以廣義高斯常數使用於深紫外光學微影之成像與照明系統設計

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國立臺灣大學電機資訊學院光電工程學研究所博士論文

Department or Graduate Institute of Photonics and Optoelectronics College of Electrical Engineering and Computer Science

National Taiwan University Doctoral Dissertation

Extreme Ultraviolet Lithography Projector and Illuminator Design with

Generalized Gaussian Constants

蕭立人 Li-Jen Hsiao

指導教授﹕林晃巖教授 Advisor: Prof. Hoang-Yan Lin

中華民國 108 年 7 月 July 2019

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Acknowledgements

To dad, thank you for the your constant support and encouragements over these long years. For the discussions, chats, and banters. And, for our coffee times.

To Prof. Lin, my advisor, teacher, and friend, thank you for taking me under your wing when I came to Taiwan. It was an honor and an enjoyment to be a part of your lab.

To my lab mates (before me), thank you guys for the many tips that made my daily PhD life much easier than I had first thought. Thank you guys for welcoming me into this small family that we call lab. (And for the short while that we gymmed together.) The time I spent here is an irreplacible part of my life, and I will treasure it always.

To my lab mates (after me), thank you guys for helping making the lab an enjoyable place to be. I wish you guys good luck in your studies, and that you would enjoy this lab part of your Master/PhD life as much as I had.

Li-Jen Hsiao

Feb. 2019

( To my fellow hunters Queen, 68, and Jemon, thank you for the adventures from 2G, P3, XX, to World. It has been a fun journey!

るー )

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Abstract

This study aims to develop a systematic design procedure for the EUV lithography (EUVL) tools, for both the projection part and the illumination part. The optical lithography is a complex process encompasses many stages. Wafer preparation, resist coating, pre-exposure bake, exposure, post-exposure bake, etching, and metrology. Through analysis using generalized Gaussian constants (GGC), relation- ships between optical properties and requirements can be obtained, and can be used to help ensuring that optical system properties required for the tool are upheld during the design process.

The GGC is closely related to the ABCD matrix method, however over and above, it is also useful in analyzing the whole system as a combination of smaller subsystems, which can then again be broken down into even smaller subsystems to any degree desired. This abstraction of raw lens data into optical properties of the sub-systems at arbitrary level of abstraction is a great help in analyzing the inter- subsystem relations, which are easily lost in the raw expansion of the ABCD matrix of even a slightly larger optical system. In fact, the development of GGC was initially intended for purpose of zoom lens design and analysis, where inside the complex optical system the optical elements are constantly moving in relation to one another.

This analytic power lends itself well to optics design of other applications, such as this case of EUVL projection systems.

As verification of the design method, this study demonstrates an eight mirror 0.4 NA projector, and its illuminator. In addition to the use of commercial design software, a simple Monte Carlo random walk algorithm is also deviced for the purpose of integrating the use of GGC into existing design software.

Keywords: EUV, lithography, imaging, non-imaging, optical system design

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中中

中文文文摘摘摘要要要

此研究的主要目的為開發及系統化極紫外光波段之光刻機之成像及照明系統的

設計。光刻為一系列頗複雜的流程的組合。其中，最為關鍵的部分之一為曝光這

一步驟。曝光機的成像品質，於其成品的製程密度與解析度甚至於此製成的生產效率中間，存在直接的影響與關係。然而，曝光機之光學系統特性，及其參數之間，存在著許多複雜且繁瑣之關係，導致於此光學系統不易分系，也不易設計。

因此，此研究最為核心的目的，為研究及開發某一系統化之分析方法，以達到簡

化此光學系統之設計之目的。此研究核心之關鍵，為將廣義高斯常數應用於光刻機之分析及設計上。通過廣義高斯常數，眾多複雜且繁瑣之光學特性及光學系統之間之聯繫能以之表達及簡化，而將所有關係結合並簡化後，從中所導出之數學關係式可用於與商用光學設計軟體之結合，以達到幫助分析及設計簡化。

原理上，廣義高斯常數與光學分析中常用之 ABCD 矩陣極為相似，唯一最大不同為當以 ABCD 矩陣進行光學系統分析時，計算其矩陣及將其展開時，其難度及複雜性與其光學系統中光學元件之數量呈指數增長，因此，不適用於較為複雜之光學系統分析。而相較於 ABCD 矩陣，廣義高斯常數適用於表達及分析由多數光學系統組合所形成之相對複雜之光學系統，而其又可將之分解成更小之光學系

統看待。此廣義高斯常數之特性，可隨心將多數光學系統視為一總光學系統，或

是將一光學系統拆開以多數小光學系統看待，可適用於分析或推導個別光學元件

之參數及整體光學系統特性之關係。

廣義高斯常數之最初用途為變焦光學系統之分析及設計，其中通常有複數多件

光學元件所組成之組合，及其為配合不同使用狀況而改變位置及光學特性。而將之強大分析能力應用於極紫外光刻機之分析及設計為此研究之重要關鍵之一。此

研究之成果之一為某一 0.4 數值孔徑之極紫外光刻機之反射式成像光學系統之分

析與設計，以及其照明光學系統之分析與設計。設計過程中所用之光學設計軟體中包含市售光學設計軟體，及一簡單 Monte Carlo 優化演算法將廣義高斯常數與市售光學設計程式結合使用以達成協助光學設計之目的。

關關

關鍵鍵鍵字字字: 極紫外，光刻，成像光學系統，非成像光學系統，光學系統設計

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List of Figures

1 Basic structure of a lithography system. . . 15 2 A conventional projection lens is a complex optical system, often consist-

ing of more than 20 to 30 lenses. . . 15 3 Early iterations of lithography projections. . . 17 4 Further developments with increasing NA, for higher resolution. . . 18 5 Introduction of aspheric lenses to reduce the number of lenses, and im-

mersion techniques are employed to further increase NA (even beyond 1.0 NA), . . . 19 6 The lens-mirror hybrid catadioptric systems. Intentional back-reflected

paths using mirrors and beam-splitters are introduced to reduce chromatic aberration and Petzval curvature. . . 19 7 Further developments on the catadioptric systems sacrifice portions of the

imaged field to eliminate the need for beam-splitters. . . 20 8 A one to one reduction ratio reflective lithographic projector curtesy of

Perkin-Elmer, 1986. The intended feature size to be printed are on the order of 1 3 µmm. [1] . . . 22 9 Completely reflective system designed to cater to the short wavelength

