region and time interval of the differential-form three-dimensional M a x w e l l ' s e q u a t i o n s a r e d i s c r e t i z e d b a s e d o n Y e e ' s a l g o r i t h m 78 . Figure 2.13 (a) shows the 3D grid I u s e t o d i s c r e t i z e t h e c o n t i n u o u s s p a c e , n a m e d t h e ' Y e e c e l l ' . T h e Y e e -cell is a small cube, which can be treated as one single cell picked from discretized model space. ¿ T AE ¿ U AE ¿ V are three dimensions of this cube and we use : E AE F AE G ; to label the point whose real coordinates are : E ¿ T AE F ¿ U AE G ¿ V ; in the model space. Instead of putting three electric field components ,B & and three magnetic field components , ,B & in the center of each cube, every electric field component, ' º AE AE , are surrounded by four corresponding adjacent magnetic field components and vice versa. For instance, as shown in Figure 2.13(a), we put component * º at : E AE FJ E s t ⁄ AE GJ E s t ⁄ ; and it is surrounded by component ' at : E AE F AE GJ E s t ⁄ ; and : E AE FJ E s AE GJ E s t ⁄ ; , component ' at : E AE FJ E s t ⁄ AE G ; and : E AE FJ E s t ⁄ AE GJ E s ; . Moreover, if the grid terminates at the boundary of the problem space, one can either employ a periodic boundary condition to ensure field continuity or apply zero fields as a hard termination. Meanwhile, ,B & field and , ,B & field components are interlaced not only in space but also in time. Similarly, we can obtain all the components of the curl expressions i n M a x w e l l ' s e q u a t i o n s 2.4 and 2.5. By updating each component, as shown in Figure 2.13 (a) and (b), with the initial field at origin as expressed by equations 2.8 and 2.9, we can finally obtain the vectorial electric

' ,B & : T AE U AE V AE P ; and magnetic * , ,B & : T AE U AE V AE P ; field components at each point and time. ¿ P H * º AE AE > 5 6 AE > 5 6 ˙ > 5 6 J F * º AE AE > 5 6 AE > 5 6 ˙ ? 5 6 I J L s ¿ V H ' AE AE > 5 6 AE > 5 ˙J F ' AE AE > 5 6 AE ˙ I J F s ¿ U H ' AE AE > 5 AE > 5 6 ˙J F ' AE AE AE > 5 6 ˙ I AE Equation 2.8 ¿ P H ' º AE AE > 5 6 AE ˙J F ' º AE AE > 5 6 AE ˙ ? 5 I J L s ¿ U H * AE > 5 6 AE AE > 5 6 ˙ ? 5 6 J F * AE > 5 6 AE AE ? 5 6 ˙ ? 5 6 I J F s ¿ V H * AE > 5 6 AE > 5 6 AE ˙ ? 5 6 J F * AE > 5 6 AE ? 5 6 AE ˙ ? 5 6 coupling structure consists of three-fold concentric grooves designed to coherently and in-phase sum the scattered SPP fields with ª ª ª J L w u r n m ,thereby enhancing their nonlinear interaction and, consequently, the signal strength. When the coupling structure is illuminated by an xpolarized ultrafast laser pulse pair, the generated SPP field emerges as a superposition of the rightand left-circularly polarized components, each carrying the opposite spin ( OEJ L J G s ). The spin of light combines with the geometric charge to generate plasmon OAM of . J L I J E OE, i.e., to form . J L { and . J L y SPP vortices. Both field components progress towards the coupling structure center superposing to form the plasmonic vortices, with L-dependent Bessel function profiles 20 . As shown by the static ( ¿ P J L r f s ) PEEM image in Figure 3.1 (c), their superposition creates an elliptical SPP vortex field surrounding a phase singularity at the center of the coupling structure. The recorded image contrast has two distinct contributions: (1) a uniform contrast originating from the ' and ' components of SPP fields, which do not interfere with the x-polarized external optical field, and sum to an elliptical profile background; and (2) periodic modulation of the ellipse due to the interference of the x-polarized components of the SPP fields and external optical field.

As for the static PEEM image ( ¿ P J L r f s ), the uniform contrast (1) dominates, because it takes 1 v r f s for the SPP fields to propagate to the vortex core, by which time most of the tail end of the optical field has propagated away as a reflected pulse such that only its trailing edge interferes weakly with the SPP field.

r d e p e n d e n c e w i t h t h e o t h e r c o m p o n e n t s t h r o u g h M a

x w e l l ' s e q u a t i o n s t 83 .

Assuming an electromagnetic wave with simple harmonic field dependence, the time dependent M a x w e l l ' s e q u a t i o n s (Equations 2.4-2.7) can be simplified to B ˇJ H J L J F E ae and B ˇJ H J L E ae . Furthermore, relations among the Cartesian components of the SPP field can be expressed as,

where ae is the SPP frequency, and and are the bulk permittivity and permeability of silver, respectively. Based on Equation 3.4, I obtain the ' component by first performing a spatial derivative of ' º along the y-direction and then integrating along the x-direction, with an assumption that ' decays to zero far away from the vortex, which is reasonable because the field distributions of the interfering vortices follow Bessel functions in real space, such that the extracted ' º field component in Figure 3.4, is 1 s r times stronger at vortex core than x I away in the xdirection. Therefore, I select a s t J H s t J m 6 region enclosing the plasmonic vortex at center, and apply Equation 3.4 to the extracted time sequence of evolution of the ' º field component in Figure 3.4 (a), in order to obtain the evolving ' component at the corresponding time step, as shown in components, I derive the transverse magnetic field components * º and * , which are necessary

where the 2 4 is the in-plane SPP field intensity that defines the Poincaré polarization sphere radius, while 2 5 , 2 6 , and 2 7 define the polarization sphere axes, as shown in Figure 3.6. Specifically, the ±45° polarizations tilt with respect to the x-axis, and 2 7 J L J G s , represent the polar axes indicating the left (J F ) and right (J E ) circular polarization, respectively. Using 4 AJ k ' º AE J o J L ' º AE : ; and + I J k ' º AE J o J L ' º AE : J E 6 ; , I obtain the Stokes parameters of SPP vortex from ITR-PEEM experiment over one optical cycle in Figure 3.4 and 3.5, as shown in the upper panel of Figure 3.7, along with the corresponding FDTD simulated Stokes parameters (the lower panels). The 2 4 distribution corresponding to the SPP in-plane intensity is prominent along the vortex ellipse, marked by the dash line in Figure 3.7(a), where the gyrating SPP field has largest amplitude, is sketched for visual emphasis. To show how Stokes parameters reveal the SPP vortex, I start from 2 7 , which evaluates the circular polarization in Figure 3.7 (d). The 2 7 distribution indicates that the in-p l a n e f i e l d s '

