Optical interactions between single emitters and nanostructures

This work studies the optical interactions between single emitters, mainly quantum dots (QD) and a sharp tip. The fluorescence intensity, quantum yield and angular emission of a single emitter can be strongly modifi ed by near- field coupling with the sharp tip. Gold, silicon, and carbon nanotube (CNT) tips are employed in order to understand the physical mechanisms which are responsible for the various near- field eff ects. Each of these materials carries diff erent properties, which modify the optical properties of QDs in unique ways. In order to maximize the amount of information accessible by our near- field scanning microscope (NSOM), a novel near-f ield tomography technique is implemented. This technique facilitates the revelation of a number of interesting three-dimensional near- field features and is instrumental in the study of the di fferent near- field mechanisms. The flexibility in the data acquisition (DAC) technique allows us to study the influence of fluorescence intermittency (blinking) in QDs on the near- field coupling with the probes. The fluorescence emission from states with high quantum yield is more sensitive to quenching due to energy transfer, while in the low-yield states, near- field signal enhancement is more pronounced. The emission fluctuations of the QDs are progressively suppressed upon approach of a gold tip due to strong near- field coupling of gold tips to the QDs. Moreover, the angular emission of QDs in proximity to gold tips is very sensitive to the exact tip-QD position but does not depend on the intrinsic quantum yield of the QD. Energy transfer dominates the interactions of single CNTs with the QDs. Precision measurements of the energy transfer exhibit unique features as a result of the one-dimensional nature of CNTs. In particular, the energy transfer efficiency saturates at ~96% for all CNTs tried, even though the CNTs are expected to have a distribution of chiralities.

OPTICAL INTERACTIONS BETWEEN SINGLE EMITTERS AND NANOSTRUCTURES by Eyal Shafran A dissertation submitted to the faculty of The University of Utah in partial ful llment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah December 2011 Copyright © Eyal Shafran 2011 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of has been approved by the following supervisory committee members: , Chair Date Approved , Member Date Approved , Member Date Approved , Member Date Approved , Member Date Approved and by , Chair of the Department of and by Charles A. Wight, Dean of The Graduate School. Eyal Shafran Jordan M. Gerton 07/13/2011 Mikhail Raikh 07/13/2011 Andrey Rogachev 07/13/2011 Michael Vershinin 07/13/2011 Ling Zang David Kieda Physics and Astronomy ABSTRACT This work studies the optical interactions between single emitters, mainly quan- tum dots (QD) and a sharp tip. The uorescence intensity, quantum yield and angular emission of a single emitter can be strongly modi ed by near- eld coupling with the sharp tip. Gold, silicon, and carbon nanotube (CNT) tips are employed in order to understand the physical mechanisms which are responsible for the various near- eld e ects. Each of these materials carries di erent properties, which modify the optical properties of QDs in unique ways. In order to maximize the amount of information accessible by our near- eld scan- ning microscope (NSOM), a novel near- eld tomography technique is implemented. This technique facilitates the revelation of a number of interesting three-dimensional near- eld features and is instrumental in the study of the di erent near- eld mecha- nisms. The exibility in the data acquisition (DAC) technique allows us to study the in uence of uorescence intermittency (blinking) in QDs on the near- eld coupling with the probes. The uorescence emission from states with high quantum yield is more sensitive to quenching due to energy transfer, while in the low-yield states, near- eld signal enhancement is more pronounced. The emission uctuations of the QDs are progressively suppressed upon approach of a gold tip due to strong near- eld coupling of gold tips to the QDs. Moreover, the angular emission of QDs in proximity to gold tips is very sensitive to the exact tip-QD position but does not depend on the intrinsic quantum yield of the QD. Energy transfer dominates the interactions of single CNTs with the QDs. Precision measurements of the energy transfer exhibit unique features as a result of the one-dimensional nature of CNTs. In particular, the energy transfer e ciency saturates at 96% for all CNTs tried, even though the CNTs are expected to have a distribution of chiralities. To my family - Jen and Ethan CONTENTS ABSTRACT : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iii LIST OF FIGURES: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viii LIST OF TABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xi ACKNOWLEDGMENTS: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xii CHAPTERS 1. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 The di raction limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Heisenberg's uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Detecting evanescent waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. OPTICAL SIGNAL DUE TO NEAR-FIELD INTERACTIONS : 10 2.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.2 Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Field enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Lightning-rod e ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Localized surface plasmon resonance . . . . . . . . . . . . . . . . . . . . . . 19 2.2.3 Optical antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.4 Interference e ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.5 AFM tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Single emitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.1 Dipole emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.2 Spontaneous decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.3 Energy dissipation in lossy material . . . . . . . . . . . . . . . . . . . . . . 28 2.3.4 Energy transfer between molecules . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.5 Quantum yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4 Collection e ciency and angular emission . . . . . . . . . . . . . . . . . . . . . . 34 2.5 The measured signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3. EXPERIMENTAL SETUP: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 43 3.1 Basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2 Types of illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.1 Gaussian illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.2 Radial illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.3 TIR illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.4 How to choose the illumination condition . . . . . . . . . . . . . . . . . . 52 3.3 Atomic Force Microscope (AFM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.1 Simple Harmonic Oscillator (SHO) . . . . . . . . . . . . . . . . . . . . . . . 54 3.3.2 Repulsive vs. attractive imaging . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.3 Making carbon nanotubes tips . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4 Producing a near- eld signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4.1 Approach curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4.2 Single Photon Near-Field Tomography (SP-NFT) . . . . . . . . . . . . 60 3.4.3 Fluorescence lifetime data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4. THREE-DIMENSIONAL MAPPING OF NEAR-FIELD INTERACTIONS VIA SINGLE-PHOTON TOMOGRAPHY: : : : : : : : : : : : : : : : : : : : 66 4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.6 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5. THE EFFECTS OF INTRINSIC QUANTUM YIELD ON THE NEAR-FIELD INTERACTIONS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86 5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.6 Supplementary information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.6.1 Polarization dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.6.2 Suppression of emission uctuations during blinking . . . . . . . . . . 96 5.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6. ENERGY TRANSFER FROM AN INDIVIDUAL QUANTUM DOT TO A CARBON NANOTUBE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 102 vi 6.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.6 Supplementary information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.6.1 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.6.2 Evidence against charge transfer . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.6.3 Possible systematic e ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.6.4 Normalization procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.6.5 Dependence on illumination polarization . . . . . . . . . . . . . . . . . . . 121 6.6.6 Nanotube buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7. CONTROLLING ANGULAR EMISSION OF QUANTUM DOTS WITH SHARP TIPS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 126 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.2 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.5 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 APPENDICES A. LIFETIME MEASUREMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 138 B. MEASURED DIELECTRIC FUNCTIONS : : : : : : : : : : : : : : : : : : : 141 C. SURFACE PLASMONS POLARITONS : : : : : : : : : : : : : : : : : : : : : : 143 D. ELECTRIC FIELD AT THE FOCUS : : : : : : : : : : : : : : : : : : : : : : : : 145 E. DEFLECTION SIGNAL : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 147 F. TIME STAMPING CIRCUIT : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 149 G. CNT GROWTH : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 151 vii LIST OF FIGURES 1.1 Apertureless NSOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Dielectric function of bulk gold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Dielectric function of doped silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Dielectric sphere in a uniform electric eld . . . . . . . . . . . . . . . . . . . . . . 18 2.4 3D electrostatic simulation of a near- eld probe made out of silicon in a uniform static electric eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Localized Surface Plasmon Resonance in di erent nanostructures . . . . . 21 2.6 An example of an optical antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.7 Dipole orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.8 Dipole radiation pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.9 F orster resonance energy transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.10 Dipole emitter near a re ecting mirror . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.11 Single molecule emission next to a gold sphere . . . . . . . . . . . . . . . . . . . 35 2.12 Electric dipole near a glass-air interface . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.13 Microscope objective collection e ciency of single molecule emission next to an optical antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1 Schematic of experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Illustration of the di erent illumination conditions . . . . . . . . . . . . . . . . 46 3.3 Gaussian beam at the focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Beam pro les at the focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.5 Total internal re ection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.6 AFM amplitude and phase signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.7 AFM scan of a CNT wafer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.8 SEM image of CNT tip and Force curves to determine the length . . . . . 58 3.9 Data acquisition procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.10 Phase histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.11 Creating a tip-sample distance dependent lifetime curve . . . . . . . . . . . . 63 4.1 Schematic of experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Tomographical reconstruction of a 20-nm diameter uorescent sphere . . 72 4.3 Comparison of approach curves at two di erent wavelengths . . . . . . . . . 74 4.4 Comparison of di erent illumination and scan conditions . . . . . . . . . . . 76 4.5 Three-dimensional tomographic reconstruction of a 20-nm diameter u- orescent sphere using a silicon tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.6 Comparison of tomographical and lock-in analyses . . . . . . . . . . . . . . . . 80 4.7 Comparison of cross-sections over the uorescent bead . . . . . . . . . . . . . 82 5.1 Experimental scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2 Vertical approach curves for di erent count-rate thresholds. . . . . . . . . . 92 5.3 Separation of enhancement and quenching. . . . . . . . . . . . . . . . . . . . . . . 94 5.4 Polarization dependence of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.5 Suppression of uorescence emission uctuations during blinking. . . . . . 98 5.6 Dependence of emission uctuations on tip-sample separation. . . . . . . . 99 6.1 Experimental scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2 Typical approach curves for bright and dark states of a QD . . . . . . . . . 109 6.3 Summary of energy transfer measurements for six CNTs of di erent lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.4 Energy transfer measurements during QD aging . . . . . . . . . . . . . . . . . . 113 6.5 Comparison of quantum dot blinking statistics with and without a carbon nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.6 Approach curves for a bare gold tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.7 Polarization dependence of energy transfer measurements . . . . . . . . . . . 121 6.8 Buckling of a CNT during an approach curve measurement . . . . . . . . . 123 7.1 Structure and emission properties of CdSe nanocrystal. . . . . . . . . . . . . . 129 7.2 Polarization changes due to gold tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.3 Polarization changes due to silicon and CNT tips. . . . . . . . . . . . . . . . . . 133 7.4 X-Z slice using gold tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 7.5 Comparison of near- eld signal with near- eld DOP. . . . . . . . . . . . . . . . 135 A.1 Emission intensity and lifetime as a function of tip-sample distance for a CNT tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 A.2 Emission intensity and lifetime as a function of tip-sample distance for a silicon tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 A.3 Emission intensity and lifetime as a function of tip-sample distance for a gold tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 ix B.1 Dielectric function of metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 B.2 Dielectric function of dielectric material . . . . . . . . . . . . . . . . . . . . . . . . . 142 D.1 The electric eld components at the focus of a microscope objective . . . 