EUV light source. [2] . . . 23 10 An early iteration of a lithographic projector. . . 23 11 Telecentricity. (a) Image space telecentricity. (b) Object space telecen-

tricity. . . 24 12 Double telecentricity. An optical system is double telecentric in the spe-

cial case that it is both telecentric in the image space and in the object space. . . 25 13 The performance of optical systems in the case that the object is deviated

from its designed position. (a) Generic non-telecentric system. (b) Object space telecentric system. . . 25

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14 Light striking a binary mask of lines and spaces forming a diffraction

pattern. . . 27

15 The maximum order of diffraction collected by the lens is limited by the physical size of the lens, which acts as an aperture. . . 28

16 The resulting electric field of the aerial image, of an illuminated binary mask through a lens. . . 29

17 The intensity profile of the aerial image. . . 29

18 The diffraction pattern formed by coherent illumination. . . 30

19 The diffraction pattern formed by partially coherent illumination. . . 30

20 The effect of partial coherence on resolution limit. (a) The aperture is able to capture the ±1^st order diffraction completely. (b) The ±1^st order diffraction moves further apart with finer mask pitch, some of the ±1^st order information are lost causing degradations in the reconstructed aerial image. (c) At this mask pitch, the ±1^st order diffraction almost moves outside the aperture completely, aerial image reconstruction of even finer mask pitch is impossible. The resolution limit. . . 32

21 A drawing of a simple gate device. Here, the CD would be how thin the walls can be made, and the pitch resolution would be how small one single unit can be printed. . . 34

22 A drawing of a grating device of lines and spaces of equal width. In this case, the CD and pitch resolution are closely related, with Λ = 2 ·CD. . . 35

23 A simple diagram of an optical lithography exposure. Light from a source is redistributed onto the mask by the illuminator (the condensor lens). The illuminated mask is then picked up by the projector (the objective lens) and imaged onto the wafer. . . 36

24 Wave optics representation of the lithographic process. . . 37

25 To convey the pattern across, the NA of the optical system (given by NA = sin θmax) must be large enough such that the ±1^st order diffraction are collected. . . 37

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26 The three major types of resolution enhancement techniques (RET). Op- tical proximity correction, off-axis illumination, and phase-shifting mask. 39 27 Off-axis illumination. The imaging resolution of an optical system can be

doubled simply by tilting the angle of the illumination optics. . . 40 28 Using a phase shifted mask effectively doubles the periodicity of the mask

pattern. . . 41 29 Optical proximity correction makes minor adjustments to the mask such

that the imaged pattern stays the same to the pattern intended as much as possible. . . 42 30 the location of the ±1^storder diffraction shifts according to the local pitch

of the mask pattern illuminated. . . 44 31 The forbidden pitch, where the 1^st order diffraction is at the center of the

pupil, with maximum OPD.. . . 45 32 Comparison between off-axis illumination and phase shift mask. . . 45 33 If the imaging target is immersed, the effective NA in air is enhancd

by a factor almost equal to the refractive index of the the immersion medium/fluid. . . 47 34 Basic schematic of EUV generation. . . 49 35 A mechanism to prevent (or reduce the amount of) byproducts of the EUV

source from entering and contaminating the lithographic optical system. [3] 50 36 Thermal expansion of Schott’s Zerodur. . . 51 37 Thermal expansion of Corning’s ULE. . . 52 38 Reflection spectrum of the 40 Mo/Si multilayer coating. [4] . . . 53 39 Anuglar reflection distribution of the 40 Mo/Si multilayer coating. [4] . . 53 40 Candidates of the reflection depth offset. (a) Without the multilayers,

the reflection occurs at the surface interface. (b) At the bottom of the multilayers. (c) At the top of the multilayers. (d) Somewhere inside the multilayers. . . 54

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41 The reflections of the EUV light over a multilayer stack. The center of the reflection lies inside the stack towards the air-multilayer interface. . . 55 42 Basic configuration of the lithographic projection system, consisting of

two subsystems separated by an aperture stop. The quantities EF, FF, BF, and thc are the effective focal length, front focal length, back focal length, and thickness of the two subsystems respectively. . . 60 43 A diagram of an illuminator in Kohler illumination configuration with a

matching imaging optical system. . . 61 44 Effect of the partial coherence RET. Top: Coherent illumination. Bot-

tom: Partially coherent illumination. Broadened diffraction spot allows the limit of the ±1 order to be collected at a larger angle (i.e. higher resolution). . . 62 45 A diagram of the Kohler integrator configuration. . . 64 46 A diagram of the ringfield mirror array, or fly eye reflector. . . 64 47 The effective focal length, back focal length, and front focal length of an

optical system. . . 71 48 Illustration of a finite conjugate optical system. . . 73 49 Upper: The paraxial representation of an optical system formulated with

GGC. Lower: The exact same optical system represented using reflective elements. . . 75 50 Example of a solution space with two local minima. . . 81 51 Flow diagram of the Monte Carlo random walk algorithm. . . 84 52 Defocus on a non-telecentric system causes blurring and image shift,

while on a telecentric system only blurring occurs. . . 87 53 A subsystem can be further expanded into multiple mirror pairs, repre-

sented by the individual lens modules. Upper: Expansion into two mirror pairs. Lower: Three mirror pair expansion. . . 93 54 A flow diagram of the Monte Carlo random walk algorithm. . . 96

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55 The amount of obstruction between a ray bundle and mirror is defined as the angle of the overlapping region. . . 97 56 The initial state of the four mirror trial run.. . . 99 57 Four-mirror system optimization using Code V. Lens module converted to

real mirror surfaces. Lens data and module properties are listed in Table 5. 100 58 Monte Carlo random walk result of an eight-mirror system. . . 102 59 Initial state of the 8-mirror system, and its spot diagram. The mirror sur-

face profiles at this point is entirely spherical. . . 104 60 The state of the 8-mirror system after optimization runs, after allowing

the conic constants of the surface profiles to vary. . . 105 61 Layout of an eight-mirror EUVL system.. . . 106 62 The MTF performance of the eight-mirror EUVL system. . . 107 63 Wavefront aberration of the six defined field positions. Top Row: Center,

top, and bottom fields on the tangential plane respectively. Bottom Row:

Center top, and bottom fields at full horizontal mask width. . . 107 64 The marginal ray angles at the grid source and at the projection tool en-

trance pupil is related to the magnification of the finite conjugate system formed by the elements in between. . . 111 65 The illumination NA is directly related to the beam size of the collimated