circulation, and therefore the red and blue contrast, periodically reverse in the radial direction from the vortex core. The vortex ellipse, however, is defined by the white contrast at the interface between counter-circulating blue and red elliptical domains. This interface is composed of the same amounts of right and left circulating components that form linearly polarized field. The interface naturally overlaps with the 2 5 domain with dominant y-polarization parallel to the major ellipse axis, and dominant x-polarization parallel to the minor ellipse axis. Naturally, the 2 6 map locates the dominant J G v w " linear polarization where the ellipse transitions between the major and minor axis segments. To understand more about the Stokes parameter maps in Figure 3.7, I plot a line profile along y-axis at the center of each Simulated Stokes parameter map, as shown in figure 3.8. As evident in Equation 3.9-3.11, 2 5 represents the difference between the intensities of horizontal (x)

and vertical (y) linearly polarized fields; 2 6 represents the difference between the intensities of linearly polarized field at J G v w " with respect to the x-axis; 2 7 shows the difference between right (RCP) and left (LCP) circularly polarized fields. Next, let us focus on the 2 7 J L r points in Figure 3.8, which evaluates that the RCP and LCP fields have equal amplitudes with the opposite sign.

At center of the SPP vortex, 2 4 declines to 0, which means the field amplitudes interfere destructively to give zero total field in this region; consequently, all Stokes parameters tend to 0 at this region. At point A in Figure 3.8, 2 7 J L r but 2 4 , 2 5 , 2 6 remain finite and near their maximum. The SPP field is linearly polarized.

The Stokes parameters that define the optical polarizations can also be considered from the perspective of singular optics, where limiting linear and circular polarizations in the surface plane are termed L-line and C-point singularities 85 . In the case of SPP vortices, for example, the observed linear polarization along the vortex ellipse that is evident in the 2 5 and 2 6 maps implies that the vortex ellipse follows an L-line contour. Furthermore, the white contour in the 2 7 mapping confirms that elliptical vortex forms as a coherent superposition of the left and right circular polarizations.

To further define where the L-line polarization singularities distribution, I compute the ellipticity of the in-plane polarizations by tracing the local polarization ellipses of the vectorial fields obtained in Figure 3.4 and 3.5. Figure 3.9 shows the ellipticity distribution of SPP vortex, which is stable when center plasmonic vortex is perfectly formed, within the same time window as in Figure 3.4 and 3.5, with the minimum and maximum values ranging from J E s for the circular and J E » for the linear polarization. Evidently, the vectorial fields sharply reverse their elliptical polarizations when passing through the linear polarization states that form the vortex ellipse. The complex ellipticity contrast within the vortex ellipse is not properly defined by the geometric charge of the coupling structure, but it appears on account of different SPP propagation lengths and concomitant field attenuation, which causes the constructive and destructive interferences to be imperfect. Similarly, the high variable experimental polarization ellipticity can also in part be attributed to such attenuation as well as insufficient photoemission yield for performing the required differential analysis of the experimental images, which introduces noise and other sources of extrinsic contrast. The L-line contour, which coincides with Point A in Fig. 3.9, consists of the coherent superposition of the left and right field circulations that dominate on either side; this reveals the that in-plane SPP field circulation reverses on the nanometer scale. 3.3 Conclusion In summary of Chapter 3, I perform ultrafast microscopic imaging on the sub-optical cycle time and J O t v n m pixel scale spatial resolution of the full vectorial near-field polarization components of plasmonic vortex fields propagating at the local speed of light. ITR-PEEM experiments based on the vectorial interference field imaging perform the ultrafast microscopy to define the space-time distribution of linearly polarized fields. Fitting of the L-line image line profile in Figure 3.10 with a Gaussian function gives the resolution of the polarization imaging of 90 nm, which is impacted by the detector pixel resolution. The recording of a single field component defines 3D Cartesian components of the electric and magnetic fields as they are related b y M a x w e l l ' s e q u a t i o n s . F u r t h e r m o r e , I obtained the complex field components by expressing with a phasor and evaluating the plasmonic polarization ellipses and their implied Stokes parameters to map out the polarization singularity distribution with sub-diffraction limited resolution. To sum up, I have developed a method to extract the vectorial SPP vortex field components from ITR-PEEM experimental movies and analyze the polarization state of the SPP vortex field, offering a new approach for interpreting ITR-PEEM images.

Calculation of the vectorial fields of the plasmonic vortex

To construct an analytical model of a plasmonic vortex with designed OAM, we need to describe its vectorial fields. According to Eqs.1.10 -12 in chapter 1, I start the simulation from the vectorial cylindrical wave SPPs field propagating in the x-direction from a point source as follows:

ª ae ª ª 4 A : º ? ; A ? Equation 4.1 ' B J L ' 4 E G ae ª ª 4 A : º ? ; A ? Equation 4.2 * B J L ' 4 A : º ? ; A AE Equation 4.3

where ' and ' are the transverse and longitudinal electric field components, * is the magnetic component parallel to the surface, and ' 4 is the field amplitude (taken as 1 in the analytical model).

Moreover, G and G are the complex and imaginary (propagation and decay) wave vectors, parallel and perpendicular to the surface, 4 is the dielectric constant (for vacuum J L s ), and

; is the time dependent phase of SPP field with the given frequency ae ª ª .