145 E.1 Illustration of the AFM de ection signal . . . . . . . . . . . . . . . . . . . . . . . . 147 F.1 Time stamping circuit diagram for tip-oscillations . . . . . . . . . . . . . . . . . 150 x LIST OF TABLES 3.1 How to choose the illumination condition. . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Information contained in each photon . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ACKNOWLEDGMENTS This work could not have been done without the help and support of many people. First of all, I would like to thank Jordan Gerton for his guidance, patience, and his persistently positive attitude. I realize that I am not always easy to work with, but Jordan has been able to get the best out of me by somehow tolerating my stubbornness and Israeli demeanor. I would also like to thank the various lab members I got to work with over the years. Analia Dall'As en, Anil Ghimire, Ben Martin, Carl Ebeling, Cassandra Hammons, Charles McGuire, Chun Mu, Jason Ho and Jessica Johnston - you have all made my life harder, but better. Seriously, thanks for making the lab a fun and happy working environment and for your feedback and insight into our research. A special thanks to Ben Mangum for taking me under his wing when I joined the lab. Our arguments about science, religion, and all of humanity's problems were my favorite part of every day. Much of this work in this dissertation was equally contributed by Ben. Nobody can succeed without a loving and supportive home environment. Thanks to my parents, who initially created this for me and continue to show interest in my work and give a lot of support even from a great distance. Thanks to my lovely wife Jen who pretends to listen and care about my research and to my friend Idan who doesn't even pretend. There are too many people to name who have been supportive and who have had con dence in my abilities, even when I didn't, thanks for everything. Finally, I would like to thank \Blueberry" for giving me a very real deadline for nishing my dissertation. CHAPTER 1 INTRODUCTION As objects get smaller, microscopic behavior becomes more signi cant and quan- tum mechanical e ects become more pronounced and dominant. Our ability to detect nanometer phenomena has increased dramatically with the invention of techniques such as electron microscopy, scanning probe microscopy (SPM), near- eld scanning optical microscopy (NSOM), optical nanoscopy and many more. Today's technology allows us to fabricate and synthesize complex objects with nanometer precision that therefore possess di erent electronic, mechanical and optical properties than objects on the macroscale. As our technological abilities improve, so does our understanding of nanoscience, and new novel nanotechnology applications have already emerged with signi cant contributions to medicine, electronics, photovoltaics and microscopy [1{6]. Of particular interest is the light-matter interaction on the nanoscale. The optical properties of structures with much smaller dimensions than the wavelength of incident light (i.e., nanostructures) can be very di erent from bulk characteristics. Most importantly, these properties vary based on the exact shape, size and composition of the nanostructure, which allows the optical properties to be tuned by carefully engi- neering the structure. Furthermore, the interaction of nanostructures with additional materials, such as other nanostructures, uorescence molecules and analyte solutions, can further change the interaction with light. For example, metallic nanoparticles pre- dominantly scatter light at di erent wavelengths when they are isolated, aggregated or surrounded by an analyte. These unique scattering properties have already made an impact in biosensing applications, such as immunoassay detection and even the home pregnancy test [7]. Other applications of metallic nanoparticles in cutting-edge elds include improved photovoltaic devices [8] and the integration into nanoscale 2 electronics [9]. The ability to precisely engineer nanoshells - dielectric core metallic shell particles - has enabled the ne tuning of the absorption resonance of these particles to the infrared and might lead to a breakthrough in photothermal cancer therapy treatments and diagnostics [10]. In semiconductor nanoparticles, the spatial con nement of the charge carriers results in the modi cation of the energy levels of the nanocrystal. This quantum con nement e ect leads to optical properties which can be tuned by the size of the nanocrystal [11]. If the quantum con nement occurs in all three spatial dimensions, the nanocrystal is referred to as a quantum dot (QD). Due to their broad absorption spectra, narrow emission band, high quantum yield and high photostability, semi- conductor nanocrystals are desirable for light-emitting devices [12] and biological imaging applications [13]. They are also useful to various other applications including photovoltaics [14{16], thin layer eld-e ect transistors (FET) [6,17] and quantum dot lasers [18]. The capability to con ne light to a subdi raction volume by nanostructures with sharp edges such as lightning rods [19,20], optical antenna [21{23], and roughened sur- faces [24{26] is particularly appealing for spectroscopic and uorescence applications. When a molecule is placed in proximity to a nanostructure, near- eld e ects become prominent and the properties of the \coupled" system need to be considered. This allows the manipulation of light by means of enhanced excitation and emission [27,28], modi ed spectra [29], and control of the angular emission [23, 30]. For example, the Raman cross-section of molecules can be enhanced by many orders of magnitude near metal surfaces, which enables single molecule Raman spectroscopy measurements via surface enhanced Raman spectroscopy (SERS) [31] and tip-enhanced Raman spectroscopy (TERS) [32]. The con ned elds near a sharp dielectric or metallic tip can also be used for high contrast optical microscopy with resolution which far surpasses conventional optical microscopy [33{36]. 3 1.1 The di raction limit In conventional microscopy light is focused by a microscope objective lens onto/from a sample. The resolving power of an optical microscope depends on the ability of the lens to focus light to a tight spot. Due to the wave nature of light, the light di racts as it encounters the lens, and as a result the light cannot be focused to an in nitely tight spot. This means that when an isolated, in nitesimally small object is imaged, the image will re ect the size of the laser spot and not the object. This phenomenon is often referred to as the di raction limit. The Abbe limit states that for light with wavelength , traveling in a medium with refractive index n, and going through a lens with a collection angle of , the smallest spot size is denoted by d = 2n sin (1.1) The denominator can be de ned by the numerical aperture of the lens NA=n sin . The NA of modern microscope objectives can be as high as NA=1.65; therefore, con- ventional optical microscopy cannot distinguish details in the specimen smaller than =2. This optical phenomenon is a major hindrance in the study of nanoscience. 1.2 Heisenberg's uncertainty One way the obtain better resolution than the di raction limit is by using near- eld optics. The improved resolution due to the detection of near- eld radiation can be understood by examining the relationship between the di raction limit and Heisen- berg's uncertainty principle. A thorough discussion on the Heisenberg's uncertainty principle in near- eld optics can be found in Ref. [37,38]. The optical representation of the uncertainty principle can be written as follows [37] x px h (1.2) where x and px are the uncertainties in the position and momentum of a particle in one dimension, respectively, and h is the Planck constant. Full knowledge of the position of a particle and its exact momentum is forbidden by the uncertainty principle. It is clear from Eq. 1.2 that a better position accuracy can be obtained on 4 the expense of a large uncertainty in the momentum. Using the de Broglie relation p = ~k, where k is the wavevector of the particle, Eq. 1.2 can be rewritten as x kx 2 : (1.3) In theory, if there was a way to extend the integrated wavevectors to in nity, there would be no limitation on the spatial resolution. However, in a classical microscope, only the radiating (i.e., far- eld) components are detected, which possess a limit on the spread of wavevectors kx collected. If the light is collected or focused by a microscope objective lens with numerical aperture of NA, only the spatial frequencies between jkxj = 0 and jkxj = n sin (!=c), where ! is the angular frequency and c is the velocity of the photons, will be collected by the lens. If we insert the expression for the spread of wavevectors kx = 2n sin (!=c) into Eq. 1.3 and use the dispersion relation for photons !=c = (2 = ), we recover a similar expression to the di raction limit x 2n sin : (1.4) Even if the lens was able to collect over all angles, i.e., sin = 1 and kmax = n!=c, the maximum collected wavevector is still limited by the fact that only the propagating far- eld components of the radiation reach the objective lens. Therefore, the spatial resolution is limited by the inability to detect the nonradiating components of the light. The detection of evanescent components (i.e., waves with an exponentially decay- ing intensity) can achieve better resolution than the di raction limit without violating the uncertainty principle. Let us consider light which has wavevector components along the x and z directions. We nd that k2x + k2 z = (!=c)2 (1.5) If we consider the light along the z-direction to have an imaginary wavevector kz (i.e., kz = ijkzj), the wavevector along the x-axis takes the following form: k2x = (!=c)2􀀀k2 z = (!=c)2+jkzj2. Therefore, the range of collected wavevectors is extended, and thus, the spatial resolution in the x-axis is improved in comparison with the 5 di raction limit. If we de ne kz as imaginary however, the propagation term in the z-direction becomes eikzz = e􀀀jkzjz and the radiation decays exponentially, as expected from evanescent light. The problem then becomes the ability to detect such strongly localized decaying light. To do so, one must bring an object with subwavelength features into the near- eld radiation zone as discussed in the next section. However, such close \invasion" leads to the coupling of the object with the sample, which can result in undesired or misunderstood outcomes. This complicates matters but also opens up the possibility to control and manipulate light by careful design and understanding of near- eld phenomena. 1.3 Detecting evanescent waves Generally speaking, near- eld components are accessible experimentally by either having a near- eld illumination and/or near- eld detection scheme. In a near- eld illumination scheme, such as an apertureless NSOM, a scatterer smaller than the wavelength of light is brought into a subwavelength separation distance from a sample. In such a case, the evanescent or near- eld components of the scattered light interact with the sample, rather than only the propagating or far- eld components. By scanning a sample relative to a scatterer and collecting the light emitted by or transmitted through the sample, an image could thus be built up that would not be subject to the limits imposed by di raction of the far- eld components. Figure 1.1 illustrates the implementation of apertureless NSOM. Another realization of near- eld illumination can be achieved by aperture type NSOM. In this scheme, the scattering particle is replaced with a subwavelength aperture in a thin metal plate or lm. Very close to the aperture, con ned optical elds re ect the shape of the aperture itself, rather than the far- eld di raction pattern. The light transmitted through the aperture would excite optical processes (e.g., uorescence, Rayleigh scattering, Raman scattering, etc.) in a subdi raction volume, and the resulting optical signal would be collected with a lens (e.g., micro- scope objective) and nally detected in the far eld with a light sensitive detector. Although a subwavelength aperture will transmit merely a fraction of the incident 6 Figure 1.1. Apertureless NSOM. A sharp probe of an atomic force microscope is illuminated with a di raction limited spot. Upon the right excitation conditions, the excitation light is strongly localized at the apex of the probe. When the probe is brought in proximity to a uorescent sample, the localized near- eld components near the apex of the probe interact with sample. The emitted light from the sample is then collected from below by a microscope objective. light, only this transmitted light will excite the sample, so background signals are low. In a near- eld detection scheme, the detector must be on a subwavelength distance from the sample. Although possible, this is a much less common approach in near- eld microscopy and is described in more detail elsewhere [39]. 1.4 Outline In this work an apertureless NSOM is used to study optical interactions between single emitters and various types of materials. The detected signal from uores- cence samples, mainly semiconductor quantum dots, can be a ected by the detection e ciency of the microscope, the coupling of the excitation light to the near- eld probe, and nally the coupling of the emitter to the near- eld probe. The physical mechanisms, which modify the total measured uorescence signal, are reviewed in Chapter 2. The experimental details concerning the integration of an atomic force 7 microscope (AFM) with an optical setup are discussed in Chapter 3. The strength of our measurements lie in our unique data acquisition (DAC) and analysis techniques; therefore, the basic DAC and analysis are also summarized in this chapter. Chapters 4-7 are a summary of the experimental results. Since near- eld mi- croscopy is a relatively new eld, new acquisition and detection techniques help reveal new physics. The extension of our detection abilities from 2D images to a complete 3D map of near- eld interactions is explained in Chapter 4. This technique has been used in the rest of the projects as well. In Chapter 5, the balance between competing near- eld mechanisms, which can result in signal enhancement and/or signal reduc- tion (quenching), is studied by utilizing photoluminescence blinking in semiconductor QDs. By measuring the near- eld signal as the QDs undergoes di erent quantum yield states, we were able to isolate the contribution of enhancement and quenching for various AFM tips materials. Chapter 6 is focused on the energy transfer between isolated QDs and single carbon nanotubes (CNT). Finally, the change in angular emission of QDs in proximity to commercial metal tips is examined in Chapter 7. 1.5 References [1] S. Tans, A. Verschueren, and C. Dekker, Nature 393, 49 (1998). [2] W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003). [3] E. Serrano, G. Rus, and J. Garcia-Martinez, Renewable & Sustainable Energy Reviews 13, 2373 (2009). [4] M. Stockman, Physics Today 64, 39 (2011). [5] M. Ferrari, Nature Reviews Cancer 5, 161 (2005). [6] D. V. Talapin, J.