LPP source. . . 112 66 The number of array elements on the pupil array determines the array

pitch and the grid source NA. . . 113 67 An initial evaluation of the governing equations. . . 115 68 The amount of obstruction between a ray bundle and mirror is defined as

the angle of the overlapping region. [5]. . . 116 69 The same initial evaluation with a 5^◦tilt introduced. . . 116 70 Obstruction resolved paraxial layout. The tilt angle required is θ_tilt =

6.991^◦.. . . 117

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71 Direct conversion from the paraxial result. The lower zoomed in part shows that aberration resultant from the plain spherical mirror offsets the illumination profile from different array elements. . . 118 72 The aberration corrected illuminator. . . 119 73 The eight mirror projector design from which the illuminator specifica-

tions are derived from. . . 120 74 The combined system of the illuminator and the projector. . . 121 75 Upper: The resultant ringfield illumination profile at the mask side. Lower:

The illumination profile at the wafer end. . . 122

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List of Tables

1 The general trend of source wavelength reduction from early lithography history. . . 22 2 Types of illumination coherence. . . 31 3 Specification of a EUVL tool design trial run. . . 97 4 Initial lens module parameters of the optimization run. The four free

variables are marked with an asterisk (*), the remaining parameters are dependent. . . 98 5 Random walk result of the four-mirror system. Left: Lens module prop-

erties. Right: Conversion to raw lens data. . . 99 6 Basic lens data of the eight-mirror system, displaying the radius of curva-

ture R, thickness T , conic constant K, and aspheric coefficients from A₄ to A₁₂. . . 108 7 RMS wavefront error and Zernike coefficients (tilt and defocus) in wave-

length unit. . . 108 8 A set of specification and initial condition of the illuminator design. . . . 115 9 Initial evaluation of the governing equations using specification and initial

conditions provided in Table 8. . . 115 10 Obstruction resolved illuminator parameters.. . . 117 11 Obstruction resolved illuminator parameters.. . . 119

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1 Introduction

1.1 Lithography Overview

The lithography process is a major part of the process chain of the manufacturing of a semiconductor device. Typically in the manufacturing of an integrated circuit chip, lithography can account for up to 30% of its manufacturing cost. As such, the progress in the development of lithography systems is of great interest to the semiconductor industry. [6]

In particularly, the optical lithography systems have received much attention, as historically the technological advancements in semiconductor mass manufacturing are often gated by the progress in optical lithography. In the past when the progress in optical lithography reaches a bottleneck, the industry has attempted to look at the alternative lithography methods to potentially replace optical lithography, such as e-beam and x-ray lithography. After all is considered however, as each of the bottlenecks were overcame the optical lithography quickly regains its place as the most important lithography method, as out of all of them the optical lithography remains the most economical and the most efficient way to mass manufacture semiconductor devices to this day. [7] [8]

The optical lithography process encompasses many stages, usually starting with wafer preparation, then resist coating, pre-exposure bake, exposure, post-exposure bake, etching, and then finishing with metrology. During the exposure stage, light from the source is first redistributed and reshaped by an illuminator before striking the mask. The patterns on the mask, now illuminated, is then picked up by the projection tool, and is transferred

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Figure 1: Basic structure of a lithography system.

Figure 2: A conventional projection lens is a complex optical system, often consisting of more than 20 to 30 lenses.

to the photoresist coated on the wafer. Figure1shows a typical illuminator and projector arrangement, using a laser as the light source.

Figure 2 is a diagram of a conventional lithography projection lens. The lithography projector is an essencial part of optical lithography process, and is conventionally composed of many lenses, spanning well over a meter in length. The designing of a lithography tool, portrayed in many textbooks as the pinnacle of imaging optical system design. The sheer number of optical elements required, coupled with the stringentness of the required imaging quality, all contributes to the overall design difficulty. Despite the difficulty, designers have little choice but to continue to tackle the task, as the design and manufacturing of this projection system will largely determine the resolution and performance of the lithography process.

Aside from the projection tool, other aspects of the lithography also have significant impact on the lithography performance. Regarding the resolution of the lithography pro-

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cess, a commonly accepted fact is that the resolution (of a straight single exposure) is determined by the numerical aperture (NA) of the projection tool, given by

R= k₁ λ

NA. (1)

1.1.1 Evolution of Lithographic Systems Early Developments

The lithographic projection system is a relatively recent development in the field of optical design. Historically, early attempts at designing lithographic projectors results in a high NA but small FOV systems. The wavelengths used are in the blue region of the visible spectrum, where conventional glass materials can be used.

Since its early stages, improvements in the lithography techniques has been partly driven by the extrapolation of Moore’s observation in 1965, that the number of compo- nents on an integrated circuit chip would double every year reaching roughly 1000 times to what it was by 1975, [9] and indeed for the first decade the semiconductor industry kept up with the trend. However for future decades to come the trend was proven to be too difficult and optimistic for the industry to follow, and Moore himself made adjustments to his observations extrapolated a gentler slopes for future semiconductor manufactur- ers. [10] [11] Of course, simply by common sense would dictate that Moore’s Law cannot continue forever, however as the years advance the market and the industry has seen an ever increasing need for higher lithography resolution.

This need for higher and higher resolution pushed later development towards shorter wavelengths, with increasing NA and FOV, and consequently tighter performance requirements. Among the designs, some special cases exist which are quite insightful in terms of optical system design. In particular, the designs of a field flattener, and the design of a chromatic aberration corrector with only one available glass material. The technology and unerstanding of several aspheres within one system, as well as the principles of catadioptric and mirror systems have been possible as a result of these developments.

More physical optical questions occur during the development of the various gen-

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Figure 3: Early iterations of lithography projections.

erations of lithographic systems. The understanding of polarization, the correction and control of birefringence, the theory of simulation for high NA and artial coherent illumination are all developed further within this context. Therefore, although the systems of this type are very situational, they will be discussed in more details.

Figure 3 are some examples of early lithographic projectors. [12] [13] Historically, early projectors consists of a retrofocus photographic lens at the front, and a scaled mi- croscopic lens in the rear. [44-6] The mask side will have a larger field due to the reduction ratio (usually between 0.2 to 0.25), and the wafer side has a smaller field but high NA.

Combined together, the two systems forms a complete lithographic system.

Following, in the years of development, the size of the lithography system gradually increased. To enhance the resolution (Equation 1), one possible way is to increase the NA of the projector, as can be seen in Figure4. [14] [15] However, systems with larger NA generally results in more severe geometric aberrations, which results in the need for lithographic systems to house more and more lenses in the hope to correct the aberrations.

To help supressing the aberrations, the angle of refraction at each interface are kept to a minimum, which results in the characteristic smooth bulges and waists in lithographic systems.