To describe the plasmonic vortex, I express the field in a cylindrical coordinate system as shown in Figure 4.1. In this analytical model, a vortex generator CS, as introduced in section 1.4, is illuminated by the circularly polarized light ( OEJ L J G s ). To emphasize, the CS structure I use in this chapter is a vortex generator with geometric charge I J L r , which is a circular CS with constant radius R, as shown in Figure 4.1. For a point source of SPPs, SPP field carries a transverse ' AE component coupled at radius N 4 as ' AE ª J L ' 4 A : B ˛ ? ; A ? A fi : E i J E ? , E i J E ; N 4 @ B ˛ V ‚ , where ' 4 is the initial amplitude, A : B ˛ ? ; is the time and radial angle dependent phase oscillating at a frequency ae ª ª J L ae ¯, A ? is the evanescent SPP field penetration into vacuum normal to the interface, and N 4 @ B ˛ is an infinitesimal linear increment along circumference of the coupling structure. According to the Huygens principle, the coupling structure is modeled as an array of point sources along its circumference from which SPP waves emanate with a varying phase profile defined by the phase of circularly polarized light ( OEJ L J G s ) and geometric charge m of the spiral CSs, as described in chapter 1. The SPP fields at the vortex core are obtained by propagating and integrating contributions from each source with its appropriate phase. The total ' field integrates contributions from all sources along the coupling structure (the time parameter is omitted), as follow 57,93 ,

where r > refers to the dielectric side, , ¯ @ > V is the R P D order of Bessel function of first kind, defined by OAM of the generated SPP vortex and all other parameters have already been defined

in Chapter 1. To obtain the longitudinal SPP components, however, requires integration of the radial and tangential vector projections by taking a dot product between the unit vectors along inplane field

and the radial (tangential) unit vector [ J J : J J ; ?, i.e. J J fi

, where the coordinate J J E is the radial unit vector pointing from the center to edge of the circular coupling structure. The longitudinal components of E field are as follows,

Similarly, I obtain the magnetic field components as follows,

oJ g Equation 4.7 * : N AE AE r > ; J L J F * 4 E ¯ ? 5 Ł A ¸ A ¯ > t E N , ¯ : G N ; J E 4 : , ¯ ? 5 : G N ; J F , ¯ > 5 : G N ; ; ? Equation 4.8 * : N AE AE r > ; J L r Equation 4.9

Please note that, the starting point of our analytical model is the SPP filed generated by a point source, which only exists for TM mode ( * J L r ) polarization 7 because of the boundary condition I mention in section 1.12. Next, using the analytical vectorial * and ' fields I obtained in Equation 4.4-4.9, I analyze the field and spin properties of the plasmonic vortex with designed OAM in following sections.

Analytical SPPs field results with designed OAM

Furthermore, to evaluate the analytical model for plasmonic vortex, I start with a design for OAM . J L J F s , which is generated by illuminating the circular CSs (Geometric charge I J L r ) by circular polarized light with helicity OEJ L J F s . Due to the symmetry of the CSs, using the circular CSs with geometric charge I J L r is possible for us to extract the vectorial field components from the experimental result.

To generate the pseudo color plots of distributions of the field components, I set . J L J F s and radius 4 J L t ª ª ª , where ª ª ª J L w u r n m is the SPP wavelength, and calculate the vectorial SPP field using Eq. 4.4 -4.6. As shown in topological spin textures. Unlike " field components, both and have a strong focus at the vortex core. The field component reaches minimum at N 5 , indicating that there the SPP fields undergo purely tangential motion orbiting around the first dashed circle, as expected from the Poynting vector in Fig.4.2 (a). The field component diminishes more slowly with increasing N than , such that its minimum occurs at N 6 J N r x s ª ª ª [2 nd dashed circle in Fig.4.2 (c)]. The Plasmonic vortex fields in Figure 4.2 show the time averaged field amplitude distributions. However, the instantaneous fields orbit around the vortex core on the femtosecond time scale, generating the SAM textures according to the right-hand rule. By multiplying the Eq. 4.4-4.9 by the dynamic factor A ? , I obtain the evolution of the instantaneous SPP vortex field components, whose circulation is captured in the plotted time windows. Evolution of the (a), in-plane, J ¥ E ~E i : ; J E E ~E i : E q E ´ ; and (b), out-of-plane, E ~E i : " ; electric field components of a E z =-1 vortex over one-half of an optical cycle. (c) In-plane magnetic field, J ¥ E ~E i : ; J E E ~E i : ; , of the same vortex over one-half of an optical cycle.

the in-plane and out-of-plane electric field components of a . J L J F s vortex over one-half of an optical cycle ( 1 r { f s ). At the center of the CS, the in-plane electric field shows a strong focus, but the out-of-plane field interferes destructively causing the field amplitude to vanish at the vortex core, as both component orbits counterclockwise as defined by the OAM.

A novel aspect of vortex fields that emerges from this analytical model, and which is likely to find important applications in magnetoelectric interactions and topology, as shown in as long as the vortex is established. Due to the strong focus of the in-plane field, ideally, the vortex core will be a focus of intense magnetoelectric coupling defined by the pseudoscalar fi . The calculated distribution of fi from the analytical model is shown in Figure 4.4, where the antiparallel a and fields are focused on the 1 ª ª ª t ⁄ length scale. Note that the and will become parallel when the sign of vortex OAM is reversed.

The anti-parallel context of the and field at the vortex core is a consequence of circular polarized light interacting with the circular CS; it is thus a profoundly simple construct for driving and interrogating magnetoelectric phenomena by ITR-PEEM and other near-field microscopic techniques. Moreover, such (anti)parallel E and B fields are a property of vortices composed of surface evanescent waves, in general, where the associated transverse field forms a singularity at regions of the strong focus due to the cycloidal field polarization in the meridional plane. And similar magnetoelectric coupling has been reported to exist in metallic plasmonic nanosphere dimers, where it drives coherently the light-nanostructure orientationdependent magnetic and electric dipole chiral Raman scattering 94 . Furthermore, the strong E B pseudoscalar field may assist in the study of dynamical axion field through the topological magnetoelectric coupling fi 95,96 , because it locally enhances the electromagnetic field strengths, breaks the TRS on a subwavelength scale, and its interaction with matter can be tuned continuously to fall within the bandgap of magnetically doped topological insulators.