-S. Lee, M. V. Kovalenko, and E. V. Shevchenko, Chemical Reviews 110, 389 (2010). [7] T. Endo et al., Analytical Chemistry 78, 6465 (2006). [8] H. A. Atwater and A. Polman, Nature Materials 9, 205 (2010). [9] E. Ozbay, Science 311, 189 (2006). [10] S. Lal, S. E. Clare, and N. J. Halas, Accounts of Chemical Research 41, 1842 (2008). 8 [11] D. E. Gomez, M. Califano, and P. Mulvaney, Physical Chemistry Chemical Physics 8, 4989 (2006). [12] V. L. Colvin, M. C. Schlamp, and A. P. Alivisatos, Nature 370, 354 (1994). [13] I. L. Medintz, H. T. Uyeda, E. R. Goldman, and H. Mattoussi, Nature Materials 4, 435 (2005). [14] P. V. Kamat, Journal of Physical Chemistry C 111, 2834 (2007). [15] P. V. Kamat, Journal of Physical Chemistry C 112, 18737 (2008). [16] R. Schaller, M. Sykora, J. Pietryga, and V. Klimov, Nano Letters 6, 424 (2006). [17] D. V. Talapin and C. B. Murray, Science 310, 86 (2005). [18] V. I. Klimov et al., Science 290, 314 (2000). [19] L. Novotny, Applied Physics Letters 69, 3806 (1996). [20] J. L. Bohn, D. J. Nesbitt, and A. Gallagher, Journal of the Optical Society of America A 18, 2998 (2001). [21] P. Mhlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005). [22] J. N. Farahani, D. W. Pohl, H. J. Eisler, and B. Hecht, Physical Review Letters 95, 4 (2005). [23] T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, Nature Photonics 2, 234 (2008). [24] M. Fleischmann, P. J. Hendra, and A. J. McQuillan, Chemical Physics Letters 26, 163 (1974). [25] D. L. Jeanmaire and R. P. VanDuyne, Journal of Electroanalytical Chemistry 84, 1 (1977). [26] M. G. Albrecht and J. A. Creighton, Journal of the American Chemical Society 99, 5215 (1977). [27] P. Anger, P. Bharadwaj, and L. Novotny, Physical Review Letters 96 (2006). [28] A. Kinkhabwala et al., Nature Photonics 3, 654 (2009). [29] M. Ringler, A. Schwemer, M.Wunderlich, and A. Nichtl, Physical Review Letters 100 (2008). [30] A. G. Curto et al., Science 329, 930 (2010). [31] K. Kneipp et al., Physical Review Letters 78, 1667 (1997). 9 [32] B. Pettinger, B. Ren, G. Picardi, R. Schuster, and G. Ertl, Physical Review Letters 92 (2004). [33] H. G. Frey, S. Witt, K. Felderer, and R. Guckenberger, Physical Review Letters 93 (2004). [34] V. V. Protasenko, A. Gallagher, and D. J. Nesbitt, Optics Communications 233, 45 (2004). [35] J. M. Gerton, L. A. Wade, G. A. Lessard, Z. Ma, and S. R. Quake, Physical Review Letters 93 (2004). [36] Z. Y. Ma, J. M. Gerton, L. A. Wade, and S. R. Quake, Physical Review Letters 97 (2006). [37] J. M. Vigoureux and D. Courjon, Applied Optics 31, 3170 (1992). [38] J. Gre et and R. Carminati, Progress in Surface Science 56, 133 (1997). [39] L. Novotny and B. Hecht, Principles of Nano-Optics, Cambridge University Press, Cambridge, 2006. CHAPTER 2 OPTICAL SIGNAL DUE TO NEAR-FIELD INTERACTIONS A near- eld scanning optical microscope combines the principles of optical mi- croscopy and atomic force microscopy. In apertureless NSOM, a di raction-limited illumination source, usually a focused laser, is used to polarize the tip of the scanning probe. When the probe is brought in the proximity of a sample, the local environment of the sample is perturbed, which results in the modi cation of the excitation rate and the emission rate of the sample. In general, the near- eld component is the result of a competition between tip-induced enhancement of the optical eld and various tip-induced suppression mechanisms. This chapter discusses ve major issues: material properties of the tip and sample, interaction of the excitation with the tip, interaction of the emission with the tip, system's detection e ciency and the total change to the measured signal. The focus is on the uorescence samples although near- eld microscopy is not limited just to the detection of uorescence. The detected signal in our setup is typically the uorescence emission rate modu- lated by the collection e ciency of the optical system. Below saturation (i.e., in the linear scattering regime) of the sample, the total detected signal can be written as: S = C(r)􀀀exc(r)q(r) (2.1) where r is the tip-sample distance, C(r) is the collection e ciency of our system, 􀀀exc(r) is the excitation rate and q(r) is the quantum yield of the sample. Each of the above parameters can change upon bringing a nanostructure in proximity to the sample. In Sec. 2.1, some general properties of materials that pertain to optical inter- actions are reviewed. The discussion covers two types of materials: metals and 11 dielectrics. Section 2.2 focuses on the interaction of the excitation light with the near- eld probe. This interaction generally leads to eld enhancement at the apex of the tip as a result of lightning-rod and plasmon e ects and results in the mod- i cation of the excitation rate 􀀀exc(r). The emission properties of a single emitter in both homogeneous and inhomogeneous environments are reviewed in Sec. 2.3. In particular, this section focuses on the changes in radiative and non-radiative decay rates of an emitter due to the environment which lead to a change in the emitter's quantum yield q(r). Although the angular emission pattern of a dipole is also related to the properties of an emitter and its environment, this e ect is discussed in a separate section since it can change the e ective detection e ciency of the system C(r). Finally, the experimentally measured uorescence signal, as written in Eq. 2.1, is revisited in more details in Sec. 2.5. This chapter is intended to provide some general theoretical background for the experiments discussed in the remainder of the dissertation. By the end of the chapter, the reader should be familiar with the general considerations that determine the measured uorescence signal from a single emitter in proximity to a near- eld probe. A thorough review of nano-optics and nanoscale optical interactions can be found in Ref. [1]. Several subsections in this chapter follow closely some sections from a recent book chapter we prepared [2]. 2.1 Material properties In order to understand the interaction of light with matter at the nanoscale, we must rst understand the properties of materials at di erent wavelengths. The response of a material to an alternating eld can be described by a complex dielectric function (electric permittivity) (!) = 0(!) + i 00(!) (2.2) The dielectric function strongly depends on the oscillation frequency of the eld, !. For example, metallic materials generally have 0(!) < 0 when ! is small and therefore are highly re ective in this regime. In contrast, when ! is large, 0(!) > 0 and the metal is mostly transparent. The imaginary part of the dielectric function is 12 associated with the Ohmic losses in the material, and resistive heating [3]. By using Ohm's law and Maxwell's equations, the complex dielectric function can be denoted by (!) = 0(!) 􀀀 i 0! (2.3) where is the AC conductivity of the material and 0 is the permittivity in vac- uum. Therefore, the imaginary part of the dielectric function is related to the AC conductivity of the material. 2.1.1 Metals Metals are good electric conductors because they have a large density of quasi-free electrons in the conduction band. The Drude-Sommer eld model [1,4,5] approximates these conduction electrons as a free-electron gas and provides an explanation for many observed features such as charge transport and optical measurements in metals. Within this model, it is assumed that the free electron response to an external driving eld (e.g., a light eld) is the dominant e ect and that the contribution of the valence electrons can therefore be neglected. The dielectric function calculated from this model is then D(!) = 1 􀀀 !2 p !2 + 2 + i !2 p !(!2 + 2) : (2.4) Here = vF =l is a damping term that arises from scattering of the electrons where vF is the Fermi velocity and l is the mean free path of the electrons. The plasma frequency is de ned by !p = p ne2=(me 0) where n is the free-electron density and e and me are the electron charge and e ective mass, respectively. The Drude-Sommer eld model successfully describes the dielectric function of metals in the infrared spectrum. At optical frequencies, ! >> , and the Drude- Sommer eld model is reduced to D(!) = 1 􀀀 !2 p !2 . However, at higher energies, the incident light can excite valence electrons into the conduction band; when this occurs, interband transitions become signi cant and must also be taken into account. For example, in the visible range, most metals exhibit a resonance behavior in the 13 imaginary part of the dielectric function, which can only be explained by the inclusion of the interband transitions. The interband transitions can be most easily accounted for using a semiclassical model where the electrons bound to the lattice sites (i.e., the lower-energy valence electrons) are considered to be classical oscillators with resonance frequencies given by the quantum-mechanical transition frequencies. An external driving eld near these frequencies can promote the valence electrons into the conduction band, where they will then contribute to the electronic and optical response, and thus the dielectric function. The dielectric function due to tightly bound electrons does predict a resonance behavior in 00 [6, 7]. Therefore, the dielectric function of metal is better described by a combination of the Drude-Sommer eld model, which accounts for free-electrons, and an interband term (!) = D(!)+ IB(!) [1, 8{10]. As the spatial dimensions of a metal become similar to the mean free path of the electrons, scattering of electrons from the particle surface changes the dielectric function due to the increase in damping (i.e., increase in ). For example, the mean free path of gold is l 38 nm. Thus, any geometry with dimensions of several tens of nm or smaller will exhibit more damping than bulk gold. For this reason, in the nanoscale regime, the dielectric function is also size dependent. A surface scattering correction term can be analytically calculated for spheres, but for more complex geometries, it is di cult to extract the appropriate damping coe cient [9]. Figure 2.1 compares experimentally measured dielectric function values for gold (solid blue line) taken from Ref. [11] to the the Drude-Sommer eld model (green dotted line). The plasma frequency and the damping coe cient used for calculating the Drude response were !p = 2:165 1015 Hz and = 17:64 1012 Hz [12], respectively. As mentioned above, at longer wavelengths, the experimental results and the model agree well. At shorter wavelengths, the Drude-Sommer eld model fails to reproduce the measured results, particularly the resonances observed in 00(!). Most metals have a negative real part of the dielectric function in the visible range and an imaginary part that is comparable in magnitude to the real part (Fig. 2.1 and Appendix B). The wavevector can be related to the dielectric function by 14 −60 −40 −20 0 200 400 600 800 1000 0 2 4 6 8 Wavelength (nm) Im(ε) Re(ε) Figure 2.1. Dielectric function of bulk gold. Measured data (solid blue line) and the Drude-Sommer eld model (dotted green line). Dielectric function values are found in [11]. k = (!=c)p , where is the magnetic permeability. Therefore, if the dielectric function has an imaginary component, so does the wavevector k. As a result, the electromagnetic elds are attenuated as they propagate in the material: the electrons in the metal move to screen the applied eld, thus decreasing the electric and magnetic elds as the light penetrates the metal. The exponential decay of the electromagnetic elds into the material has a characteristic decay length called skin depth: = 4 p 00 (2.5) where is the wavelength of the incident light. For example, gold has a skin depth of 31:5 nm at = 550 nm. This is crucial for nanoscale optics because the size of nanoparticles may be on the same order of magnitude as the skin depth (at optical frequencies). 2.1.2 Dielectrics Dielectrics have a very di erent response to an electromagnetic eld compared to metals. For dielectrics, the main response to the alternating electric eld is the 15 result of bound electrons. The AC conductivity in dielectrics is usually very small, indicating a minute amount of resistive losses in the material. Although silicon is a semiconductor, it has very small AC conductivity at optical frequencies and thus behaves largely as a dielectric. Silicon is a widely used material for near- eld applications due to the large 0(!) and small 00(!) (relative to metal). The measured dielectric function for silicon can be seen in Fig. 2.2. In the UV range, the real part (solid blue line) has a dispersion lineshape and the imaginary part (dotted green line) has a strong resonance peak. In the visible spectral range, the dielectric function does not change much. Typically, commercial silicon probes are doped. The type and concentration of dopant can in uence the dielectric properties of silicon. However, the doping concentration and type of the dopants is di erent for each manufacturer and therefore the results shown here are only in qualitative agreement with that expected for silicon probes. Other dielectric materials typically do not have an imaginary component in the 200 400 600 800 1000 −20 −10 0 10 20 30 40 50 ε Wavelength (nm) Figure 2.2. Dielectric function of doped silicon. In the optical range, the real part (solid blue line) is high while the imaginary part is fairly low (dotted green line). Data points taken from Ref. [13]. 16 dielectric function, but their real part is substantially smaller than that of silicon (Appendix B). 