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Figure 4: Further developments with increasing NA, for higher resolution.

Aspheric Lenses and Immersion

Beginning around year 2000, the need to reduce the size of the lenses and progress in manufacturing technology allows the inclusion of aspherical surfaces inside the lithographic projection lenses. This considerably reduces the number of lenses required, and the necessary lens diameter of the high NA lenses compared to purely spherical systems of the same NA, as can be seen in Figure5. [16] With aspheres, the NA of dry systems can be enhanced to roughly 0.95, and if immersion fluids are used (e.g. water), increasing the NA beyond 1.0 is possible. [17]

Catadioptric Systems

Since high NA system leads to large lens diameters in the rear lens groups, problems occur in correcting aberrations and obtaining a good uniformity in the material. Immersion systems are particularly different in their behavior. For this reason, the catadioptric systems, shown in Figure6, have been introduced. [18] [19] [20] The mirrors help in correcting the Petzval curvature, while the Schupman principle can be used to correct axial chromatic aberrations. The size of the systems can be reduced by a considerable factor, and a NA of 1.35 can be achieved. In Figure7, a further development upon the catadioptric systems removes the need of a beam splitter cube (which are used at the cost of a 75% decrease

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Figure 5: Introduction of aspheric lenses to reduce the number of lenses, and immersion techniques are employed to further increase NA (even beyond 1.0 NA),

Figure 6: The lens-mirror hybrid catadioptric systems. Intentional back-reflected paths using mirrors and beam-splitters are introduced to reduce chromatic aberration and Petz- val curvature.

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Figure 7: Further developments on the catadioptric systems sacrifice portions of the imaged field to eliminate the need for beam-splitters.

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in the aerial image brightness), by trading in half of the usable FOV of the lithographic system. [21] [22] [23] [24]

Reflective Systems

The use of all reflective projection system coupled with a short wavelength source in optical lithography can be dated back over 30 years ago. Figure8shows one such early iteration of an all reflective projector, with 1:1 reduction ratio for printing features with size on the order of ∼1 µm. [1]

Table 1 shows a general trend of source wavelength usage from the early history of lithography. For higher resolution, the natural course of action would be to further reduce the wavelength of the light source, since the optical system NA can only be increased to so much, to a maximum of 1 in vacuum or ∼1.3 by immersion. Following the trend in choosing shorter and shorter wavelengths, the next wavelength after the 193 nm line is the 157 nm line. The problem of using the F₂ laser line at 157 nm wavelength is to be used in projection lithography to obtain a better resolution, only calcium flouride CaF₂can be used as the transparent material due to the absorptions in other materials.

At 157 nm, the corresponding spectral line has a poor efficiency and therefore the line narrowing cannot be achieved as it can for other longer wavelengths. Therefore the need to achromatize the projection lenses has generated some investigation into how to use diffractive elements to correct the system for chromatic aberrations. In principle, this is a possibility, however no current industrial system employs this method. The problem of stray light and the manufacturing of the microstructure tolerances to guarantee the high performance are a severe problem.

Therefore, instead of progressing to the 157 nm lithography, the advent of the immersion method using 193 nm lithography became the next milestone. However, this solution is only temporal, as the NA increase provided by the immersion method is limited. As the search continued for shorter wavelengths, the next possible source discovered is the 13.4 nm line in the EUV region. In the past, the decrease in wavlength has been grad- ual, however this time the wavelength is an entire magnitude lower from the previous

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Figure 8: A one to one reduction ratio reflective lithographic projector curtesy of Perkin- Elmer, 1986. The intended feature size to be printed are on the order of 1 3 µmm. [1]

Table 1: The general trend of source wavelength reduction from early lithography history.

Wavelength (nm) Source

405 Hg lamp h-line

365 Hg lamp i-line

248 KrF excimer laser (DUV)

193 ArF excimer laser (VUV)

157 F₂excimer laser

(193) In conjunction with immersion technique 13.4 Syncrotron radiation or plasma source (EUV)

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Figure 9: Completely reflective system designed to cater to the short wavelength EUV light source. [2]

Figure 10: An early iteration of a lithographic projector.

generation.

Due to the high absorption of light in the EUV region exhibited in most mediums, the optical system is forced to use only reflective elements (i.e. mirrors). Figure9is an example of such an all reflective optical system. [2] Aside from having only reflective elements, the system must also be placed inside a vacuum environment, since the absorption at the EUV wavelengths is high even in air.

1.1.2 Lithography Systems

The basic operation of a lithography system are as follows. Light from the source is first redistributed and reshaped, by an illuminator system, before striking the mask. The pattern on the illuminated mask is then picked up, by a projector system, and imaged onto the wafer. Shown in Figure10 is a typical projection lens during the early stage in the history of optical lithography, a complex optical system often consisting of more than 20

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Figure 11: Telecentricity. (a) Image space telecentricity. (b) Object space telecentricity.

to 30 lenses.

During the manufacturing of a device, overlaying multiple exposures are often needed in lithography processes. Therefore one crucial requirement on the projection system is that, small deviations in the axial direction do not influence the lateral position of the image. This leads to the following conditions:

• Telecentricity in both the mask (object) space, and the wafer (image) space.

• Extremely well corrected distortion aberration in the aerial image.

Telecentricity

There are three types of telecentric systems. Image space, object space, and double telecentricity. An optical system is image space telecentric when the chief rays of the optical system emerge parallel to the optical axis before arriving at the image in the image space.

On the other hand, an optical system is object space telecentric when the chief ray from the object is parallel to the optical axis prior to entering the optical system, in the object space. Figure11is a diagram showing the setup of the optical system when the optical system is telecentric in the image sapce, and in the object space.

In the special case that the optical system is telecentric in both its image space and object space, the optical system becomes double telecentric. For an optical system to be doubly telecentric, the system must be afocal. The advantage of a telecentric system is that the magnification of the optical system is invariant in the presence of defocusing.

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Figure 12: Double telecentricity. An optical system is double telecentric in the special case that it is both telecentric in the image space and in the object space.

Figure 13: The performance of optical systems in the case that the object is deviated from its designed position. (a) Generic non-telecentric system. (b) Object space telecentric system.

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Even if the optical system is designed and manufactured perfectly, the image quality will suffer nonetheless if during operation the object and image plane is offsetted from its designated positions. Figure13shows a comparison of a non-telecentric system to an object space telecentric optical system. For non-telecentric optical systems, two major effects of defocus are blurring (image quality degradation) and change in image magnification. If the system is telecentric, then the optical system have one less thing to worry about since the change in the image magnification is eliminated.