Plasmonic SAM Skyrmion

In chapters 1.3 and 1.4, I discuss the SAM of SPP and the topological spin texture. Having defined the vectorial SPP field components in Eq.4.5-4.9 for . J L J F s , I use them to calculate the associated SAM ( ), based on Eq.1.17 in chapter 1. 5 : N AE AE r > ; J L t Ł 6 4 6 F G 6 ae ª ª 6 4 J E 4 GJ c , 4 6 J k G ª ª NJ o J F , 6 6 J k G ª ª NJ oJ g Equation 4.12 Based on Eq. 4.10-4.12, we could obtain the normalized vacuum side Skyrmion SAM texture, S J L in Fig.4.5 (a). In this texture, vacuum SAM points up (J E V) at the vortex core, aligning it with the SAM direction of the excitation light; moving away from the core, it continuously winds to lie in the surface plane pointing away from the core at N 5 [small dashed circle in Fig 4.5 (b)], and finally pointing down (J F V) at its periphery at N 6 97 , as illustrated in Fig.4.5 (b). Progressing outward, at N 6 , " AE S , the z-component S , has a Ł-phase reversal, such that

the " AE S vectors on each side of N 6 are surface normal, but point in the opposite directions. We note, however, that the unnormalized spin, " , is continuous as it smoothly reverses direction upon crossing through zero at N 6 98 .The anti-parallel " AE S vector across the N 6 reflect the counterrotation of the in-plane fields that originate from the evanescent properties of SPPs. The superposition of oppositely circulating in-plane fields at N 6 necessitates a transition through linear polarization, which is a chirality discontinuity catastrophe that occurs on the screening scale. As for the in-plane component of the normalized SAM, AE S ,as shown in the arrow of Fig. 4.5 (c), its amplitude is zero at the vortex core and at N 6 , but reach maximum at N 5 . Note that there is also a non-zero AE S

proportional to the dimension of vortex through R, which is an evident as a twist of AE S from the radial direction of in Fig. 4.5 (c). It appears because r is not negligible with respect to R in deriving the Bessel function.

Because of the non-zero 5 components of SAM, for small structure where r 1 R , the nonzero 5 component twists the SAM texture away from the Néel-type Skyrmion spin texture, where the in-plane component is point directly in or out toward the center without rotating, as shown in Figure 1.5 in chapter 1.5. To quantify the twist due to 5 , I use the helicity angle V, the angle of the interface projection of SAM, with respect to the horizontal direction (x-axis), concerning the angle of the in-plane spin orientation 97 . Clearly, the helicity angle has radial dependence within N 6 , the Skyrmion boundary, and is plotted by the red curve in Fig. 4.6 as function of distance r from the vortex core, for the coupling structure with R J L t ª ª ª . Note that for the experiment I describe in next section, I use the coupling structure with R J L w ª ª ª to make the experimental result clearer and easier to analyze, while keeping the radius small enough to maintain a non-zero helicity angle and generate a twisted SAM texture. According to Eq. 4.11, V 1 r at the vortex core, but far away from the core V becomes negative and approaches J F v { " when NB \ N 6 . Next, I define the helicity angle at the Skyrmion boundary ( N 6 ) as a characteristic angle, which yields a value of 1 J F v { " for R J L t ª ª ª . Next, I consider the characteristic Skyrmion helicity angle as 4 increases (blue curve). A transition from a twisted to a Néel-type Skyrmion, where 5 B \ r , occurs for N ' 4 , as the radius of the coupling structure increases, i.e., 4 B \ » . A clear evolution of the characteristic helicity angle is observed from V 1 J F x x " for R J L ª ª ª , to V 1 J F s " for R J L s r r ª ª ª , confirming that a the Skyrmion texture converges from a twisted to a Néel-type as the coupling structure radius increases with respect to ª ª ª . Similar to magnetic Skyrmion lattices, where a Skyrmion unit is defined by a soliton solution of interfering waves with a defined wavelength 99,100 , I define the Skyrmion boundary based on the SPP interferences, and how SPP polarization maps onto the Poincaré sphere 51,52 . The tiling of SPP polarization onto Poincaré sphere involves examination of SPP polarization distribution. The C-points of the polarization distribution defines the spere poles, while the L-lines wrap the equator. A Skyrmion field is expected to wrap the Poincaré sphere integer number of times. The Skyrmion polarization field will, therefore, be centered at a C-point, pass through an L-line singularity band, and terminate at another C-point region. The vortex center coinciding with the phase singularity is the C-point origin. From there the polarization becomes elliptical until it reaches the first L-line singularity at N 5 is the sphere equator. 85 The L-line band is defined by linear field polarization, such that the SAM gradually passes through " J L r . From the L-Line the elliptical field circulation direction reverses, until it reaches the Skyrmion boundary at the dashed circle of radius N 6 in Figure 4.7 (a) and (b). The reversal of the C-point circulation causes a p h a s e shift (sign change) in " . As shown in Figure 4.7(a), the L-line and C-point 85 distribution is computed by calculating the polarization ellipticity, which is proportional to the s " ⁄ , of the fields; the minimum value s " ⁄ = 1 defines a C-point singularity (e.g., at the vortex core), while s " ⁄ = » defines an L-line loop.

To evaluate the topological charge of the plasmon SAM texture, according to Equation 1.24 in section 1.5, I calculate the charge density, D, as shown in

ITR-PEEM Experimental of Plasmonic Skyrmion

As shown schematically in a 1MHz excitation rate with either LCP or RCP light. The opposite helicity excitation images show superficially identical signal distributions that are centered on the vortex core, i.e., the circular region of 1 ª ª ª diameter, from where the most intense 2PP signal is excited by ' field. The admixture of the in and out-of-plane field-induced signals, however, shrouds the SAM Skyrmion, because one cannot immediately separate them.

In ITR-PEEM experiment, as I mentioned in section 2.4.2, the optical field consists of a pump and probe pair with delay . While both pulses generate SPP wave packets, the ultrafast imaging with varying delay involves the interference of the probe pulse with the in-plane component of the pump-generated SPP field. In this chapter, because the coupling structure is a simple circular grating, which has a perfect cylindrical symmetry, the pump-probe dependent signal captures dominantly the ' evolution of the pump induced SPP vortex. To extract the interference signal between the SPP field and incident probe pulses, I Fourier filter the raw ITR-PEEM data at the laser driving frequency ( s ae ae ¯)104 for each pixel. Note that we use a simple circular CS with perfect symmetry, the ' component of pump excited SPP field will ideally not interfere with the prob field because it is always perpendicular to the probe field, therefore the modulated filed after Fourier filter is dominated by ' component. Figure 4.9 (a) presents the s ae ae ¯ Fourier filtered ITR-PEEM image at 1 t u s f s . Furthermore, to improve contrast quality for deriving the remaining vectorial components of the electric and magnetic fields, I normalized t h e a m p l i t u d e a l o n g t h e t a n g e n t i a l d i r e c t i o n w i t h r e s p e c t t o t h e m e a n v a l u e , where an additional v "-wide Gaussian filter for tangential smoothing is applied, taking advantage of the rotational symmetry of the vortex field, as shown in Figure 4.9 (b). Finally, I rescale the magnitude of the s ae ¯ component, which is almost proportional to ' ª Ł ª 6 ' ¯ AE ª 6 because of the perfect symmetry of our CS, where ' ª Ł ª is the pump excited plasmon field and ' ¯ AE ª is the prob light field, by the square root of its local value, to obtain a linear amplitude of the radial component dominate of the SPP vortex field, as shown in Figure 4.9 (c).