2.2 Field enhancement The working principle of a near- eld microscope is usually to use a probe to enhance the local eld in a nanometer-scale region to produce high-resolution images. The interaction between the excitation light and the near- eld probe leads to the polarization of the probe, which can result in a locally con ned spot with a much larger (local) amplitude than the incident light. Essentially, much like an antenna, the probe is used in order to concentrate the light source. The general demand for most near- eld experiments is to increase the local intensity I(r; r0) compared with the far- eld intensity I0(r0) as the tip approaches the sample. Here, r0 signi es the position of the emitter and r is the distance between the emitter and the probe. It is useful to de ne an enhancement factor, which describes the ratio between the local intensity and the far- eld excitation intensity (r) I(r; r0) I0(r0) : (2.6) If there is no tip-induced change in the absorption cross-section of the sample, the intensity is related to the excitation rate: I(r; r0) I0(r0) = 􀀀exc(r) 􀀀0 exc (2.7) where 􀀀0 exc is the far- eld excitation rate (i.e., when the tip is far away). Note that the enhancement factor, , depends on the tip-sample separation distance and is an indicator of how much intensity is gained by bringing the probe close to the sample. To optimize the near- eld contrast, should be as large as possible, and to optimize resolution, should decay over as short a range as possible. This section will discuss eld-enhancement mechanisms and their dependence on di erent types of near- eld probes. The most important eld-enhancement e ects are the lightning-rod e ect and localized surface plasmon resonances. These two mechanisms are strongly dependent on the precise geometry of the near- eld probe 17 and on the tip material, the excitation wavelength and the light's polarization with respect to the tip. 2.2.1 Lightning-rod e ect In the most general sense, eld enhancement is the result of a discontinuity in the perpendicular (to the interface) component of the electric eld amplitude on either side of an interface with di erent dielectric constants, as required by Maxwell's equations. To understand the basic mechanism of eld enhancement, we rst consider the simple example of a dielectric sphere in a uniform static electric eld. In this case, the applied electric eld causes the electrons and ions to migrate toward opposite ends of the sphere, thus polarizing the sphere. This creates an internal eld inside the sphere resulting in a total electric eld outside the sphere which is a superposition of both the applied and induced elds. If the sphere's size is much smaller than the wavelength, retardation e ects can be neglected and a quasi-static approximation is justi ed even in the case of an oscillating electromagnetic eld. The electric eld inside and outside the sphere can be obtained by applying Maxwell's equations at the boundary of the sphere. In particular, the tangential component of the electric eld and the normal component of the displacement eld must be continuous at the boundary. In this case, the electric eld can be expressed analytically: ~E out(r; ) = E0 cos 1 + 2 r 􀀀 1 r + 2 R3 r3 ^r + E0 sin 􀀀1 + r 􀀀 1 r + 2 R3 r3 ^ (2.8) ~E in(r; ) = E0 3 r + 2 h cos ^r 􀀀 sin ^ i = E0 3 r + 2 ^z (2.9) where ~E 0 = jE0j^z is the applied eld, R is the radius of the sphere and r = dielectric= medium is the relative permittivity between the dielectric material and the surrounding material. Inside the sphere, the electric eld is uniform and is in the same direction as the applied eld. Outside the sphere, the eld is a superposition of the applied eld with the eld of an electric dipole. The eld intensity for a 10 nm radius silicon sphere is plotted in Fig. 2.3. At the poles along the direction of the polarization, the intensity is enhanced whereas 18 1 2 3 4 5 6 7 10nm Figure 2.3. Dielectric sphere in a uniform electric eld. The color scale illustrates the eld intensity (I / jEj2). A 2D slice through the center of the sphere shows that the sphere causes a redistribution of the eld, where it is strongest along the direction of the polarization (in this case the z-axis). The dielectric function used in this case is Si = 17:6 + i0:1 for silicon at = 532nm. the intensity is reduced near the equatorial plane. The maximum enhancement is achieved at the surface (r = R) and does not depend on the sphere size. Emax = E0 1 + 2 r 􀀀 1 r + 2 (2.10) For dielectric materials, the maximum eld enhancement is Emax = 3 for r ! 1 which results in an intensity enhancement of Imax = 9. For nonspherical geometries, the enhancement e ect is ampli ed in regions of high curvature where the eld line density is largest. This redistribution of the electric eld is known as the lightning-rod e ect, where elds are enhanced most strongly at the regions of highest curvature. For example, Bohn et al. [14] have shown that as the long axis of a prolate spheroid approaches in nity, the intensity enhancement adopts the analytic form: Emax = E0 r: (2.11) 19 Due to the high curvature at the ends (tips) of such a spheroid, the maximum eld enhancement is no longer independent of r and thus can be much greater than in the case of a sphere. The lightning-rod e ect can be very strong for any geometric shapes with sharp edges. Therefore, an AFM tip, which has a very sharp apex, is well designed to utilize the enhancement from the lighting-rod e ect. Figure 2.4 illustrates the strong intensity enhancement at the apex of an AFM tip placed in an uniform static electric eld. The intensity enhancement is much greater than that of the Si sphere (Fig. 2.3). 2.2.2 Localized surface plasmon resonance A further contribution to the eld enhancement can occur due to plasmon reso- nances in the material. For example, the solution for the electric eld near a metallic sphere placed in a uniform electric eld is similar to Eq. 2.8 as long as the diameter 5 nm Figure 2.4. 3D electrostatic simulation of a near- eld probe made out of silicon in a uniform static electric eld. The color scale illustrates the magnitude of the eld (i.e., jEj). Only a small portion of the simulation area is shown here to emphasize the eld enhancement near the apex of the tip. 20 of the sphere is much smaller than the skin depth (Eq. 2.5). If the sphere is small enough, light is not attenuated much as it travels through the sphere, and the eld inside the sphere can be approximated to be uniform, similar to the dielectric case. However, since for most metals 00(!) < 0 at optical frequencies, the denominator in Eq. 2.10 vanishes when <( r) = 􀀀2 and as a result, the eld enhancement diverges. This is called Localized Surface Plasmon Resonance (LSPR) because it results from collective oscillations of the charge carriers within the metal nanoparticle - in essence, nonpropagating (standing wave) oscillation of charges con ned to a highly localized region. LSPR can generate enormous eld enhancement. The strength of the resonance depends sensitively on material, excitation wavelength, and geometry of the illumi- nated nanostructure. Moreover, given a certain geometry, the nanostructure size also matters. If the nanostructure is larger than the skin depth, the inner electrons will be shielded, and thus will not participate in the resonant oscillations, resulting in reduced enhancement. For smaller particles with large surface area-to-volume ratios, electron collisions with the surface become a large source of plasmon damping, thus reducing plasmonic eld enhancement [9]. Figure 2.5 illustrates the importance of size, shape and composition in the light- scattering properties of metallic nanoparticles. The scattering for these particles is much stronger for incident wavelengths at the plasmon resonance, and therefore the scattering color is an indication of the plasmon resonance wavelength. Silver spheres scatter predominantly at di erent wavelengths as a function of sphere size, with a redshifted plasmon resonance for larger sphere diameters. For silver nanoprisims, the plasmon resonance is even more redshifted. The LSPR phenomenon is often used to create huge eld enhancements for Surface Enhanced Raman Spectroscopy (SERS) [15, 16]. It is also possible to fabricate an AFM tip with a metallic sphere at its apex. In fact, these tips have proven to have very large signal enhancement factors [3, 17, 18]. The correlation between sphere size and signal enhancement for gold has been studied experimentally [3, 19]. 21 Figure 2.5. Localized Surface Plasmon Resonance in di erent nanostructures. The color of the scattered light depend on the nanosructure's size, shape and composition. Reprinted with permission from [20]. Copyright 2005 American Chemical Society. 2.2.3 Optical antenna The combination of the lightning-rod e ect and plasmon resonance have led re- searchers to design nanostructures with strong, shape-speci c resonances in order to drastically enhance the optical eld. This is, in fact, a description of an opti- cal antenna, which like their radio or microwave analogs, can be used to convert free propagating electromagnetic waves into localized elds and vice versa. Many di erent optical antenna geometries have been theoretically proposed, and recently due to improvements in fabrication abilities, many have now been experimentally demonstrated [21{24]. An example of an asymmetrical bowtie antenna, which was fabricated in James Schuck's group, can be seen in Fig. 2.6. Optical antenna theory di ers from antenna design in the radio and microwave spectral regions because at optical frequencies the skin depth is of the same order of magnitude as the antenna's dimensions [25]. Further information about optical antenna can be found in Ref. 22 [26, 27]. 2.2.4 Interference e ects Another important e ect that can in uence the local eld intensity, and thus , is interference of the direct excitation eld with that scattered by the probe. The total excitation eld , E(r; r0), at the location of a particular feature of the sample, r0, is the superposition of the incident light and the scattered light. The resulting interference pattern strongly depends on illumination conditions, the shape and material of the probe, the position of the probe relative to the sample feature, r, and the polarization of the incident eld with respect to the probe. Destructive interference can result in an enhancement factor smaller than unity, and constructive interference results in > 1. Figure 2.6. An example of an optical antenna. The antenna was fabricated in James Schuck's group and was scanned under our AFM. This unique antenna design enables spectral ltering and steering of optical elds [28, 29]. 23 For elongated geometries like an AFM probe, light polarized parallel to the long axis of the tip (vertical polarization) leads to much stronger scattering than light polarized perpendicular to the tip (horizontal polarization). This correlates to the fact that vertical polarization produces a much larger eld enhancement. Therefore, interference e ects are stronger for vertically polarized excitation [30]. It has been shown that Total Internal Re ection (TIR) illumination creates parabolic shaped interference patterns that persist up to several wavelengths in the vicinity of the tip, and whose shape is due to shadowing of the incident eld behind the tip. The direction of the shadow depends on the prorogation direction of the light [31]. Reduction in the measured signal due to destructive interference of the excitation eld at tip-sample separation distances of roughly a wavelength has been observed for gold spheres [17]. The interference pattern depends on the particular tip geometry, as di erent shapes will cause di erent scattering [32]. Furthermore, the material of the tip also plays a crucial role because the phase shift of the scattered light with respect to the direct light depends on the re ection coe cient of the speci c material [6, 7]. While the interference e ects are secondary in importance and are typically much weaker than any eld enhancement, they should not be overlooked. 2.2.5 AFM tips For most probe geometries employed in near- eld microscopy, an analytical so- lution for the eld enhancement cannot be obtained. Furthermore, the long axis of a typical near- eld probe is much larger than an optical wavelength; thus, a full electrodynamic approach must be used. More precise predictions of the eld en- hancement for realistic probe geometries, such as a cone or pyramid, can be obtained by solving Maxwell's equations using numerical simulations. Such calculations have predicted eld enhancement values as high as 3,000 for metallic tips and 225 for dielectric tips [14, 33]. However, such large enhancement factors have never been observed experimentally. One possible reason for this is that the actual measured signal includes a few di erent e ects that are di cult to deconvolve and some of which contribute to signal reduction (e.g., quenching). Another possible reason is 24 that (!) for geometries such as an AFM tip might deviate signi cantly from bulk values due to electron scattering on the surface and local defects [34]. An AFM geometry is not ideal for exploiting LSPR. The long axis of the AFM tip is a few orders of magnitude larger than the skin depth of any metal at optical frequencies and therefore the majority of electrons will be shielded from the eld e ect. On the other hand, due to the sharp edge of the AFM tip and its elongated geometry, enhancement due to lightning-rod e ect is very strong. Numerical simulation for metallic geometries which resemble an AFM geometry, predict higher enhancement factors for closed geometries compared with an open geometry [35] indicating the importance of LSPR. It is di cult to experimentally measure or extract the eld enhancement factor. However, signal enhancement from metallic spheres [3, 17{19, 36] has been shown to be higher compared with commercial AFM tips with elongated geometry [32, 37{40]. This suggests that the eld enhancement is stronger for metallic spheres than for commercial AFM probes. 2.3 Single emitters Optical interactions of an emitter with a material are determined by both the properties of the emitter and the domain in which its located. In the rst section, some general properties of materials, mainly the response of the material to an electric eld, were discussed. These properties play an important role in determining the optical properties such as the quantum yield and angular emission of an emitter. In this section, the properties of single emitters in a homogeneous environment and the changes induced by an inhomogenous environment will be reviewed. 2.3.1 Dipole emission In the classical sense, a two-level molecule can be described by an electric dipole moment oscillating at the emission frequency. If the dipole moment of the molecule is denoted by ~p, then the electric and magnetic eld generated by the dipole in a homogeneous environment are [6] 25 E = eikr 4 0 k2 r (n p) n + [3n (n p) 􀀀 p] 1 r3 􀀀 ik r2 (2.12) H = ck2 4 (n p) eikr r 1 􀀀 1 ikr (2.13) where n is a unit vector, k is the wavevector, 0 is the permittivity in vacuum and r is the distance from the dipole. If we assume that the dipole is oriented in the z-axis, as illustrated in Fig. 2.7, and write the expressions for E and H in spherical coordinates, we have: Er = eikr 2 0 cos 1 r3 + ik r2 jpj E = eikr 4 0 sin 1 r3 + ik r2 􀀀 k2 r jpj (2.14) H = eikr 4 0 sin ik r2 􀀀 k2 r jpj where is the permitivitty of the medium. The elds behave quite di erently close to the dipole (i.e., in the near- eld) and far away from it (far- eld). In the near- eld zone, where kr << 1, all the terms, but the 1=r3 term, can be neglected. The contribution of the propagating term eikr is small and the electric eld has components in both radial and transverse directions. ϕ θ x y z r Figure 2.7. Dipole orientation. A dipole oriented in the z axis, i.e., p = jpj^z, in spherical coordinates. 