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Figure 14: Light striking a binary mask of lines and spaces forming a diffraction pattern.

1.2 Aerial Image Formation

The aim of optical projection lithography, is to print an image of the reticle onto the wafer.

To this end, we must first understand how an image is formed in the first place. For a start, consider the simple example of light striking a binary mask pattern of lines and spaces of equal width.

First, as depicted in Figure14, light from the exposure system strikes the mask, forming a diffraction pattern, given by

M( f_x) =

∞

∑

n

sin(πw f_x)

π f_x δ ( fx−n

p), (2)

where f_x is the lateral distance in space perpendicular to the direction of propagation, w is the angular frequency of the phase, p is the pitch of the mask, and n is the order of diffraction. The projection lens then picks up the diffraction pattern.

Due to the finite size of the lens, after passing through it the lens acts as an aperture, effectively limiting the number of the diffraction orders collected, as shown in Figure15.

Typically, when considering the resolution limit, only the 0^th order and the ±1^storder are collected. Therefore, by limiting the diffraction function (Equation2) to within the ±1^st

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Figure 15: The maximum order of diffraction collected by the lens is limited by the physical size of the lens, which acts as an aperture.

order, the pattern after the aperture is

M( f_x) = sin(πw fx) π f_x

δ ( f_x) + δ ( f_x−1

p) + δ ( f_x+1 p)

. (3)

After travelling through the lens, the collected diffraction orders then recombines at the image plane and superpositions to form the aerial image, which can be obtained by an inverse Fourier transform of the aperture function (Equation3), with electric field

E(x) =1 2+2

πcos(2πx

p ) (4)

and intensity

I(x) = 1 4+2

πcos(2πx p ) + 4

π²cos²(2πx

p ), (5)

iluustrated in Figures16and17.

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Figure 16: The resulting electric field of the aerial image, of an illuminated binary mask through a lens.

Figure 17: The intensity profile of the aerial image.

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Figure 18: The diffraction pattern formed by coherent illumination.

1.2.1 Partial Coherence

In lithography, the coherence of illumination refers to the range of the angle of the illumination. Different degrees of coherence affects both the resolution and depth of focus of the resultant aerial image. Depicted in Figure18is a diffraction pattern resultant from the mask illuminated by a parallel beam of light (coherent illumination), captured by the aperture. If the illumination is partially coherent, the light strikes the mask over a range of angles. This results in a broader diffraction pattern, as can be seen in Figure19.

The degree of partial coherence is measured by a partial coherence factor σ (also referred to as the pupil filling factor), given by

σ = d

D (6)

Figure 19: The diffraction pattern formed by partially coherent illumination.

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Table 2: Types of illumination coherence.

Illumination Type Partial Coherence Factor σ Source Geometry

Coherent σ = 0 Point

Partially Coherent 0 < σ < 1 Finite size and circular

Incoherent σ ≥ 1 Infinite size

where d is the diameter of the source illumination on the entrance pupil, and D is the diameter of the entrance pupil of the lithographic projector. A partial coherence factor of zero (σ = 0) means that the illumination is completely coherent, which is possible only with a singular point source. A partial coherence factor between zero and one (0 < σ < 1) means that the illumination is partially coherent, and that the source is of a circular shape with finite size. A partial coherence factor of one or greater (σ ≥ 1) signifies that the illumination is incoherent. The types of coerence above are also listed in Table2.

By employing partial coherence, the pitch resolution can be further improved. Con- sider the case where some amounts of partial coherence σ is present. First, in Fig- ure 19(a), the entrance pupil is sufficiently large that the information necessary to reconstruct an image of the reticle, the 0^th and the ±1^st order diffraction, are collected.

Following in Figure19(b), as the mask pitch becomes finer, the diffraction orders move further apart, and parts of the ±1^st order diffraction failed to be picked up by the lens entrance pupil, lowering the image constrast. However, the pitch resolution is extended.

In Figure19(c), at this point the ±1^st order diffraction almost moved completely outside the lens entrance pupil. As such, the reconstruction of the aerial image of even finer mask pitch is impossible, because without the±1^st order diffraction, the 0^th order contains es- sentially only a constant level of background energy, therefore no image can be formed.

The resolution limit.

Mathematically, the pitch resolution limit of the mask is given by

1

p = NA

λ (7)

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(a)

(b)

(c)

Figure 20: The effect of partial coherence on resolution limit. (a) The aperture is able to capture the ±1^st order diffraction completely. (b) The ±1^st order diffraction moves further apart with finer mask pitch, some of the ±1^st order information are lost causing degradations in the reconstructed aerial image. (c) At this mask pitch, the ±1^st order diffraction almost moves outside the aperture completely, aerial image reconstruction of even finer mask pitch is impossible. The resolution limit.

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under coherent illumination.

For partially coherent illumination, the pitch resolution limit is modified to

1

p = (1 + σ )NA

λ , (8)

yielding an improvement in the pitch resolution, with increasing value of the partial coherence factor σ .

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Figure 21: A drawing of a simple gate device. Here, the CD would be how thin the walls can be made, and the pitch resolution would be how small one single unit can be printed.

1.3 Resolution

For lithographers, the term resolution is often ambiguous. In general, two different mean- ings are often associated with the term resolution. It can either mean how small one individual feature can be made, in which case the proper term to refer to it would be critical dimension (CD); or how many units of repeating pattern that can be printed on the wafer, in which case the proper way to call it would be pitch resolution Λ, though in most cases just simply as the resolution.. While both are important, the distinction between the two must be made clear, because the influence of the two and underlying limitations are very different.

Critical Dimension

The CD is simply, how small lithographer is able to print one single individual feature.

Figure 21 is a diagram of a simple gate device. In this case, the CD would be how thin the walls of the device can be made. The CD influences many important electrical functions and properties of the device, such as the drain current, driving voltage, and device efficiency etc. In device manufacturing, as of now, the most limiting factor to the

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Figure 22: A drawing of a grating device of lines and spaces of equal width. In this case, the CD and pitch resolution are closely related, with Λ = 2 ·CD.

CD of a device feature has been the process control. There are many tricks in printing features with CD well below what the lithography tool is normally capable of. For one single exposure, the resolution is limited by the diffraction limit, as defined previously in Equation1. However, by combining multiple exposure processes, with techniques such as double imprint, double resist imprint, quadruple imprint etc, features with CD beyond the resolution of a single exposure can be printed. Although, there is a catch, and that is all require extremely high presicion control.