Introduction

The previous research on plasmonic vortex has primary focused on their field, spin, andvector as single entities or in 2D periodic arrays 20,110,111 . In this chapter, I report how such topological structured fields can be designed by coherent superposition of SPPs fields that are generated by pairs of plasmonic vortex CSs where their spatial separation is a tunable parameter that defines the topological fields of their interaction. Plasmonic vortices have complex structured

Poynting ( -vector) distributions that can potentially drive nanoscale transport by design of their momentum flow direction and strength. I develop an analytical model capable of predicting and tuning complex SPP momentum flow between two vortices and confirm the results by FDTD simulations and ultrafast PEEM imaging of the generated SPP fields. I examine how elementary vortices generate emergent phenomena arising from their coherent interaction. Specifically, by tuning the separation between two vortices of identical geometric charge, I generate an additional emergent vortex through interference of their oppositely gyrating fields with a distinct half-integer meron topological charge. Furthermore, by adjusting the relative distance between the primary vortex pairs, the emergent vortex of their interaction can form a conveyor belt-like texture for momentum transfer between the primary vortices. Therefore, the design provides an additional tunable degree-of-freedom for steering, rotating 112 , and translating 113,114 of vortex field on the nanoscale spatial and femtosecond temporal scales without additional complexity and dispersion of beam shaping optics 115

Design of dual vortices

As I introduced in section 1.3, SPP vortices are generated by illuminating a pair of lithographically defined CS. The CS are designed such that incident light carrying SAM launches SPP fields with Bessel function profiles that dress 2D space in periodic or quasiperiodic vortex field tilings with delineated geometrical charge 110 . For example, As shown in Figure 5.1(a), the Archimedean spiral CSs with defined geometric charge, I (i.e. I J L s in Figure 5.1(a)), when illuminated with circularly 116 or linearly 20,36 polarized light carrying SAM, OE, define the OAM,

. J L I J E OE , of SPP fields they generate with specific topological spin texture.

In the previous chapter, I elaborated the analytical model of SPP vortex, which is generated by illuminating Archimedean spiral CSs with circularly polarized light. The SPP vortex electric field at a flat metal/dielectric interface is mathematically described by . th order Bessel function of

the first kind with a spiral phase profile of . 117 . The vectorial field profile in the interface plane is given by Equations 4.4-4.9 in section 4.2. Next, I extend the single vortex analytical model to a system of dual vortices, generated by the coupling CS in Figure 5.1(b), which consists of two Archimedean spirals with same I and their individual centers 1 AE offset by @ ª J L 1 J F 1 .

The fields of dual SPP vortices are modeled as a coherent sum of two vortex sources,

where the suffixes l and r denote the left and right CS in Figure 5.1(b), respectively, separated by @ ª , and : N ae AE ; are the coordinates for the field simulation. Hence, the spiral radius 4 AE is expressed as 4 AE J L J § N ae 6 J E :

-6 ; 6 J G N ae B @ ª B c o s : ; . AE are the corresponding azimuthal angles given by, J L c o s ? 5 : ¸ . > : ˇ -. ; . ? . -B ¸ ; and similarly as J L N J F c o s ? 5 : ¸ . > : ˇ -. ; . ? . -B ¸

;

Based on this design, I obtain the longitudinal components of the field and the corresponding components of the magnetic H field, as follow,

Furthermore, with the analytical field given by Equation 5.1 to 5.6, I design the momentum circulation between dual plasmonic vortices, as will be shown next.

Case II: Odd E x

When . is odd, one can readily deduce that Equation 5.9 will have a trivial solution.

Therefore, to generate a vortex with odd . , I could introduce a relative Ł phase to the right relative to the left vortex by increasing the start radius ( 4 ) of the right Archimedean spiral CS by r w ª ª ª .

This causes in SPP waves launched from the two vortex-generator CSs add with a relative Ł phase shift. This additional Ł phase of the ' field allows to reproduce the non-trivial solution Equation 5.9 for vortex pairs with even . values. Please note that the Case II, and its modification, is introduced for completeness; I will not, however, investigate this scenario further in this chapter.

Table 5.1 gives solutions of Equation 5.9 for several values of Q and . as well as the necessary separation, @ ª , between the dual vortices (in units of ª ) to produce the third central vortex between them. In this chapter, I will focus on the dual vortex system with defined OAM . J L J F t , but I give additional solutions for both the even and odd values of . . Please note the Bessel function follows the relation , ? ¯J L : J F s ; ¯ , ¯.

Table 5.1 The separation in units of i E E E =530 nm between two vortex CSs to generate the emergent E q E field singularity based on the Case I scenario. . Q 0 1 2 3 1 0.77 1.22 1.63 2.03 2 1.76 2.23 2.68 3.11 3 2.75 3.24 3.70 4.14 4 3.75 4.24 4.71 5.16 5.2.3 Analytical and simulated field of plasmonic dual vortices Figure 5.2 shows the evolution of the plasmonic " field in the dual vortex system from analytical model each with an OAM, . J L J F t , for two different separations (E E E i E ) corresponding to Q J L s and t ( s x u ª ª ª and t x z ª ª ª ; Table 5.1). The series of image frames show the SPP field evolution as the phase (time) of the field is advanced in Ł v ⁄ ( 1 r t u B O ) steps, where one optical cycle is 1 s z u f s . It is evident from the time evolution of the purple spots and arrow over r { t f s in Figure 5.2, that the SPP field interference pattern, which is launched from the left and right vortex generators, undergoes counterclockwise gyration in forming two primary vortices at the center of each CSs. Furthermore, to establish the interaction between two plasmonic vortices, I just focus on the interference of their SPP fields in between them. In addition, as shown in Figure 5.2, between two primary vortices, there is an additional vortex gyration at the dual CS origin in the clockwise direction that is indicated by time evolution of the blue spot and the dotted arrows.