26 The magnetic eld in this region is much smaller than the electric eld and therefore can be neglected. On the other hand, in the far- eld region, i.e., kr >> 1, the term 1=r is dominant. Therefore, only the transverse electric eld component exists with a magnetic eld perpendicular to both the electric eld and the propagation direction. Furthermore, the radiation pattern in this region is quite di erent from the near- eld zone. Figure 2.8 illustrates the emission intensity pattern by a dipole oriented along the z-axis. Three di erent values of kr were chosen (kr = 0:01; 2; 400) to illustrate the eld intensity pattern in the near- eld, intermediate eld and the far- eld, respectively. Close to the dipole (2.8a), the eld intensity is distributed in all directions with a higher intensity along the dipole's axis. In the far- eld eld (2.8c), the dipole does not radiate along its axis at all. The pattern in the intermediate region is highly a ected by the choice of kr. Only the far- eld term contributes to the total energy transport. The Poynting vector, including only the far- eld radiation, is 90 270 180 0 90 270 180 0 90 270 180 (a) (b) (c) 0 Figure 2.8. Dipole radiation pattern. The dipole radiation pattern for di erent kr values. Panel (a),(b) and (c) illustrate the intensity radiation pattern of a dipole in the near- eld (kr = 0:01), intermediate eld (kr = 2), and far- eld (kr = 400), respectively. The dipole's direction is illustrated in red. In the near- eld, the dipole radiates in all directions whereas in the far- eld, the dipole does not radiate along its axis. The magnitude of the intensity is much stronger close to the dipole and is not illustrated in this gure. 27 S(t) = E(t) H(t) = 1 16 2 0 sin2 r2 n3!4 c3 ^r (2.15) The time-averaged power radiated by an electric dipole can be determined by inte- grating the Poynting vector over a spherical surface [7], which yields the following value P 0 = ck4 12 0 jpj2 (2.16) 2.3.2 Spontaneous decay Spontaneous emission from a quantum emitter depends in part on the intrinsic properties of the emitter and, in part, on the environment which surrounds the emitter. Purcell [41] discovered that by placing an atom in a resonant cavity, the emission rate of the atom can be modi ed. This discovery led to the realization that the surrounding environment can strongly a ect the spontaneous decay rates of molecules. To calculate the spontaneous decay rate of a two level quantum emitter, one is required to use quantum electrodynamics (QED). From the Fermi golden rule the following spontaneous decay rate can be obtained [1]: 􀀀 = 2! 3~ 0 jpj2 (r0; !) (2.17) where r0 denotes the position of the emitter, ! is the transition frequency and (r0; !) is the electromagnetic local density of states. An emitter in the excited state can go back to the ground state via a few mechanisms: emission of a photon, internal nonradiative relaxation processes, or by virtual energy transfer to the environment. The rst mechanism is radiative and is characterized by the radiative rate (􀀀r) while the last two are nonradiative and denoted by a nonradiative rate (􀀀nr). The total decay rate is always a sum of the two e ects, i.e., 􀀀 = 􀀀r + 􀀀nr. The spontaneous decay of an emitter is typically characterized by intrinsic radiative and nonradiative rates. Once the emitter is placed in an inhomogeneous environment, both the intrinsic radiative and nonradiative rates may be modi ed. Often, the emitter's spontaneous decay rate in an inhomogeneous environment (􀀀) is compared to the rate in a homogeneous environment (􀀀0). For weak coupling 28 between the emitter and the environment, the results obtained for 􀀀=􀀀0 using QED and classical theory have been shown to be the same [1]. In the classical picture, the spontaneous emission from a dipole in a homogeneous environment can be described by a simple harmonic oscillator [42] p + 􀀀0p_ + !2p = 0 (2.18) where ! is the oscillation frequency and 􀀀0 is the damping rate. The energy of the oscillator decays exponentially with a characteristic time of 0, where 0 = 1=􀀀0 is the uorescence lifetime of the molecule and is de ned as the inverse of the spontaneous decay rate. In an inhomogeneous environment, some of the emitted light is scattered back to the molecule and acts as a driving force on the molecule [42, 43]. Equation 2.19 can be modi ed to account for the driving force p + 􀀀0p_ + !2p = q2 m Es(r0) (2.19) where m and q are the e ective mass and the electric charge of the molecule and Es(r0) is the scattered electric eld at the position of the molecule. Assuming 􀀀 << ! and weak interaction with the scattered eld, the solution to Eq. 2.19 follows 􀀀 􀀀0 = 1 + q0 6 0 jpj2 1 k3Es(r0) (2.20) where q0 is the intrinsic quantum e ciency of the emitter and k is the wavevector. The problem of calculating the modi cation to the molecule's spontaneous decay rate is now reduced to nding the electric eld re ected from the environment. This is not an easy task and can only be done analytically for very speci c cases [42, 44], that will be discussed in the following section. 2.3.3 Energy dissipation in lossy material The energy of an emitter in the excited state can be dissipated without the emission of a photon, or in other words, through nonradiative processes. Some processes are intrinsic to the emitter, for example, energy loss due to vibrations or Auger recombination in nanocrystal quantum dots [45]. Other processes, such as 29 energy transfer to the environment or to other molecules, are also nonradiative and strongly depend on the surrounding environment. A lossy material placed close to a uorescence molecule may open up additional channels for photo-excited uorophores to relax back to the electronic ground state nonradiatively, thereby quenching the uorescence. In this process, it is thought that energy is transferred from uorophore to environment via exchange of a virtual photon, which in turn causes electron movement in/on the lossy material. The electrons then dissipate the energy rapidly in the form of resistive heating within the lossy material. In near- eld microscopy, as a near- eld probe approaches a uorescent sample, the local nonradiative relaxation rate may increase and as a result lead to suppression of the detected emission [44, 46]. This uorescence quenching may be accompanied by a change in the radiative rate, where both the radiative and nonradiative rates depend on the orientation of the molecule transition dipole moment relative to the probe geometry [44, 47]. If the losses are due to Ohmic losses, the nonradiative decay rate can be determined by [3, 48] 􀀀0nr(r) = 1 ~! 1 2 Z V Re fj (r) E(r)g dr3 (2.21) where V denotes the volume of the lossy material, j(r) is the current density and E(r) is the electric eld emitted by the molecule. The current density can be expressed in terms of the electric eld and the imaginary part of the dielectric function j(r) = 0! 00E(r) (2.22) Using equations 2.21 and 2.22, it is apparent that the nonradiative decay rate will be much higher for materials with a high imaginary part of the dielectric function. For this reason, metals usually induce a high nonradiative rate, while dielectrics do not. Furthermore, the nonradiative decay rate will be much higher when the lossy material is close to the molecule where the emitted electric eld is stronger. Since the dielectric function of a material is wavelength dependent, the uorescence quenching e ciency depends on the emission wavelength of the uorophore. The maximum nonradiative decay rate for metals is at the plasmon resonance frequency where 00 is the highest. 30 If the geometry and the dielectric function of the lossy material are known, it is possible to calculate the nonradiative decay rate analytically [42, 44] or by numerical simulations [3, 48]; this will be discussed further in the next section. 2.3.4 Energy transfer between molecules F orster resonance energy transfer (FRET) describes the energy exchange between two molecules and is also a form of nonradiative decay. Within this model, a u- orescence molecule in the excited state (donor) may relax back to its ground state by transferring energy to a nearby molecule (acceptor). In order for the molecules to transfer energy, the emission spectrum from the donor must overlap with the absorption spectrum of the acceptor. This is depicted in Fig. 2.9a, where the emission from a green uorescent protein (GFP) overlaps with the obsorption of a yellow uorescent protein (YFP). The overlap opens up an energy transfer channel from the donor to the emitter with an energy transfer rate of 􀀀et. Typically, excitation of a uorescence molecule is followed by thermal vibrations, which cause a small loss in energy. Once some of the acceptor's energy is lost, there is no longer an overlap in energy between the two molecules and the acceptor can not transfer energy back to the donor. If the rate of vibrational energy dissipation in the acceptor (􀀀vib) is much faster than the energy transfer rate from the donor to the acceptor, i.e., 􀀀vib >> 􀀀et, the coupling between the molecules is called weak coupling. Figure 2.9b illustrates the FRET cycle. First, the donor molecule is excited by an external source and relaxes via vibrations to a lower band in the excited state. The excited molecule can go back to the ground state either via emission of a photon, internal nonradiative relaxation, or by transferring energy to a nearby acceptor molecule. The FRET process usually ends with the emission of a photon from the acceptor or via nonradiative relaxation of the acceptor to its ground state. F orster calculated the energy transfer rate from the donor to the acceptor in a weak coupling regime based on classical considerations. However, similar equations are derived from a quantum mechanics stand point [49]. The energy transfer rate found by F orster is: 31 D S0 S1 A S0 S1 Γet Γ (a) (b) vib Figure 2.9. F orster resonance energy transfer. Panel (a) is the emission of GFP (blue) and absorption of YFP (yellow). Panel (b) shows the cycle of the FRET process. The donor molecule is excited from the ground state to an excited state. If an acceptor molecule is nearby the donor, an addional energy channel between the two is opened. 􀀀et 􀀀0 = R0 r 6 (2.23) where r is the distance between the molecules and R0 is the F orster radius. The F orster radius can be calculated by [50] R6 0 = q0 9000 ln(10) 2 128 5Nn4 J( ) (2.24) where q0 is the intrinsic quantum yield of the donor, N is Avogadro's number, n is the refractive index of the medium and J( ) is the spectral overlap integral which is de ned as follows: J( ) = Z 1 0 fD( ) A( ) 4d (2.25) where fD( ) is the emission spectrum of the donor with total intensity normalized to unity and A( ) is the absorption spectrum of the acceptor. The factor depends on the dipole's orientation and is denoted by 2 = [nA nD 􀀀 3 (nR nD) (nR nA)]2 (2.26) 32 where nD and nA are the transition dipole orientation of the donor and acceptor respectively, and nR is the direction of the vector between the acceptor and donor. The energy transfer e ciency between the molecules can be calculated by E = 􀀀et 􀀀et + 􀀀0 = 1 1 + (r=R0)6 (2.27) Therefore, the F orster radius indicates the distance between the molecules where the energy transfer e ciency drops to 1/2 of the peak value. The stronger the molecules are coupled together, the longer that distance. For FRET between single molecules, the F orster radius is typically between 3􀀀6 nm depending on the speci c combination of donor and acceptor molecules. 2.3.5 Quantum yield The quantum yield of an emitter changes in an inhomogeneous environment due to changes in the radiative and nonradiative rates. These changes lead to a quantum yield which is di erent than the intrinsic one. One must take into account that the modi cation in the rates depends on the exact position of an object, which introduces the inhomogeneity, relative to the emitter. The intrinsic quantum yield can be described by q0 = 􀀀r 􀀀0 = 􀀀r 􀀀r + 􀀀nr = 􀀀r 0 (2.28) where 􀀀r is the intrinsic radiative rate of the molecule, 􀀀nr in the intrinsic nonradiative rate, 􀀀0 is the total intrinsic decay rate and 0 is the uorescenece lifetime. The quan- tum yield due to an interaction with a nearby object can be described by introducing additional nonradiative (􀀀0nr) and radiative decay (􀀀0r) rates to the intrinsic rates [50]. 􀀀r ! 􀀀r + 􀀀0r(z) (2.29) 􀀀nr ! 􀀀nr + 􀀀0nr(z) (2.30) If the above substitutions into Eq. 2.28 are made, the apparent quantum yield takes the following form q(r) = 􀀀r + 􀀀0r(r) 􀀀r + 􀀀nr + 􀀀0r(r) + 􀀀0nr(r) = 􀀀r + 􀀀0r(r) 􀀀0 + 􀀀0(r) = [􀀀r + 􀀀0r(r)] (r) (2.31) where we de ne (r) [􀀀0 + 􀀀0]􀀀1 as the uorescence lifetime in the presence of a proximate object and 􀀀0 􀀀0r + 􀀀0nr as the modi ed relaxation rate. 33 Chance, Prock and Sibley calculated the decay rate 􀀀 of a single molecule in the proximity of a re ecting mirror [42]. The electric eld of a single molecule is found using Eq. 2.12 and the total electric eld in space can be calculated by taking into account re ections from the mirror. The presence of the mirror modi es both the radiative and nonradiative decay rates, which can be found separately. Figure 2.10 summarizes the results for the decay rates and the quantum yield for a dipole perpendicular to the interface of the mirror, as a function of the distance from the mirror. In this case, the mirror is made out of gold, the emission wavelength is = 605 nm and the intrinsic quantum yield of the molecule is unity. The radiative rate changes most dramatically on a wavelength length scale. The nonradiative rate monotonically increases as the mirror gets close to the molecule. At short distances, 􀀀0nr / z􀀀3 as can be calculated from Eq. 2.21 for this geometry. At these distances, the nonradiative rate dominates, as is apparent from the reduction of the quantum yield. The same calculation can be performed for a dipole oriented parallel to the mirror interface and yields very di erent results from the one shown here. For a dipole placed near a small metal sphere, Carminati et al. [44] calculated the modi cation to the decay rates and eld enhancement assuming only dipole- dipole interactions. Anger et al. [3] calculated the same thing for a single emitter oriented in the z-axis by means of a numerical simulation. The results of the di erent approaches for the quantum yield and the eld enhancement are shown in Fig. 2.11a. While the dipole approximation and the numerical simulation yield similar results for the eld enhancement from a gold sphere, the dipole approximation underestimates the quantum yield by a cosiderable amount. The total emission rate for spheres of di erent sizes is shown in Fig. 2.11b. There is an obvious competition between eld enhancement and quantum yield (which leads to emission reduction in this case). Issa et al. [48] modeled the changes in the rates of an emitter close to an object geometry that resembles an AFM tip. Interestingly, their work suggests that even in the absence of damping (i.e., 00(!) = 0), a uorophore can couple to a metal tip via intermediate range (10􀀀50nm) excitation of surface plasmon traveling waves, which for an open-geometry tip (e.g., metal coated cone or pyramid) results in a reduction 34 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 z/λ Γr / Γ0 0 0.2 0.4 0.6 0.8 1 q(z) 10-3 10-2 10-1 100 100 102 104 106 108 Γnr / Γ0 10 -2 Figure 2.10. Dipole emitter near a re ecting mirror. The decay rates for a perpendicular dipole as a function of distance from a gold mirror. The blue line in the main gure (inset) indicates the changes in the radiative (nonradiative) rate of the emitter. These changes lead to a variation in the quantum yield (green line). The intrinsic quantum yield is assumed to be unity and the emission wavelength is 605 nm. of the quantum yield and the uorescence signal. By using pulsed lasers and time-correlated single photon counting (TCSPC), it is possible to experimentally measure the uorescence lifetime of a single emitter. The lifetime measurements can even be extended to produce a tip-sample distance dependent lifetime data [51, 52]. However, since the radiative decay rate cannot be measured directly, it can be quite challenging to measure the quantum yield or even the intrinsic quantum yield of single uorophores. 