Pitch Resolution

This is the ability of the lithography tool to print repeating features. Or, how many devices can be printed in a finite given area (i.e. one wafer). Figure22shows a simple diagram of a grating of equal lines and spaces, in this case the pitch resolution Λ is the size of one repeating unit. The pitch resolution affects aspect of the device directly related to its packaging and manufacturability, such as the cost per function of the device and functions per chip. Here, however, this aspect of the resolution has a hard limit of according to Equation1

R= k₁ λ

NA (9)

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Figure 23: A simple diagram of an optical lithography exposure. Light from a source is redistributed onto the mask by the illuminator (the condensor lens). The illuminated mask is then picked up by the projector (the objective lens) and imaged onto the wafer.

1.3.1 Resolution Limit

The root of the cause of the resolution limit, is due to diffraction. Consider an optical system shown in Figure 23. From the point of view of Fourier optics, when the light from the illuminator strikes the mask, the resultant diffracted field interferes to form a diffraction pattern. The pattern is then received by the projection system, and recombined at the image plane. [25] Figure24shows a diagram of the process.

Mathematically, this can be expressed as

U( f_x, f_y)_Image=F⁻¹

A(x⁰, y⁰)_Lens·F [ U(x,y)Mask ]

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where U (x, y)_Mask is the mask pattern, A(x⁰, y⁰)_Lens is the aperture function of the optical system, and U ( f_x, f_y)_Imageis the resultant imaged field.

Ideally, to perfectly reconstruct the mask pattern, the entire diffracted field from the maskF [U(x,y)Mask] must be transfered through. To achieve this, however, would require an infinitely large aperture function (or A(x⁰, y⁰)_Lens= 1) Obviously, this is not possible, nor realistic. Therefore, since the aperture function A(x⁰, y⁰)_Lens is finite, during transmission some part of the the diffraction will always be obstructed, causing a lose of higher order information.

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Figure 24: Wave optics representation of the lithographic process.

Figure 25: To convey the pattern across, the NA of the optical system (given by NA = sin θmax) must be large enough such that the ±1^storder diffraction are collected.

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Given some patterns with pitch Λ on the mask, light from the pattern interferes and diffracts into orders given by the well known grating equation

sin θ_m= sin θ_i+mλ

Λ (11)

In the fourier reconstruction of the image, the more orders are present, the more the image reconstructed adhere accurately to the original object

u= A₀+ A₁sin(α) + A₂sin(2α) + A₃sin(3α)... (12)

However, at least one order of the diffraction must be present in order for information to be conveyed, since the 0^th order only carries energy, but contains no information (i.e. DC offset). Therefore, the size of the lens must at least be big enough to collect one order of diffraction, in order for the image to be constructed at all (Figure25), so

NA = sin θmax

= mλ

Λ (m = 1)

= λ

Λ, (13)

which can be rearranged to obtain

R ≡ Resolution

= Λ

= λ

NA. (14)

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Figure 26: The three major types of resolution enhancement techniques (RET). Optical proximity correction, off-axis illumination, and phase-shifting mask.

1.3.2 Resolution Enhancement Techneques

Recall that the resolution limit previously stated at the beginning

R= k₁ λ

NA (15)

has a different form to the one obtained in the previous section

R= λ

NA. (16)

The two differs by a factor k₁. This parameter k₁ is the Process Parameter, known as such because it is quite literally a parameter that depends on the process. The value of k₁usually has a lower limit of 1, but over the years researchers have developed tricks to further lower k₁ to as far as 0.5, effectively doubling the resolution to what is normally possible. These tricks are referred to as RET (Resolution Enhancement Techniques).

As of currently, there are three main types of RET in industrial use. Off-axis illumination (OAI), optical proximity correction (OPC), and phase shifting mask (PSM). Each of them is able to enhance the resolution of the aerial image through different physical mechanisms.

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Figure 27: Off-axis illumination. The imaging resolution of an optical system can be doubled simply by tilting the angle of the illumination optics.

Off-Axis Illumination

Recall that previously when defining the resolution limit, the criterion is set such that at least the ±1^st order diffraction are received by the optical system, since the 0^th order contains only background energy but no information. Here, by the same logic, one could argue that only one of the ±1^st order are necessary for the reconstruction of the image since the two diffraction orders are symmetric. Therefore, one way to improve resolution is to simply tilt the illumination to one side [26], hence the name off-axis illumination, as shown in Figure27. The resulting diffraction patterns will undergo a shift in one direction.

This way, by sacrificing one diffraction order, the available angular space is now twice as large, effectively doubling the resolution.

This method is not without faults, however. Since information of the mask pattern is contained in the ±1^storder and the 0^thorder contains only background energy, eliminating one of the ±1^st order diffraction results in a significant decrease in the contrast of the intensity of the aerial image. Also, since the enhancement of resolution is achieved by tilting the illumination to one side, the enhancement is therefore in one direction only.

These demerits must be taken into consideration if OAI is to be utilized.

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Figure 28: Using a phase shifted mask effectively doubles the periodicity of the mask pattern.

Phase Shifting Mask

Another method to improve the resolution is to use a specialized mask. By covering the mask pattern with small blocks of material in an alternating fashion as portrayed in Figure28, a phase shift of half the wavelength can be introduced. Doing so doubles the effective periodicity of the mask pattern, and therefore the resolution can be increased to twice as that of an unshifted mask. [27]

One other advantage of using a PSM is that in this configuration, the 0^thorder diffraction is minimized. Looking at the comparison of the transmittance in Figure28reveals that the average of the amplitude function of the transmittance of the unshifted mask is 0.5, which means a significant portion of the total energy is directed to the DC offset (0^th order). In the case of the PSM however, the average of the amplitude function of the transmittance is zero, therefore no energy is wasted to the 0^th order diffraction. This means that in comparison to a unshifted mask, using the PSM is advantageous in that it doubles the resolution, and at the same time improves the aerial image intensity contrast.

However, one obvious setback of the PSM is that it can be applied to transmission masks only, and that it is difficult to apply to irregular and non-repeating features.

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Figure 29: Optical proximity correction makes minor adjustments to the mask such that the imaged pattern stays the same to the pattern intended as much as possible.

Optical Proximity Correction

Since realistic optical systems are finite in size, the higher frequency information outside of the lens aperture are lost in the imaging process. The most pronounced consequence of the loss of high frequency information is that the blurring of the edge sharpness in the intensity of the imaged pattern, causing an inevitable distortion. OPC attempts to make minor alterations to the mask, in order that the final reconstructed image stay as close to the intended pattern as possible. [28] An example of this is shown in Figure29.