The field patterns of Figure 5.2, therefore show that there is one counterclockwise rotating plasmonic vortex with OAM . J L J F t centered on each CSs, and interference between them generates an additional SPPs field that gyrates in the opposite direction. For higher order solutions ( Q J P t ) in Table 1, the emergent SPP field between the two primary plasmonic vortices is similar to the one with Q J L t J k @ ª J L t x z ª ª ª J o but the larger separation forms a longer rotating conveyor belt-like SPP field in between them.

To test reliability of the analytical model, I also simulate the nano-femto evolution of the SPPs fields by the FDTD method, that I introduced in section 2.5. I perform the FDTD simulations for the vacuum/silver interface with two geometric charge I J L J F s vortex generator CSs having 4 J L s { ª ª ª radius, which is illuminated at normal incidence by LCP light with ª ¯J L w w r n m wavelength and helicity OEJ L J F s . Figure 5.3 shows the FDTD calculated distribution of the " for comparison with the same field calculated from the analytical model in Figure 5.2. The agreement between simulated and analytical out-of-plane SPPs field confirms the analytical model. Note that the comparison is not perfect. Specifically, the FDTD model does not reproduce perfect phase singularities at the vortex cores, and the field strengths lack the full symmetry of the analytical model. The reason for this minor discrepancy is: The analytical model assumes the CS consists of an array of point sources, whereas for the FDTD model to produce the SPP field, the CS must be modeled as having a finite width. The asymmetry of the dual vortex CS system affects the constructive and destructive interferences causing the destructive interference to be incomplete at the phase singularities, as well as the simulated fields to reflect the structure asymmetry.

Analytical momentum flow between the dual plasmonic vortices

To depict the vortex field distributions as well as the relative momentum (energy) flow between the dual plasmonic vortices of Figure 5.4, I further calculate the associated Poynting vector ( -vector) distribution that are shown in Figure 5.4 for the same dual vortices system as in distribution in Figure 5.4 (a) supports the formation of two primary counterclockwise gyrating vortices, which are generated by the respective CSs, and a clockwise gyrating vortex created in between them by the dual vortex SPP field interference. Moreover, the color scale in figure 5.4 (a) denotes the phase singularity of the " field which locates the phase singularities by white shading where the out-of-plane amplitude vanishes at each of the three vortex cores. In Figure 5.4(b), where @ ª is increasing to Q J L t , the momentum transport given by the Poynting vectors has a conveybelt like structure with the yellow arrows indicating linear flow between the left and right primary vortices generating overall a clockwise circulation, where the primary vortices function as the head and tail pulleys. t h e e m e r g e n t v o r t e x . A s s h o w n i n F i g u r e 5 . 5 ( b ) a n d ( d ) , w h e n @ ª equals solutions of Equation 5.9 given in Table 5.1, the momentum flow between the two primary vortices will form another emergent vortex rotating clockwise; for other non-integer values of Q the center momentum flow is disordered. Furthermore, the phase map shows that a singularity in the emergent vortex is necessary to obtain ordered momentum flow between the primary vortices. As stated in section 1.3, the helicity of the evanescent SPP fields causes their quantized SAM 34,38 to be transversely locked to their -vector. This generates SPP spin textures that are topological invariants of the plasmonic vortex vector fields. Vortices are defined by a quantized topological charge in integer multiples of one-half or one, generally corresponding to the meron 20 and Skyrmion 116 vector spin textures. The topological charges are defined by how many times their Stokes parameters wrap a Poincaré sphere 52,118 , as already explained in Chapter 3. Having defined the vectorial SPP electric and magnetic field components of the dual-vortex system in Equations 5.1-5.6, I use them to calculate the associated SAM ( ) texture, as given by Equation 1.17 in section 1.3. center vortex. The same distinct boundary, however, cannot be defined for the two primary vortices because the interference of SPP fields from the dual CS sources is not fully destructive, and thus does not produce a defined L-line singularity. Going further out from the center, clear L-line, including the primary and emergent vortices is the surrounding 2 nd L-line domain (red dashed line).

Experimental setup and time integrated measurement

To confirm the temporal evolution of the dual vortex fields, we record the SPP space-time movies by experimental imaging 2PP signal by ITR-2P-PEEM as a function of pump-probe pulse delay, ¿ P, as it is advanced in ~100 as time steps. The plasmonic vortex host sample is generated by depositing a 200 nm thickness polycrystalline Ag film by thermal evaporation onto an n-type Si substrate with a native oxide. Subsequently, the dual CS is lithographically fabricated by focused ion beam milling of a coupled pair of vortex generators with 4 J L s { ª ª ª , geometric charge I J L J F s , and Q J L s J k s x u ª ª ª J o separation. The coupling structure, as shown in Figure 5.7, consists of threefold concentric grooves forming Archimedean spiral structures separated by ª ª ª for coherent in-phase summation of the SPP fields from each groove to enhance their combined nonlinear interaction. polarized OEJ L J F s laser pulses of 1 u r f s duration and ª ¯J L w w r n m center wavelength incident normal to the surface illuminate the dual plasmonic vortex generation CS, launching the SPP fields toward each CS center. The delay ¿ P between a pair of identical pulses is scanned by a Mach-Zehnder interferometer (MZI) in ~100 as steps to record movies of space-time evolution of the generated SPP fields. Photoelectrons emitted by the 2.2 eV photons excite them to above the vacuum level to generate the nonlinear 2PP signal, are collected by the PEEM objective lens. Further transmission through the aberration corrected PEEM optics column onto a multichannel plate/CCD camera imaging detector generates photoelectron spatial images. The images are recorded by accumulating the 2PP signal over multiple excitations for each delay time.