2.4 Collection e ciency and angular emission The collection e ciency of the system is determined by a few factors: the amount of light collected by the microscope objective relative to the total light emitted, the optical signal losses due to optical components, and the detection e ciency of the photon detector. The last two are constant for a speci c emission wavelength. 35 Figure 2.11. Single molecule emission next to a gold sphere. Panel (a) plots the increase in excitation rate due to enhancement (red) and the decrease in quantum yield (blue). The solid lines where calculated using numerical simulation while the dotted lines are the analytic results usint the dipole approximation taken from Ref. [44]. Panel (b) shows the total emission rate for di erent sphere sizes. The emission rate is determined by the product of the excitation rate and the quantum yield. The dipole orientation of the molecule is in the z-axis and its intrinsic quantum yield is unity. Reprinted gure with permission from [3]. Copyright (2006) by the American Physical Society. However, the percentage of light collected by the objective depends on the angular emission of the emitter which in turn depends on the orientation of the emission dipole and the environment. It is useful to de ne the angular directivity of the emitted light which re ects on the directionality of the emitted power: D('; ) = R4 P('; ) P('; )d (2.32) where P('; ) is the radiated power and ' and are the same angles de ned in Fig. 2.7. For isotropic emission D('; ) = 1, while for a radiating dipole, the directivity is similar to the far- eld emission pattern from Fig. 2.8 with a maximum directivity of Dmax = 1:5. 36 The environment of the emitter plays a crucial role in the angular directivity of an emitter. When an emitter is placed close to an interface between two mediums with a di erent dielectric function, most of the emission will be radiated into the medium with the higher dielectric function. In a typical experiment a dipole is located right above a glass-air interface. In these cases, about 85% of the light is radiated into the glass; this can be determined analytically [42]. The introduction of the glass also causes a change in the angular emission pattern as depicted in Fig. 2.12. Futhermore, if a near- eld probe is nearby the molecule, the radiation pattern can change even further, thus modifying the amount of light radiated into the glass medium. In an experiment, the only detected light is that which radiates into the microscope objective. The collection e ciency of the objective can be obtained by 90 270 180 0 Figure 2.12. Electric dipole near a glass-air interface. A dipole placed 1 nm above a glass interface with perpendicular orientation (with respect to the interface) depicted in green and parallel orientation in blue. The radiation pattern depends on the orientation of the molecule, and in both cases, the majority of the (propagating) radiation is emitted into the glass medium (lower half). 37 coll = 1 4 Z2 0 Z NA 0 D('; ) sin( )d d' (2.33) where NA is the maximum collection angle of the objective. The higher the numer- ical aperture of the microscope objective (which means larger NA), the better the collection e ciency will be. Signi cant changes in the angular emission of a single molecule may occur due to strong near- eld coupling with an optical antenna [21, 24, 54, 55], which can lead to di erent collection e ciencies. Taminiau et al. [53] calculated the collection e ciency of di erent microscope objectives when a molecule is placed close to an optical antenna. Their results are shown in Fig. 2.13 for horizontally and vertically oriented dipoles and for two di erent microscope objectives. The dipole was placed 10 nm below an aluminum antenna and the collection e ciency was calculated as the antenna x [nm] x [nm] ηColl ηColl Figure 2.13. Microscope objective collection e ciency of single molecule emission next to an optical antenna. The collection e ciency for the low NA is very low and also depends on the orientation of the molecule. For the high NA, the relative changes in the collection e ciency as a function of lateral distance from the antenna are reduced. Furthermore, the collection e ciency for di erent dipole orientations is much closer. Reprinted gure with permission from [53]. 38 moves in the lateral direction. While the detection e ciency of the high NA objective exhibits relatively small changes between the di erent dipole orientations as a function of distance from the antenna, the low NA objective exhibits quite large changes. The total collection e ciency of the system is: C(r) = coll(r) (2.34) where the collection e ciency of the objective now depends on the distance of the sample from the near- eld probe and is the collection e ciency of the detection path, including losses in the optical components and the quantum e ciency of the photon detector. 2.5 The measured signal In this section, the experimentally measured signal is derived using the parameters de ned in the previous sections. For light focused by a microscope objective, assuming one uorophore in the focal area and excitation intensities far from saturation, the detected far- eld emission count rate can be denoted by Sff = C0􀀀0 excq0 = C0I0(r0) 0 hc q0 (2.35) where C0 is the collection e ciency of the system, 0 is the absorption cross-section of the molecule and hc is the energy of a detected photon. For a molecule in the proximity of an object, the emission count rate can be quite di erent from Eq. 2.35. The intensity I0 at the molecule position can either increase or decrease depending on the illumination conditions, the distance of the molecule from the near- eld object, and the shape and geometry of the object. The radiative rate 􀀀r can alter due to a change in the local density of states (i.e., the Purcell e ect). At short distances, the object may introduce additional nonradiative channels, thus, increasing the nonradiative relaxation rate. Finally, the collection e ciency of the system can vary due to the presence of the object. One can denote the signal in the near- eld as Snf = C(r)􀀀exc(r)q(r) = C(r)I(r; r0) 0 hc q(r) (2.36) 39 It is often useful to normalize the near- eld signal (Eq. 2.36) to the far- eld signal (Eq. 2.35). In such a case, the normalized signal can be denoted by Snorm = (r) q(r) q0 C(r) C0 (2.37) If the collection e ciency is similar in the far- eld and near- eld (C(r) C0), which is true for high numerical aperture objectives, the normalized signal takes an even simpler form Snorm = (r) q(r) q0 (2.38) Since the uorescence lifetime is a measurable quantity, it is useful to rewrite Eq. 2.38 in terms of the tip-induced uorescence lifetime and the intrinsic uorescence lifetime. Using equations 2.28 and 2.31 the normalized uorescence signal can be denoted by: Snorm = (r) (1 + 􀀀0r=􀀀r) 0 (2.39) The goal of this chapter was to examine equations 2.36, 2.38 and 2.39 and to under- stand how these equations determine the experimentally measured signal. Therefore, it is also essential to comprehend the mechanisms which impact and alter these equations. 2.6 References [1] L. Novotny and B. Hecht, Principles of Nano-Optics, Cambridge University Press, Cambridge, 2006. [2] B. D. Mangum, E. Shafran, J. Johnston, and J. M. Gerton, Near- eld scanning optical microscopy, in Optical Techniques for Solid-State Materials Characteri- zation, edited by R. P. Prasankumar and A. J. Taylor, CRC Press, 2011. [3] P. Anger, P. Bharadwaj, and L. Novotny, Physical Review Letters 96 (2006). [4] P. Drude, Annalen der Physik 306, 566 (1900). [5] A. Sommerfeld and H. Bethe, Elektronentheorie der Metalle, Springer, Verlag, 1967. [6] J. D. Jackson, Classical Electrodynamics, John Wiley and Sons, Hoboken, third edition, 1999. 40 [7] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propa- gation, Interference and Di raction of Light, Cambridge University Press, New York, seventh edition, 2002. [8] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders College, Philadelphia, 1976. [9] S. Link and M. A. El-Sayed, Journal of Physical Chemistry B 103, 4212 (1999). [10] C. Kittel, Introduction to Solid State Physics, John Wiley & Sons, Hoboken, N.J., eighth edition, 2005. [11] L. G. Schulz, Journal of the Optical Society of America 44, 357 (1954). [12] N. K. Grady, N. J. Halas, and P. Nordlander, Chemical Physics Letters 399, 167 (2004). [13] S. Adachi, Physical Review B 38, 12966 (1988). [14] J. L. Bohn, D. J. Nesbitt, and A. Gallagher, Journal of the Optical Society of America A 18, 2998 (2001). [15] K. Kneipp et al., Physical Review Letters 78, 1667 (1997). [16] K. A. Willets and R. P. Van Duyne, Annual Review of Physical Chemistry 58, 267 (2007). [17] S. Khn, U. Hkanson, L. Rogobete, and V. Sandoghdar, Physical Review Letters 97, 017402 (2006). [18] P. Bharadwaj and L. Novotny, Journal of Physical Chemistry C 114, 7444 (2010). [19] H. Eghlidi, K. G. Lee, X. W. Chen, S. Gotzinger, and V. Sandoghdar, Nano Letters 9, 4007 (2009). [20] N. L. Rosi and C. A. Mirkin, Chemical Reviews 105, 1547 (2005). [21] A. G. Curto et al., Science 329, 930 (2010). [22] J. N. Farahani, D. W. Pohl, H. J. Eisler, and B. Hecht, Physical Review Letters 95, 4 (2005). [23] T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, Nano Letters 7, 28 (2007). [24] T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, Nature Photonics 2, 234 (2008). [25] L. Novotny, Physical Review Letters 98, 266802 (2007). 41 [26] P. Bharadwaj, B. Deutsch, and L. Novotny, Advances in Optics and Photonics 1, 438 (2009). [27] L. Novotny and N. van Hulst, Nature Photonics 5, 83 (2011). [28] Z. Zhang et al., Nano Letters 9, 4505 (2009). [29] A. McLeod et al., Physical Review Letters 106 (2011). [30] F. H'Dhili, R. Bachelot, A. Rumyantseva, G. Lerondel, and P. Royer, Journal of Microscopy 209, 214 (2003). [31] H. Hamann, M. Kuno, A. Gallagher, and D. J. Nesbitt, Journal of Chemical Physics 114, 8596 (2001). [32] S. Kuhn and V. Sandoghdar, Applied Physics B 84, 211 (2006). [33] L. Novotny, R. X. Bian, and X. S. Xie, Physical Review Letters 79, 645 (1997). [34] L. Novotny and S. J. Stranick, Annual Review of Physical Chemistry 57, 303 (2006). [35] J. T. Krug, E. J. Sanchez, and X. S. Xie, Journal of Chemical Physics 116, 10895 (2002). [36] P. Bharadwaj and L. Novotny, Optics Express 15, 14266 (2007). [37] B. Mangum, E. Shafran, C. Mu, and J. Gerton, Nano Letters 9, 3440 (2009). [38] F. Huang, F. Festy, and D. Richards, Applied Physical Letters 87, 183101 (2005). [39] A. Kramer, W. Trabesinger, B. Hecht, and U. Wild, Applied Physical Letters 80, 1652 (2002). [40] W. Trabesinger, A. Kramer, M. Kreiter, B. Hecht, and U. P. Wild, Applied Physics Letters 81, 2118 (2002). [41] E. M. Purcell, Physical Review 69, 681 (1946). [42] R. R. Chance, A. Prock, and S. R., Advances in Chemical Physics 37, 1 (1978). [43] H. Kuhn, Journal of Chemical Physics 53, 101 (1970). [44] R. Carminati, J. J. Gre et, C. Henkel, and J. M. Vigoureux, Optics Communi- cations 261, 368 (2006). [45] M. Nirmal et al., Nature 383, 802 (1996). [46] L. Novotny, Applied Physics Letters 69, 3806 (1996). [47] S. Vukovic, S. Corni, and B. Mennucci, The Journal of Physical Chemistry C 113, 121 (2008). 42 [48] N. A. Issa and R. Guckenberger, Optics Express 15, 12131 (2007). [49] R. M. Clegg, Fluorescence resonance energy transfer, in Fluorescence Imaging Spectroscopy and Microscopy, edited by X. F. Wang and B. Herman, John Wiley & Sons, New York, 1996. [50] J. R. Lakowicz, Principles of Fluorescence Spectroscopy, Springer, New York, third edition, 2006. [51] E. Yoskovitz, D. Oron, I. Shweky, and U. Banin, The Journal of Physical Chemistry C 112, 16306 (2008). [52] E. Yoskovitz, G. Menagen, A. Sitt, E. Lachman, and U. Banin, Nano Letters 10, 3068 (2010). [53] T. H. Taminiau, F. D. Stefani, and N. F. Van Hulst, New Journal of Physics 10 (2008). [54] H. Gersen et al., Physical Review Letters 85, 5312 (2000). [55] R. J. Moerland, T. H. Taminiau, L. Novotny, N. F. Van Hulst, and L. Kuipers, Nano Letters 8, 606 (2008). CHAPTER 3 EXPERIMENTAL SETUP The use of our near- eld microscope requires expertise in several di erent areas and involves many experimental details. First, the user must be an expert in the oper- ation of the Atomic Force Microscope (AFM). Second, there are various ways in which the optical setup can can be con gured, each of which can alter the physics of the near- eld interaction. Finally, the optical data are acquired by home-written software designed for maximum capability and exibility in both hardware and software and there are a number of di erent data analysis techniques which have evolved over the years. Thus, the user must understand the operation of the data acquisition software and hardware, and recognize the advantages and limitations of the postanalysis algorithms. In this chapter, the basic layout of the near- eld microscope is described and the di erent illumination schemes are summarized. The basic physics of an AFM microscope is also examined by modeling the AFM as a simple harmonic oscillator. The operation of the AFM probe, which is relevant to the data acquisition and analysis, is reviewed. The last section of the chapter focuses on the basic techniques used to produce a near- eld optical signal and the advantages of using single-photon sensitive detection. The subsequent chapters review the experimental results using the single photon sensitive detection scheme; most of these results would not be possible without our unique data acquisition scheme. Since much of my e ort has been focused on setting up the data acquisition and postprocessing software, I will touch brie y on the capabilities of the technique in this chapter. The main intent of this chapter is to introduce the reader to some of the technical details and considerations along with some basic theoretical concepts important to 44 the experimental setup. For a person familiar with Near- eld Scanning Optical Microscopy (NSOM), this chapter should provide a review of important experimental considerations. For a person who is new to NSOM, this chapter along with Ref. [1] and the AFM manual should provide a good technical background. 