OPC differs from OAI and PSM in the sense that it does not alter the pitch resolution at all, however it still comes under the RET catagory because it helps to maintain the correctness of the image.

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RET Enhanced Resolution

Taking RET into account and partial coherence into account, a more complete description of the resolution are given below [6]

• Coherent Illumination

R= k₁ λ

NA (17)

• Partially Coherent Illumination

R= k₁ λ

NA(1 + σ ) (18)

• Off-Axis Partially Choerent Iluumination

R= k₁ λ

NA+ NAσ + sin θ (19)

Depth of Field and Depth of Focus

The depth of field (DOF) and depth of focus (also DOF) refers to a range of depth in which the mask and wafer can be placed while still maintaining the image quality above a given threshold, and is related to the NA of the optical system, given by

DOF = k₂ λ

NA² (20)

similar to the pitch resolution.

Both the depth of focus and depth of field are abbreviated as DOF, with depth of field on the mask (object) side and depth of focus on the wafer (image) side. Both are related to one another by a factor proportional to the square of the magnification of the optical system, given by

DOF_Focus= M²· DOF_Field (21)

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Figure 30: the location of the ±1^st order diffraction shifts according to the local pitch of the mask pattern illuminated.

Forbidden Pitch

For a lithography tool optimized with OAI, there is a pitch resolution where the projection tool performs exceptionally poor, and is therefore referred to as the forbidden pitch.

In designing the mask pattern, lithographers must inform the designer to avoid placing features at the forbidden pitch to ensure optimum result.

When employing OAI optimized for one feature CD, one inevitable consequence is that at larger feature pitch, the location of the 1^th order diffraction shifts closer to the 0^th order diffraction as depicted in the right part of Figure30. This induces an optical path difference between the two orders when they recombine at the wafer.

The worst case scenario occurs at the point where the diffraction order shifts to the midpoint between where the two orders were, as shown in Figure 31. As the feature pitch increases to the point where the 1^th order diffraction is at the center of the lens NA, maximum OPD is reached. At this point, the resultant image quality at the wafer is at its poorest. This pitch is referred to as the Forbidden Pitch, and should be avoided in the mask design where possible.

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Figure 31: The forbidden pitch, where the 1^st order diffraction is at the center of the pupil, with maximum OPD.

Figure 32: Comparison between off-axis illumination and phase shift mask.

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As a side note, although the PSM have similar enhancement effects to OAI, the two rely on different physical mechanisms, as shown in Figure32. The OAI technique sac- rifices one of the ±1^st order for the gain in resolution, while the PSM introduces a half wavelength phase shift to every second feature in the pattern. As such, the PSM does not suffer from the effect of the forbidden pitch as OAI does.

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Objec ve

Wafer

Figure 33: If the imaging target is immersed, the effective NA in air is enhancd by a factor almost equal to the refractive index of the the immersion medium/fluid.

1.4 EUV Lithography

As previously mentioned, the resolution of a projection lens is given by

R= k₁ λ

NA (22)

where R is the resolution, k₁is the process parameter usually ranging from 0.5 to 1.0, λ is the operation wavelength of lithography process, and NA is the numerical aperture of the projectiopn lens. From this relationship, there are two possible methods to decrease the pitch size even further. Either increase the NA of the projection lens, or decrease the operation wavelength.

The NA of an existing optical system can be further enhanced by the method of immersion technique. Figure 33 shows a basic setup of the immersion method [29]. By submerging the imaging target in a medium of high refractive index, the NA of the projection can be increased above what is normally possible in air, by a factor almost equal to the refarctive index of the immersion medium.

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1.4.1 Extreme Ultraviolet Source

In lithography, constraints on the projection system comes from the desired requirements on the image (wafer) side. The illumination system then accomodates to the properties of the projection system accordingly, such that the constraints on the projection system can be met. And once the property of the exposure system is determined, the light source in question must then comply with the constraints on the exposure system.

Therefore, some considerations must be given regarding the properties of the source [30], since they will directly influence the quality of the image at the wafer end. Some points to consider are:

• Operation Wavelength

The wavelength of the light source. As previously mentioned, this affects the resolution limit directly.

• EUV Power

The power of the light source. This affects the throughput of the lithography process, as well as the sensitivity of the lithography process to noise.

• Hot Spot (Source) Size

The size of the source, together with the illuminator optics, influences the degree of coherence of the illumination.

• Collection Angle

The larger the collection angle, the more light from the source is collected and used.

• Pulse-to-Pulse Repeatability (Consistency) The fluctuation of the source output power.

• Debris Induced Component Lifetime (Damage and Contamination) Longer lifetime means less downtime, and less cost spent on maintainance.

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Influences on Lithography Operation

These properties of the lithography process influence the lithography operation in different ways.

• Process Throughput

Operation Wavelength, EUV power, Source Size, Collection Angle, Pulse-to-Pulse Repeatability

• CD (Imaging) Control Pulse-to-Pulse Repeatability

• Cost of Operation

Debris Induced Component Lifetime

Laser Produced Plasma

The schematic illustrated in Figure34shows the basic generation of EUV light [31]. First, stream of liquid xenon is injected from the top of the chamber. A high powered laser then heats the xenon to plasma state, which then emits EUV radiation. The stream of xenon stream is then collected and recirculates back to be reused.

Fast-Ion Mitigation

As a byproduct of LPP, the Xe ions from Xe¹⁺ up to Xe⁶⁺has been observed in the EUV generation process. Due to the high energy nature of the laser bombardment, these ions

Figure 34: Basic schematic of EUV generation.

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Figure 35: A mechanism to prevent (or reduce the amount of) byproducts of the EUV source from entering and contaminating the lithographic optical system. [3]

possess high kinetic energy, which cause contamination and damage to the multilayer coating when striking the collector mirror, and to the subsequent optical elements.

As such, preventive measures are necessary to eliminate or reduce the damage. Fig- ure35demonstrates an example of mechanism using a strong magnetic field to deflect the ions into a collector cup to prevent damage to the reflective mirror [3].

1.4.2 Mirror Optics

At the EUV wavelengths, absorption in almost all media (air included) is high. This leads to a drastic change in the lithography tool, in that conventional refractive optics must be abandoned for reflective systems made entirely of mirrors. Aside from that, some other consequences of EUV are that:

• The entire process must be performed in vacuum.

• Resolution enhancement techniques requiring transmission (e.g. phase shifting mask) cannot be used.

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Figure 36: Thermal expansion of Schott’s Zerodur.

• Immersion technique cannot be used.