For static measurements with single pulse excitation, the PEEM records sub-diffraction limited the nonlinear field intensity images as described by Equation 2.1 in section 2.4 81 . The total field of the LCP incident light together with the SPP wave packet that it generates, excite nonlinear order J J L t two photon photoemission (2PP) signal. The experimental static image in Figure 5.9(a) shows an intense signal from the dual CS core, and two interference rays extending in the y-direction. The signals for the x-and y-direction add constructively, but the former is stronger and appears brighter due to its stronger nonlinear 2PP excitation. The signals propagating in other directions add destructively to the dual vortex CS causing them to be weaker in the nonlinear process. The experimental static image signal in Figure 5.9 (a) can be compared to the analytically calculated static image obtained by integrating " , shown in Figure 5.2 (a), over one optical cycle, as shown in Figure 5.9 (b). The comparison confirms the origin of the diverging signals in the experiment. As I discuss in section 2.4.3, in time resolved experiments, the pump and prob pulses generate separate SPP wave packets, the ultrafast imaging with varying delay ¿ P involves the interference of the in-plane components of the probe pulse with the pump generated SPP field. As I mention in section 4.3.1, time domain Fourier analysis can remove the out-of-plane SPP field to extract the purely delay dependent signal. However, because the CS has less symmetry compared with the circular CS I use in the last chapter, it is hard to extract each component of SPP field as I did in last chapter. Therefore, I will compare the Fourier filtered ITR-2P-PEEM images with reproduce FDTD images in following part. Such Fourier filtered ITR-2P-PEEM images recorded at sub-optical diffraction limit are presented in Figure 5.10 together with the calculated images from the FDTD simulation, where the CS structure is shown in Figure 5.7. When the CS with geometric charge . J L J F s is illuminated by LCP laser pulse carrying photonic SAM with helicity OEJ L J F s , the generated SPP fields undergo SOI of light to form two distinct plasmonic vortices with OAM . J L J F t at the center of the CS, as shown in static PEEM image in Figure 5.9(a). at the laser driving frequency ( s ae ae ¯) of the 2PP signal at each image data pixel. The figures show a series of images as the delay is advanced by a total of ½ of an optical cycle ( 1 r { f s ) or a phase delay of Ł. The starting time 1 u y w fs corresponds to 1 t s optical cycles, which is the time it takes the SPP field to propagate from the CS region to their centers to form the dual vortices and J L r is the time when pump and prob light overlap with each other space and time. 69 The series of Fourier filtered images show two SPP vortices having formed at the center as they gyrate

Introduction

In previous chapters, I have introduced theoretically and experimentally single and dual plasmonic vortices with normally incident optical excitation and dynamical PEEM imaging.

Establishing the complete model for SPP interference imaging by laser excitation at normal incidence and 2P-PEEM imaging of SPP fields, however, requires a quantitative model to examine the interference patterns of SPP fields that are generated by elementary CS. For obliquely incident optical excitation, PEEM images the total local field, which is the sum of the incident electric and its generated SPP fields 33,35,73 . For the light at normal incidence, PEEM also images the interference between the incident optical and generated SPPs fields, but the imaging mechanism can also involve SPP-SPP wave interference producing the 2PP signal, which has not been quantitatively investigated. Analyzing the SPP-SPP interference residual coherent signal is essential for a full understanding of PEEM imaging with normal light incidence, and potentially other plasmonic phenomena. Most of the previous studies, however, only focus on the intensity terms 102 (e.g., ' D 8 , ' 8 and ' D 6 ' 6 ), which likely capture the dominant optical field-SPP interference for simple CSs with perfect symmetry and when the SPP excitation and detection properties are completely understood. For complex CSs, such as dual vortex CSs, however, the cross terms involving the coupling among the in-plane and out-of-plane SPP fields to jointly contribute to the nonlinear response, may play a significant role in affecting the PEEM imaging.

In this chapter, therefore, I perform a simple experiment with double line CSs to explore how different interference processes may affect the PEEM image. My effort outlines a more in-depth analysis aimed at achieving a more quantitative understanding of the ultrafast PEEM microscopy of SPP field driven coherent processes.

Asymmetric SPPs fringes in double lines experiment

The discussion of SPP imaging is motivated by Figure 6.1 where I record the 2PP signal spatial distribution by 2P-PEEM between a pair of linear CS. The CS composed of two nearly parallel lines that are separated by u r J m is excited with linearly polarized, 550 nm, normally incident laser pulses of 1 u r fs duration. The excitation field is polarized orthogonal to the pair of CS lines. To show more detail, I plot a line profile along the propagation direction of SPPs [Figure 6.1 (b)]. From the line profile, I demonstrate that from each CS line symmetric SPP fringes with a period of ª ª J L w u r n m emanate and decay in intensity as they propagate towards each other.

These intense ª ª fringes are interpreted as due to interference between the generated SPP and the surface parallel fields of the external plane wave excitation light, as in most PEEM images in this thesis. When the SPP field propagates to the middle of the CS pair, surprisingly fringes with period of ª ª t appear, which may be expected for the SPP-SPP field interference at the center of the double lines structure. While I can justify the ª ª t spatial period, what is surprising is that the near center fringes possess distinct asymmetry with a period of ª ª , which modulates the peak heights of the ª ª t fringes. To understand the origin of these unexpected asymmetric fringes I explored different factors that influence the SPP imaging that become evident at the center between the line pair CS. In the experiment [Figure 6.1(b)] the asymmetry is only observed at the center of the CS pair, while near each CS, the intense fringes oscillate symmetrically. To explore the possible conditions for this to happen, I performed FDTD simulations reproducing the setup I used to obtain the PEEM image in Figure 6.1, and I plot the line profiles of the instantaneous in-plane and outof-plane field components at different time delays, shown in Figure 6.3 (a) and (b). As shown in Figure 6.3(a), when the SPPs are launched ( 1 r f s ), a strong external light plane wave illuminates the entire surface between the CS elements, therefore, external-in-plane interference (red line) shows symmetric fringes with ª ª period near the edge of CS. The out-of-plane field components, however, will not be modulated by the external light because they are orthogonal. When out-ofplane SPPs fields emanating from the two CS meet near the center of the CS, as shown in Figure 6.3 (b), the SPP-SPP interference will modulate the 2PP signal but with a with ª ª t period associated with even powers of the excitation field. Moreover, to show how the external-in-plane field and SPP-SPP interference of out-of-plane field affect the PEEM contrast, I plot line profiles of time-average of the in-plane ' 8 and out-of-plane ' 8 components, as shown in Figure 6.4 (a). Near the edge of CS, the in-plane component (red line) will dominate producing symmetric fringes due to the external-in-plane interference. When the SPP fields propagate to the center of the CS, however, the SPP-SPP interference of out-of-plane SPPs fields appear with ª ª t fringes. The asymmetric fringes can be generated near the center by superposition of signals from the externalin-plane interference with ª ª fringes and the out-of-plane SPP-SPP interference with ª ª t fringes.