3.1 Basic setup Our near- eld scanning optical microscope is comprised of two essential parts: an Atomic Force Microscope (AFM) and an inverted confocal microscope. Figure 3.1 depicts the basic layout of the setup. A laser is coupled to an optical ber via a ber coupler. The output of the coupler is magni ed to the desired size by a simple two lens beam expander. The expanded beam goes through a half wave plate which enables control of the polarization. Typically one of following three focused illumination types is used: evanescent, radial, or Gaussian. Each has its own advantages and disadvantages as will be discussed in Sec. 3.2. For radial and evanescent illumination, the beam is sent through a beam mask. The excitation beam is then re ected by a dichroic mirror onto a scanning mirror. Excitation light is then focused by a high numerical objective (NA = 1.4) onto the sample surface. Emitted light is collected by the same objective and directed onto an avalanche photodiode (APD) via the scanning mirror, a steering mirror and a focusing lens. The 1:1 telescope and scanning mirror de-scan scheme to ensure alignment with the APD during movement of the scanning mirror. To ensure that only the emission light is detected, an emission lter is positioned right before the APD and the focusing lens to reject scattered light wavelengths. The AFM sits on top of the inverted confocal setup. For regular imaging the sam- ple is scanned laterally via a piezo actuator while the AFM head and the microscope objective are stationary. In order to achieve a near- eld signal, the AFM tip must be aligned within the laser's focal spot. The scanning mirror plays an important role since it allows movement of the excitation beam independently from the AFM and is therefore used for tip-laser alignment. 45 Figure 3.1. Schematic of experiment. A laser beam is directed through a beam mask (BM), producing either a radially polarized laser beam or a 60 section (wedge) of an annular beam. A microscope objective (OBJ: NA = 1.4) focuses the laser beam and collects emitted uorescence, which is focused onto an avalanche photodiode (APD). The laser focus is positioned onto an AFM tip using a scanning mirror (SM). The sample is raster-scanned laterally, where by convention the X-axis corresponds to the fast-scan direction. The inset shows the tip-sample interaction region in more detail. Other important components include an optical ber (OF), a dichroic mirror (DM), and a spectral lter (SF). Reprinted with permission from [2]. Copyright 2009 American Chemical Society. 3.2 Types of illumination The excitation light plays a crucial role in near- eld microscopy. Most important is the ability to control the polarization of the light at the laser focus; light polarized along the axis of the AFM tip typically results in strong eld enhancement at the tip apex while light polarized perpendicular to the tip axis does not (and might even result in a reduction in the eld intensity). Other parameters, such as the beam waist and the point spread function (PSF) at and above the sample, can also alter the resulting near- eld optical signal. In our experiments we typically use one of '" " \ \ '. \ AFM 46 three types of illumination conditions: Gaussian excitation, radially polarized light and total internal re ection (TIR) that results in an evanescent illumination. 3.2.1 Gaussian illumination The output of a laser beam is typically speci ed by a TEMmn mode, where TEM stands for Transverse Electric and Magnetic. The lowest mode, TEM00, results in a Gaussian beam pro le. Moreover, the output light is typically strongly polarized along a particular axis (i.e., linearly polarized). This illumination mode is the most commonly used type of excitation in microscopy and most commercial lasers operate in this mode. Therefore, to achieve this type of illumination condition is rather straightforward. In our setup, a TEM00 laser light is coupled to a polarization maintaining single mode optical ber which preserves the polarization of the Gaussian light. After the light is focused by the objective, this illumination scheme results in horizontally polarized light at the center of the focus (Fig. 3.2a). Adjacent to the center of the focus, the light polarization is no longer perfectly horizontal. Following Ref. [3], for a linearly polarized Gaussian beam focused by a high numerical aperture lens, the di erent electric eld components at the focal spot can be found. Figure 3.3 shows the (a) (b) (c) Figure 3.2. Illustration of the di erent illumination conditions. (a) Gaussian illumination results in horizontal polarization at the focus. (b) Radially polarized light yields vertical polarization at the focus. (c) By letting in only critical rays from one side, TIR yields an evanescent wave, which can also result in vertical polarization. Notice that in (c), the incoming light is illustrated on the left side and the re ected light is on the right. 47 results of the intensities for the di erent eld components (i.e., jExj2, jEyj2, jEzj2). For this calculation, it is assumed that the beam waist perfectly lls the back aperture of an objective with NA = 1:4, n = 1:518 (immersion oil/glass) and the incident light is polarized along the x axis. Notice that the eld component that corresponds to the initial polarization of the light is much stronger than the other components (256 and 8 times greater than the y and z components, respectively). Therefore, for a sample which does not have a particular dipole orientation (e.g., a collection of randomly oriented molecules), the far- eld emission will look similar to jExj2 which has a slightly elongated Gaussian pro le in the direction of the polarization [3{5]. The measured beam pro le for 20nm uorescence beads can be seen in Fig. 3.4a. The characteristic decay length of the light intensity above the sample (i.e., Rayleigh length) is typically a few m. Although this is the easiest illumination condition to achieve, it is rarely used be- cause the horizontal polarization at the focus does not result in any eld enhancement at the apex of the tip. 3.2.2 Radial illumination One way to achieve vertical polarization at the focus is to use a radially polarized light. Radially polarized light is a superposition of TEM10 laser mode with the same 200nm |Ex |² |E y |² |E z |² Figure 3.3. Gaussian beam at the focus. The intensity of the di erent electric eld components is plotted. The intensity scale for the ^y and ^z components were magni ed by 256 and 8, respectively. The incident polarization before focusing is along ^x. 48 mode rotated by 90 [5]. When a radial excitation pro le is focused by the objective, the horizontal components of the electric eld will cancel each other out to some degree, thus creating light with mostly vertical polarization (Fig. 3.2b). To the side of the focus, the strength of the horizontal components increases and the strength of the vertically polarized light components is reduced. The actual eld components for radially polarized light can also be computed using Ref. [3] (Appendix D). The vertical-polarization part of beam pro le at the focus for this mode is well described by a Gaussian (Fig. 3.4b). In fact, the focal spot is tighter for radial excitation by nearly 40% in comparison to the Gaussian excitation [5, 6]. In practice, the radially polarized light is created through a commercial device (Arcoptix) using twisted nematic crystals. The output signal from the crystal can be spatially ltered by passing the beam through a pinhole which results in a high quality radially polarized light. 3.2.3 TIR illumination When a light beam is incident on an interface between two di erent mediums, the re ected and transmitted light behaves according to Snell's law. Above some critical incident angle, the light is totally internally re ected at the incident medium and strongly decays above the interface. It is therefore important to consider the size of a beam waist before it enters the objective. The combination of a high numerical aperture objective (NA = 1:4) with a glass/air interface at the sample results in some TIR light. Using Snell's law: n1 sin 1 = n2 sin 2 (3.1) At the critical angle the equation reduces to sin c = n2 n1 (3.2) The diameter of the back aperture of the microscope objective is given by [8] D = 2NAfTL Mag (3.3) 49 0 Photon Count (a.u.) Position (nm) 30 60 90 120 0 100 200 300 400 500 (c) (a) 200 nm 0 100 200 300 400 500 0 30 60 90 120 150 Photon Count (a.u.) Position (nm) (d) (b) 200 nm Figure 3.4. Beam pro les at the focus. Far- eld scanned images of uorescence beads. (a) The Gaussian beam focuses to a Gaussian pro le. The light focuses to a larger spot at the polarization axis. (b) Radial polarization results in a symmetric focal spot. The beam focus is slightly tighter than the Gaussian beam. Reprinted with permission from [7] (© 2008 IEEE). where fTL is the focal length of the tube lens and Mag is the magni cation of the microscope's objective. The diameter at the critical angle can be obtained using equations 3.2, 3.3 and the relation NA = n1 sin to give: Dc = 2n2fTL Mag (3.4) For a Nikon objective with Mag = 100, NA=1.4, n1 = 1:518, n2 = 1 and fTL = 200 mm, the back aperture and the critical diameter are 5:6 mm and 4 mm, respectively. Thus, the parts of a beam with a diameter larger than Dc will be totally internally re ected. The critical angle can be calculated via Eq. 3.2 to be c = 41:8 . The incident light can be described by a plane wave: ~E = ~E 1ei( ~ k1 ~r􀀀!t). The 50 electric eld can be separated to s-polarized (parallel to the interface) and p-polarized (perpendicular to the interface) components ~E 1 = ~E (s) 1 + ~E (p) 1 (3.5) When light hits an interface at an angle larger than the critical angle, the z component of the wavevector ~k becomes imaginary. Figure 3.5a illustrates a case of a plane wave impinging on a at interface with an incidence above the critical angle ( 1 > c). The transmitted electric eld in this case can be expressed as [3]: ~E 2 = 2 64 􀀀iE(p) 1 tp( 1) (k2z0)􀀀1 E(s) 1 tp( 1) E(p) 1 tp( 1) (k2z0)􀀀1 q 1 + (k2z0)2 3 75 ei sin 1k1xe􀀀z=z0 (3.6) where tp is the Fresnel coe cient for p-polarized light, k1 is the wavevector in the incident medium, ~n = p 1 1 p 2 2 is the relative index of refraction with 1 and 2 being the permittivity of the incident and transmitted medium, respectively. The magnetic permeability is assumed to be unity for both mediums. The decay length z0 is de ned by z0( 1) 1 k2 p ~n2 sin2 1 􀀀 1 (3.7) where k2 is the wavevector in the transmitted medium. Two important observations can be made from this derivation. First, an expo- nentially decaying (evanescent) electric eld is established above the interface, which results in a tightly con ned excitation. For the case of a glass/air interface ( 1 = 2:25, 2 = 1) and an incidence angle of 1 = 45 the decay length is: z0 = 2:22 . For Gaussian and radial illumination, the intensity of the excitation light decays on a typical length scale of about a few m above the focus (i.e., Rayleigh range). In comparison, the intensity of the laser in the case of evanescent illumination decays to 1=e after a few hunderd nm (assuming visible illumination wavelength). Second, the light polarization above the interface is easy to manipulate via the incident light polarization. If the incident light is s-polarized (i.e., E(p) 1 = 0), then the x and z components of the transmitted electric eld disappear (Eq. 3.6), thus, creating s-polarized light above the interface. If the incident light is p-polarized, 51 E(p) E(s) kx k1 E(s) E(p) Ez Ey Ex E(s) E(p) (a) (b) θ1 Figure 3.5. Total internal re ection. (a) An incident light above the critical angle results in total internal re ection. Above the interface the light decays exponentially along the z axis. This is called an evanescent eld. (b) An illustration of the beam mask which is being used to create the TIR illumination. The polarization of the incidence eld determines the polarization of the light at the focus. the light above the interface will have both s-polarized and p-polarized components with the p-polarized component always being stronger. Notice that the decay length, the amplitude, and the ratio between the di erent amplitude components of the transmitted eld depend on the incident angle 1. For a case of NA=1.4, a glass/air interface and a beam waist similar or larger in size than the back aperture, there will be a distribution of incidence angles between 41:8 < 1 < 69:0 . To create an evanescent illumination we use a Gaussian beam and an opaque (metal) screen with a wedge-shaped window. This mask blocks all the light except the supercritical rays, namely those light rays that enter the back aperture of the microscope objective at a diameter smaller than Dc. The wedge-shaped mask is illustrated in Fig. 3.5b. The green arrows in Fig. 3.5b represent the polarization of the incident light. The 60 opening ensures that the oppositely directed vertical vector components originating from opposing sides of the beam do not cancel each other at the focus. An example of such a case is illustrated in Fig. 3.2a where the vertical component of the light coming from the left and the right cancel the vertical component at the focus. The wedge shape also suppresses the horizontal components 52 passing through the wedge. A bigger opening angle results in more s-polarized light. 3.2.4 How to choose the illumination condition The ideal choice of an illumination condition depends on the particular experi- ment. Gaussian illumination can be used if vertical polarization is not a necessity (for example see Chap. 6). The main advantage of the Gaussian illumination is that it is easy to set up. For any other experiment, either radial polarization or TIR illumination should be used. The TIR scheme enables fast switching between horizontal to vertical polarization and vice versa. On top of that, the beam size in the case of TIR is elongated along one dimension, which makes it is easier to align the tip into the laser spot. The evanescent wave resulting from the TIR con guration, decays quickly above the sample surface. Therefore, scattering of excitation from the tip will be reduced in comparison to Gaussian and radial polarization. The reduced scattering also results in interference e ects (Sec. 2.2.4) which are con ned to short tip-sample vertical distance. This results in easier determination of where the far- eld region starts (i.e., where the tip has no e ect on the signal). On the other hand, when measuring high density samples, a larger laser spot will result in increased background and therefore reduced contrast [9]. In such a case, it is advantageous to use radial polarization due to the reduced laser spot at the focus. Radial polarization should also be used when imaging thick samples. The fast decay of the TIR illumination above the sample will result in excitation of only the lower part of the sample. Table 3.1 summarizes the conditions in which each illumination con guration should be used. Table 3.1. How to choose the illumination condition. Illumination condition: When to use: Gaussian No need for vertical polarization. Radial High density sample and/or thick sample. TIR All other. 