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Optical Physics for Nanolithography
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Book Description

This book provides an in-depth, self-contained introduction of partially coherent imaging theory for researchers and engineers working on optical lithography for semiconductor manufacturing, including those in the EDA industry. It is mathematically complete: the opening chapters discuss the essential principles, and all derivations are presented with their intermediate steps. For increased accessibility, simplified and consistent notations are used throughout the text. Full-color pages illustrate the connections between figures and equations.

Book Details

Estimated Publication Date: 16 April 2018
Pages: 352
ISBN: 9781510617377
Volume: PM284

Table of Contents
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Table of Contents

1 Mathematical Preliminaries
1.1 Mean-Value Theorems of Integral Calculus
1.2 The Delta Function
     1.2.1 Representations of the delta function
     1.2.2 Properties of the delta function
1.3 Weierstrass' Approximation Theorem
1.4 Fourier Transform
     1.4.1 Fourier transform theorems
     1.4.2 Useful Fourier transform pairs
     1.4.3 Fourier series
     1.4.4 Nyquist sampling theorem for a band-limited signal and its reconstruction
1.5 Vector Algebra
1.6 Vector Calculus
     1.6.1 The del operator
     1.6.2 The gradient of a scalar field
     1.6.3 Gauss' theorem
     1.6.4 Additional forms of Gauss' theorem
     1.6.5 Green's identities
     1.6.6 Stokes' theorem
     1.6.7 Additional forms of Stokes' theorem
     1.6.8 Useful vector identities
1.7 Curvilinear Coordinates
1.8 Solution to Poisson's Equation
1.9 Helmholtz's Theorem
Appendix 1.1
Appendix 1.2
Appendix 1.3
Appendix 1.4

2 Electromagnetic Waves in Free Space
2.1 Maxwell's Equations in Free Space
2.2 Electromagnetic Radiation
2.3 Systematic Solution of Inhomogeneous Wave Equations with Boundary and Initial Conditions
2.4 Solution Inhomogeneous Wave Equations with Harmonic Time Dependence
Appendix 2.1
Appendix 2.2
Appendix 2.3
Appendix 2.4

3 Electromagnetic Waves in Material Media
3.1 Macroscopic Fields in Material Media
3.2 Energy in Electromagnetic Fields
3.3 Hertz Vectors
3.4 Fields of an Electric Dipole
3.5 Dielectric Properties of Matter
3.6 Physical Origin of Reflection and Transmission; Ewald-Oseen Extinction Theorem
3.7 Reflection and Transmission upon Incidence on a Multilayer Stack
     3.7.1 Wave vector inside each layer
     3.7.2 Reflection and transmission at the interface between two layers
     3.7.3 Derivation of the reflection and transmission coefficients
     3.7.4 Derivation of the electric field in the resist layer
     3.7.5 Derivation of the Poynting vector
Appendix 3.1
Appendix 3.1a
Appendix 3.2
Appendix 3.2a

4 Elements of Geometrical Optics
4.1 Geometrical Wavefronts and Light Rays
4.2 Fermat's Principle
4.3 The Characteristic Functions of Hamilton
4.4 The Sine Condition
4.5 The Herschel Condition
4.6 The Paraxial Approximation
4.7 The Smith-Helmholtz Invariant
4.8 The Thin Lens Formula
4.9 Stops, Pupils, and Windows
4.10 Chief and Marginal Rays; Meridional, Tangential, and Sagittal Planes; Meridional and Skew Rays
Appendix 4.1
Appendix 4.2

5 Diffraction of Electromagnetic Waves
5.1 The Scalar Theory of Diffraction
5.2 Fresnel and Fraunhofer Approximations
5.3 Representation of the Diffracted Field by the Angular Spectrum of Plane Waves
5.4 The Vector Theory of Diffraction
Appendix 5.1
Appendix 5.2
Appendix 5.3a
Appendix 5.3b
Appendix 5.4

6 Image Formation in an Optical System
6.1 Heuristic Imaging Theory
6.2 The Lithographic Imaging System
6.3 Imaging by a Point Source: Coherent Imaging
6.4 Imaging by an Extended Source: Partially Coherent Imaging
6.5 Yamazoe's Stacked Shifted Pupil Formulation
6.6 Imaging with a Higher Numerical Aperture: Radiometric Correction
6.7 3-D Point Spread Function
6.8 Imaging with a Higher Numerical Aperture: Polarization Effect
     6.8.1 Change of polarization due to focusing
     6.8.2 Change of polarization after entering the resist layer within a thin-film stack
Appendix 6.1
Appendix 6.2
Appendix 6.3
Appendix 6.4