The in-plane field produces ª ª fringes as long as it can interfere with the external field.

When the external field is gone, the in-plane field would also produce ª ª t fringes through SPP-SPP interference. Therefore, the ª ª fringes in the center can be attributed to the remnant presence of the external field, as described in equation 2.2 in section 2.4.2. (a) FDTD simulation of the CS separation dependent interference with 20 fs pulse duration. (b) FDTD simulation of the pulse duration dependent interference with E ˘E CS separation.

To verify this interpretation of the origin of asymmetric fringes, I performed FDTD simulations for different separations between double line CS and for different pulse durations. The goal of this simulation was to distinguish between contributions of the external and in-plane-SPP field and out-of-plane SPP-SPP interferences to the PEEM signal. When the separation is sufficiently large or the pulse duration is sufficiently short, so that by the time the SPP wave packet has reached the CS center, the external field strength has diminished so that the external and inplane SPP field no longer interfere, and only the SPP-SPP interference remains. Figure 6.5 shows the changing of the interference period by changing the separations between double line CS and the pulse durations, where I normalized the intensity of the simulated signals. ª ª t . Similarly, as shown in Figure 6.5(b), decreasing the pulse duration of the external pulse with the same v r J m double lines structure, I find the period of the in-plane field component to change from ª ª to ª ª t . Both increasing the CS line separation and decreasing the pulse duration decreases the external-in-plane SPP interference with respect to the SPP-SPP interference resulting in the appearance of asymmetric fringes under intermediate conditions and ultimately in the dominant ª ª t fringes. While such a transition from the ª ª to ª ª t fringes could explain the asymmetry the CS, according to the simulation of Figure 6.5, the separation ( u r J m ) and the pulse duration ( u r f s ) do not sufficiently suppress the external-in-plane SPP interference to explain the asymmetric fringes appearing in Figure 6.1. For the u r J m separation, the asymmetric fringes could only appear if the excitation pulse were substantially shorter than 30 fs. Therefore, I consider how the asymmetric fringes could appear while the external light is illuminating the CS middle for unrelated reasons.

To proceed, if we only consider the PEEM signal to be defined by the quadratic and quartic (intensity) terms + ª J L : ' 6 J E 6 ' 6 ; 6 ,where J L w z , as in previous research 102 , the asymmetric fringes will not appear, as shown in Figure 6.6 (blue line). The asymmetry could arise, however, if we take the cross terms with odd exponents of the fields, such as ' 7 ' term into consideration. Calculating the signal assuming contributions of odd power terms produces the asymmetrical fringes with a period of ª ª t , as shown in Figure 6.6 (red and green lines).

Experimental setup

To understand how asymmetric cross terms contribute to the PEEM imaging, I performed a simple experiment by using a normally incident PEEM, as shown in along their length from u y ª ª ª to u z ª ª ª , which corresponds to 1 t r J m , such that the externalin-plane interference still exists at the center of CS. As shown in Figure 6.8, I report line profiles for different separations together to show the changing of asymmetry at the center from u y I q n n separation to u z I q n n separation (bottom to top). Note that the line profiles in Figure 6.8 are offset for clarity. The change in asymmetry can be best seen in the left peak near the dashed line (the center of the parallel lines structure), which changes continuously from the larger to smaller separations. The separation of the parallel lines structure is changed by s I q n n , causing the propagation length of the SPPs from one side to the other to change by half the SPP wavelength. Consequently, if the asymmetric cross terms contribute to the signal shown in the PEEM image, the asymmetric response at the center will change over the 2 SPP wavelength cycle.

Furthermore, I performed Fourier filtering of the PEEM signal as in Figure 6.2 to extract the s ae and t ae components. As shown in Figure 6.8 the signal shows the external-in-plane interference signal, with ª ª ª period, and out-of-plane ' 8 signal, with s t ⁄ ª ª ª period, respectively. The external-in-plane interference component at the center of CS, as shown in Figure 6.9 (a), changes from peak to trough (along the dashed line) by changing separation from u y I q n n to u z I q n n . By contrast, as shown in Figure 6.9 (b), the out-of-plane field component at the center of CS is always a trough (destructive along the dash line) by changing the separation between parallel lines. Therefore, considering the contribution of asymmetric cross terms between s ae and t ae signals, the continuous changing of in-plane field will get a similar result shown in Figure 6.8.

Calculation result with simulated signal

To confirm the asymmetric phenomena shown in Figure 6.8, I performed the FDTD simulation for the same CS to get the in-plane ' º and the out-of-plane component ' . As predicted in Fig. 6.5, the interference at the center of a structure with CS separation of u y I q n n to u z I q n n when excited by a 1 u r fs laser pulse will still show the I q n n interference between in-plane SPPs field and external light plane wave at center of CSs. Therefore, in all the following FDTD simulations, I capture the field at the top of the surface to include the external light information.

Furthermore, after I obtain all the field components, modulated in-plane components between the external and in-plane SPPs field ' º and out-of-plane SPPs field ' by FDTD simulation, on the top of the sample surface, I use Equation 6.1 to model how the even and odd power field terms contribute to the total PEEM image contrast. By fitting to Equation 6.1, I can analyze how the odd power terms could contribute to the asymmetric images. in Figure 6.8, the trend of the asymmetric fringes, where the higher peak changes to a lower peak by changing the separation from u y I q n n to u z I q n n in calculated result in Figure 6.10 is similar to the experimental result in Figure 6.8. Both the experimental result in Figure 6.8 and the calculated result by using the field components from FDTD simulation in Figure 6.10 confirm that the asymmetric cross terms with odd powers of in-and out-of-plane fields play a significant role in PEEM imaging for normal optical field incidence. The nonlinear coupling of orthogonal field components can be anticipated from other nonlinear spectroscopy measurements such as surface second harmonic generation 134 .