53 3.3 Atomic Force Microscope (AFM) In NSOM, the ability to scan a surface accurately with minimal damage to the sample is highly desired, while the main physical quantity of interest is the optical scattering signal. In order to prevent near- eld optical artifacts or misinterpreted data, it is important to understand the advantages and limitations of di erent scan- ning techniques. The AFM used in our lab is a commercial microscope (Asylum Research), which comes with its own software (Igor Pro). The AFM consists of a few important parts: a very precise piezo actuated scanning stage, AFM head, AFM control box and a cantilever with a sharp probe at its end. The scanning stage is used to scan the sample laterally (i.e., X 􀀀 Y ). The cantilever is attached to the AFM head which controls the vertical motion of the tip. The AFM head itself consists of a few mechanical and optical components which keep the tip moving in the desired motion while scanning the surface. All the signals end up in the AFM control box where the signals are ltered and recorded on the computer. Access to real time signals is available through the control box. The most important signal regarding the optical measurements is the de ection signal which provides information on the vertical motion of the tip (see Appendix E). Typically the scanning techniques used in AFM can be separated into an inter- mittent mode (tapping) and a contact mode, although in our lab we exclusively use tapping mode. In this mode, the tip oscillates vertically and only intermittently touches the sample. Tapping mode holds a few main advantages over contact mode: it reduces damage to the sample because the contact is only intermittent, large amplitude oscillation are sometime desired to achieve higher optical contrast [9] and it allows us to obtain vertical tip-sample distance dependent statistics in an elegant way. During a typical near- eld experiment, the sample can either be raster scanned or the X 􀀀 Y scan can be halted while the tip maintains its vertical Z oscillations above a speci c chosen location in the sample. 54 3.3.1 Simple Harmonic Oscillator (SHO) As mentioned above, the probe will oscillate in a typical distance-control system. The AFM's driving signal is harmonic, and for the most part, so is the actual motion of the tip (especially when imaging in air). It is therefore bene cial to model the oscillations of the tip as a Simple Harmonic Oscillator (SHO) [10]. This simple, approximated model gives valuable insight into the physics involved. Within this model, the motion of the tip can be approximated by the following equation: m z + z + m!0 Q z_ = Fts + Ad cos(!t) (3.8) where is the spring constant of the AFM cantilever, m is the mass of the cantilever, !0 is the resonance frequency, Q is the quality factor of the oscillator, Fts is the net tip-sample force, Ad and ! are the amplitude and the angular frequency of the driving force and z is the distance between the tip and the sample. The tip-sample forces are usually approximated to rst order, and their contribution can be included as an e ective spring constant e = 􀀀 @Fts @z (3.9) The resonance frequency of the probe then becomes !e = q e m . By assuming a solution of the form: z(!) = A(!) cos(!t + (!)), the steady state amplitude and phase can be easily found: A(!) = Ad!2 q (!2 e 􀀀 !2)2 + !2!2 e Q2 (3.10) tan( ) = !e! Q(!2 e 􀀀 !2) (3.11) These simple solutions are fundamental for the understanding of the operation of the AFM tip. Any changes in the gradient in the tip-sample interaction forces @Fts @z lead to a change in the e ective spring constant (Eq. 3.9) therefore shifting the actual resonance frequency of the tip, !e. This in turn leads to variations in both the amplitude and the phase. Therefore, any of the three values, !e ,A(!) , (!) , can be used as a feedback loop. 55 In our setup, the feedback loop detects changes in the oscillation amplitude and reacts by changing the position of the tip above the sample to ensure constant amplitude throughout a scan. Important physical information can be extracted from these three channels, and all three data sets should be recorded during an experiment. 3.3.2 Repulsive vs. attractive imaging Tip-sample forces are also essential for understanding the di erent operating modes. The Leonard-Jones potential qualitatively describes the repulsion between the sample and tip at short distances and their mutual attraction at longer distances via van der Waals forces. In tapping mode, the AFM can be operated in two di erent tip-sample interaction regimes called repulsive and attractive modes. The main di erence between the two is that the cantilever is given enough energy to overcome the attractive tip-sample forces in the repulsive mode but not in the attractive mode. As a consequence, the cantilever will touch the sample in the repulsive mode but typically stay a few nm away from the sample in attractive mode. It is very important to know which force regime is dominant for any particular experiment. Figure 3.6 is an illustration of the amplitude and phase (Eq. 3.10 and Eq. 3.11) signals for realistic tip parameters. In this example, the tip's operating frequency is set to be at !0 (vertical black line). This is the tip's natural resonance in the absence of any tip-sample interactions (i.e., Fts = 0). This is a realistic scenario when the tip is su ciently far from the surface. The amplitude and phase for this case are plotted in black. As the tip is lowered to the surface, if the repulsive forces are dominant, the e ective resonance frequency !e becomes larger, e ectively shifting both the amplitude and the phase to the right (dotted red line). As a consequence, the phase drops below 90 and the cantilever is driven at a frequency ! below the new resonance frequency, !00 . These are indicators of operating in the repulsive regime. In the attractive regime, ! is above the resonance !00 and is larger than 90 as demonstrated by the blue lines. Therefore, we can determine which forces are more dominant during a measurement by inspecting the direction of the phase shift. The user does have some control of the imaging mode. Operating at lower oscillation amplitudes and using sti er tips with bigger radii can increase the success 56 0 10 20 30 Ad 70 72 74 76 78 80 0 50 100 150 Frequency (kHz) Phase (deg) Figure 3.6. AFM amplitude and phase signals. The black curves are the amplitude (a) and phase (b) signals plotted for typical tip values following equations 3.10 and 3.11. The dotted black line depicts the resonance frequency for no tip-sample interactions. Attractive tip-sample interactions result in lower e ective resonance frequency (dotted blue curves) while repulsive interactions have a higher e ective resonance (dotted red curves). of operating in attractive mode. This mode is indeed best for minimizing shear forces on the sample. However, the repulsive mode yields better topography tracking. In addition, in order to demodulate the signal into the far- eld, as discussed in Ref. [9], it is sometimes necessary to operate in repulsive mode. 3.3.3 Making carbon nanotubes tips CNTs are grown on oxidized silicon substrates using methane-based chemical vapor deposition (CVD) and ferric nitrate catalyst nanoparticles (Appendix G). The growth recipe adopted has been shown to produce mostly single-wall CNTs of both semiconducting and metallic chiralities [11, 12]. Following growth, CNT substrates are imaged with the AFM using gold-coated probes, and vertically oriented CNT whiskers can be lifted o the substrate by adhering to the sidewalls of the AFM probe. The mechanistic details of the pickup process are not fully understood, although ,-,, ,, , ,: " ' ..,' , '""' ... _.. . " ... \ , .... ... -;, ... ,,, , ,,, '' ' , I , , ' ... ; ,.. . - ......~.' 57 experimental and theoretical studies suggest that relatively large diameter CNTs (3 - 5 nm) are more likely to attach due to the increased CNT-probe interfacial area [12, 13]. The two main indicators that a CNT has been picked up by the tip are a change in the topography image resolution while scanning (Fig. 3.7a) and a phase change (Fig. 3.7b). Long CNTs have a tendency to buckle under large compression and also exhibit large vibration at the distal end. Thus, a change in the scanning resolution can either be for the better (if a short CNT was picked up) or worse. As mentioned above, the phase signal re ects on the tip-sample interaction forces. Once the CNT has been picked up, the tip-sample interactions are drastically modi ed leading to a substantial shift in the phase. The large shift in the phase is an excellent contrast mechanism and it is usually easier to detect CNT pickup using the phase image. To reduce the chances of multiple CNT pickup, the scan needs to be stopped as soon as a CNT has been picked up. Following pickup, the CNT length is measured by pressing the CNT against a smooth Si substrate while measuring the de ection of the AFM cantilever (force curve). When the distal end of the CNT touches the substrate, the cantilever initially begins to de ect. As more force is applied, the CNT will elastically buckle and the cantilever de ection relaxes somewhat resulting in a kink in the approach curve. 2!m (a) (b) Figure 3.7. AFM scan of a CNT wafer. (a) Height topography. (b) Phase signal. Once a CNT has been picked up, as can be seen at the bottom of the images, both the height and phase images change. Reprinted with permission from [7] (© 2008 IEEE). 58 Depending on its length, a number of additional kinks are possible until nally the apex of the AFM probe comes into contact with the substrate after which a linear de ection of the cantilever is observed as the tip is further pressed into the substrate. The measured distance between the rst kink and the linear onset gives the CNT length. Figure 3.8b is an example of two AFM force curves for a long (red curve) and short (blue curve) CNT. In theory, the length of the CNT can be determined by SEM imaging (Fig. 3.8a). However, this is a time consuming process and can also leave carbon deposits on the CNT. When initially attached, CNTs are generally too long to possess su cient axial sti ness for use in AFM imaging and thus they must be shortened to <200 nm. This is achieved by application of short ( 10 s) voltage pulses of 10 V amplitude between the AFM probe and a conductive substrate. These pulses induce electrochemical etching of the distal end of the CNT, which leads to shortening in quasi-controllable steps of 10-15 nm and removes any fullerene or catalyst cap. 3.4 Producing a near- eld signal To record the measured signal we utilize a single-photon detection technique. Every photon that is detected contains several pieces of information. Most impor- 0 100 200 300 400 -4 -2 0 2 4 6 pyramid contact Tip-Sample Distance (nm) Deflection (nm) (a) nanotube contact (b) 500nm Figure 3.8. SEM image of CNT tip and Force curves to determine the length. (a) An SEM image of a CNT tip. After pickup the CNT are generally too long to be used and need to be shortened. (b) Force curves determine the CNT length before (red) and after shortening (blue). Reprinted with permission from [7] (© 2008 IEEE). 59 tantly, the position of the AFM tip is calculated at the time of emission of each photon collected. All of the information is stored on the computer which enables application of any algorithm on the data and therefore this technique allows for maximum exibility in the post-analysis. This detection method is based on the technique used by Gerton et al. [14] with the addition of a few new features. 3.4.1 Approach curve The most basic form of near- eld data taken in our lab is called an approach curve where the sample's movement in the lateral X􀀀Y plane is halted and the tip oscillates vertically above the sample. To record the signal from the AFM, the de ection signal (Appendix E) is rst converted from an analog signal to a TTL-compatible signal by a simple electronic circuit (Appendix F). The resulting TTL signal is then used to generate time stamps using a Data Acquisition Card (DAC) by a technique called period measurements or edge detection. The tip-ocsillation TTL signal will be referred to as the \tapping" signal. The APD generates a TTL pulse for each photon detected which is also time stamped by the same DAC used to record the tapping signal. It is important that the two signals (tapping and photon) are recorded by the same card to insure synchronization in time. A time delay ( ) between the arrival time of each photon and the preceding probe-oscillation time stamp is computed and converted to a phase delay in the post analysis. The data acquisition procedure is illustrated in Fig. 3.9. The phase delay values are tabulated into a histogram of phase delays as shown in Fig. 3.10a. The phase corresponding to minimal tip-sample distance ( 0) can be found if some knowledge about the near- eld signal exists a priori. For example, silicon tips typically enhance the signal and therefore 0 should be located at the center of the enhancement peak (dashed gray line in Fig. 3.10a). The phase delays are converted into a tip-sample distance z by assuming harmonic probe oscillation z = A[1 􀀀 cos( 􀀀 0)] where A is the tip's oscillation amplitude (not peak to peak amplitude!). This analysis yields a tip-sample distance dependent signal which we refer to as an approach curve (Fig. 3.10b). This is the basic data acquisition technique 60 t Δ t t t t T T Δ Δ Δ Δ Figure 3.9. Data acquisition procedure. The de ection signal from the AFM (blue line) is converted to a digital signal (dashed green line) which is then time stamped (depicted by Ti). Each photon (red circles) is also time stamped (ti). For each photon recorded, a phase delay between the photon arrival time and the preceding tapping signal is computed in the post analysis. Thus, for each photon we now have an absolute arrival time and a phase delay which will later be converted to tip's height. used in our lab and is implemented in almost every type of experiment. In the case of anharmonic tip oscillations (typical for water scanning), instead of converting the de ection signal into a TTL, the de ection signal can be recorded by an analog to digital converter with a high sampling rate (1MHz). It is then possible to correlate the height of the tip with the photon time stamp by directly comparing the de ection signal (blue curve in Fig. 3.9) with the time stamp of each photon (red circles in Fig. 3.9). 3.4.2 Single Photon Near-Field Tomography (SP-NFT) The approach curve can be easily extended to a full 3D tomography map of probe- sample interactions [2]. In this acquisition mode, the sample is raster scanned in the X 􀀀 Y plane while the tip oscillates vertically. Every time the sample scanner advances a line, a signal marker is generated and time stamped by the same DAC used to record the approach curve. This enables us to correlate each approach curve with • • .1 • I • .I 1 • • • • 61 0 2 4 6 0 0.5 1 1.5 2 x 10 5 Phase (rad) Count Rate (cts/sec) 0 25 50 75 100 Tip−Sample Distance (nm) (a) (b) Figure 3.10. Phase histogram. (a) The phase delays ( i) are accumulated into a histogram. Each phase re ects a di erent tip-sample separation distance. (b) Once the minimal tip-sample distance is found (dashed gray line), i.e., 0, the phase histogram can be transformed to a distance dependent curve by assuming harmonic tip oscilla