7 Aberrations in Optical Imaging Systems
7.1 Design of Aspherical Surfaces
     7.1.1 Attainment of axial stigmatism using one aspherical surface
     7.1.2 Attainment of aplanatism using two aspherical surfaces
7.2 Connection between Ray and Wave Aberrations
7.3 The Wave Aberration for Rotationally Symmetric Optical Systems: Seidel Aberrations
7.4 General Form of the Aberration Function for Rotationally Symmetric Optical Systems
7.5 Digression: Gram-Schmidt Orthogonalization Process and Orthogonal Polynomials
7.6 Orthogonal Polynomials for Expanding the Aberration Function: Zernike Polynomials
Appendix 7.1
Appendix 7.2


This book is written for anyone who wishes to learn the underlying physics of imaging in photolithography applied to the fabrication of integrated circuits. While it is our desire to present the beauty of this physics, the fundamentals of which were established more than 140 years ago yet the applications of which still flourish, the objective here is to present the fundamentals of optical physics to lithography engineers coming from diverse fields with no or little background training in this subject. Our desire is to level the playing field, so to speak, with the hope that, after studying the book, they are better prepared to do cutting-edge work. Over the past five decades, progress in the fabrication of integrated circuits has been made possible largely by advances in photolithography - the patterning technology whereby circuit patterns are formed in the photo-sensitive-material-coated semiconductor wafer, by reductionimaging of larger versions of them on the mask delineated nowadays by beams of electrons. As these projection-imaging tools called scanners continue to be refined (through the reduction of the operating wavelength, increase of the numerical aperture with continual refinement of the imaging optics, and continuous improvement in mechanical precision and accuracy), finer and finer images have been patterned on the wafer. High-volume manufacturing of the 28-nm generation of logic integrated circuits is accomplished by scanners operating at the 193-nm wavelength and employing water-immersion optics. The same technology has been extended to the 10-nm generation by patterning it multiple times when defining a single physical layer of the integrated circuit. Starting with the 7-nm generation, the world will see a switch to a new wavelength at 13.5 nm in the soft x-ray or extreme ultraviolet (EUV) region of the electromagnetic spectrum. EUV scanners used in high-volume manufacturing are the most precise instruments in the world.

Building on the inherent capabilities of scanners, lithography engineers have been advancing photolithography's practical resolution in integrated-circuit manufacturing by increasing the usable depth of focus and by mitigating such issues as linewidth variation between dense and relatively isolated lines, foreshortening of line ends, and distortion of feature shapes, commonly referred to as proximity effects, to name just a few. Therefore, our goal here is to better prepare our readers for the difficult tasks of extending this practical resolution by applying our knowledge of optical physics.

We start Chapter 1 with a few important mathematical topics needed for the rest of the book. Chapter 2 starts with a deductive introduction of the fundamentals of electromagnetism, culminating in Maxwell's equations and the solution of the generalized wave equation, a preparation for Chapter 5. In Chapter 3, we first discuss Maxwell's equations in material media and then go on to examine the generation of electromagnetic waves (with the help of Hertz vectors) and the microscopic description of reflection and refraction, and end with a discussion on reflection and transmission of light incident on a multilayer stack. In Chapter 4, we digress to discuss geometric optics whose many powerful but simple results are essential for understanding projection imaging. Chapter 5 is devoted to the theory of diffraction in both scalar and vector formulations. Chapter 6 deals with the theory of imaging with its various formulations and (computation-wise) efficient approximations. Imperfections in imaging systems are the subject of Chapter 7, which starts with a description of Seidel aberrations and concludes with a detailed derivation of the Zernike polynomials from the first principles. We endeavor to include several pedagogical "innovations" in this book. First of all, we make copious use of mathematics and are concise with our words, for we believe physics is best described using the language of mathematics. To enhance the reader's comprehension, we include detailed intermediate steps in all derivations. However, in order to not disturb the flow, these detailed derivations are printed in blue. We also make use of colors other than black to mark more complicated mathematical derivations. The reader may find them useful or be reminded to skip them at the first reading.

These notes are a refined and substantially expanded version of the original hand-written material delivered in weekly evening lectures by the first author for the good part of the year 2012. He wishes to express his gratitude to Dr. Burn Lin for his support of the project. He also acknowledges the heavy debt owed to Born and Wolf's Principles of Optics (referred to as B&W in these notes), which was assigned as the primary reference book to accompany the lectures.

Finally, the authors would like to thank SPIE Press for their dedication and effort in improving the presentation of the text and its timely publication.

Anthony Yen
Shinn-Sheng Yu
January 2018

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