Mirror Substrate

Due to the choice of operation wavelength at EUV, at the production of the mirrors one very important requirement or trait of the mirror substrate material is that the thermal expansion of the material must be small. In EUVL, the mirrors are under constant concen- trated illumination from a high energy source. This means that which causes the mirrors to heat up quickly.

Presently, from well known vendors, two such materials are available. ZERODUR^R from Schott, and ULE^R from Corning. Figures 36and37are thermal expansion curves for Schott’s ZERODUR^R glass [32] and Corning’s ULE^R (Ultra Low Expansion) glass [33]

respectively.

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Figure 37: Thermal expansion of Corning’s ULE.

Multilayer Coating

In EUVL, the illumination is highly absorbed in all materials, conventional refractive optics cannot be used, and the optical system must consist entirely of reflective optics. Due to the extremely short wavelength, uncoated mirrors still exhibit very poor reflectance.

As such, multiple layers of reflection coatings, typicall molybdenum (Mo) and silicon (Si) coating pairs, are employed. The Mo/Si multilayer acts as Bragg reflector, and is designed to operate at the EUV wavelength, at near-normal incidence angles. Figures38 and 38 shows the reflection spectrum and angular reflection distribution of a designed Mo/Si multilayer coating with 40 layer pairs, at EUV wavelengths. The Mo layer have thickness of 2.76 nm and Si layer of 4.14 nm. [4]

Reflection Depth Offset

There is, however, the issue of where the reflection occurs exactly. In many cases with optical coatings, the difference is minute due to the coating thickness being very small.

With the multilayer in EUVL however, where the the multilayer number easily exceeds

“a few”, the multilayer thickness becomes a serious concern. The 40 layer pairs of Mo/Si are of 2.76 and 4.14 nm thickness respectively, and totals up to 276 nm for the entire stack, more than 20 times longer than the wavelength of EUV. With the multilayers, in

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Figure 38: Reflection spectrum of the 40 Mo/Si multilayer coating. [4]

Figure 39: Anuglar reflection distribution of the 40 Mo/Si multilayer coating. [4]

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Figure 40: Candidates of the reflection depth offset. (a) Without the multilayers, the reflection occurs at the surface interface. (b) At the bottom of the multilayers. (c) At the top of the multilayers. (d) Somewhere inside the multilayers.

actuallity, reflection occurs at every multilayer interfaces, tapering off to zero as the light propagates further into the multilayers. This ambiguity is not useful for optical designers, there are simply not enough resources to allow the inter-mirror distances to split over a range values each time a mirror is encountered. There must be one reference point to say that this is where the reflection occured. Figure 40 shows some possible candidates of positions to define where the reflection occured.

Figure40(a) shows the situation where the multilayers are not present, in which case the reflection can be defined to be at the air-mirror interface. In Figure 40(b), in the presence of the multilayer stack, it seem illogical to use the multilayer-mirror interface at the bottom as the reference plane of reflection, that the reflection occurs only after propagation through the entire multilayer stack. In fact, at the bottom of the stack there are hardly any EUV radiations left to reflect. Likewise, in Figure40(c) the air-multilayer boundary is also illogical. Since the reflections occur over a range of depths, the first interface is unlikely to be representative of where the reflection occurs. Therefore, lastly in Figure40(d), the most appropriate representation seems to be somewhere between the

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Figure 41: The reflections of the EUV light over a multilayer stack. The center of the reflection lies inside the stack towards the air-multilayer interface.

top and bottom of the stack.

As each coating layer contributes to the total reflection, where the actual reflection occurs is ambiguous. In this situation, an effective reflection depth z_{e f f} can be best used to approximate the depth where the reflection occurs [34].

Multilayer Imperfections

In theoretical calculation, the Bragg reflector configuration of the Mo/Si multilayer coating has a theoretical limit of reflectivity of 75.5%, at the EUV spectrum [4]. However due to the multilayer imperfections, the measured reflectivity of the manufactured coating do not exceed 69%. Some of the imperfections are associated with the manufacturing, while some are inevitable physical phenomenons.

• Interdiffusion

This imperfection comes from a third intermixing zone formed at the interface between the Mo and Si layers due to interdiffusion. Insertion of thin barrier layer of B₄C or C can allow consistent production of multilayer coating of reflectivity of 70%.

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• Mirror Substrate Roughness

Manufacturing errors on the mirror profile leads to errors in the multilayer coating profile, and hence further impacts the total reflectivity.

Aside from optical coating, another affecting factor that impacts the mirror reflectivity in EUV lithography is the mirror roughness itself.

Marechal’s Criterion

Typical definition of diffraction limited optical system requires that the peak wavefront error OPD_peak be smaller than

OPD_Peak<1

4λ , (23)

this is known as Rayleigh’s criterion.

For EUV lithography, the requirement for precision is so high that Rayleigh’s criterion is not enough warrant performance. Another more useful difinition is Marechal’s criterion, which states that the RMS wavefront error OPD_rmsmust be contrained under

OPD_rms< 1

14λ . (24)

As an optical designer, it is easy to be tempted into thinking that more optical elements leads to better optical performance. However a consequence of Marechal’s criterion is that, since the total wavefront error is spread across the entire optical system, the allowable error on one particular element in a system with N elements is

OPD_individual< OPD_rms

N . (25)

As an example, for a 6 mirror EUVL projection system operating at 13.4 nm, the maximum allowable RMS error on the surface figure of each element is approximately 0.2 nm.

Increasing the number of elements means to further reduce the allowable RMS error on the mirror surface.

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Power Spectrum Density Analysis

The impacts of mirror surface roughness can be seen clearly in the spatial frequency domain, by taking a Fourier transform of the roughness profile. In general, the magnitude of the roughness at different frequencies affects the result in different ways.

• Low Spatial Frequency Roughness

LSFR impacts the overall shape of the mirror surface profile, and hence reduces the aerial image sharpness and worsens the resolution.

• Mid Spatial Frequency Roughness

Overall shape of the mirror is unaffected, but the MSFR causes increased scattering at the mirror surface, which results in flares in the optical system.

• High Spatial Frequency Roughness

HSFR at the scale of the illumination wavelength disrupts the constructive inter- ference mechanism that the multilayer coating relies on, and reduces the resultant reflectivity.

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1.5 EUV Lithographic Tool Design

1.5.1 Projector Design

As previouly mentioned, the optical lithography process encompasses many stages. Dur- ing the exposure stage, a mask of the desired pattern to be transferred is first illuminated by a light source, then a projector system picks up light from the mask, and images the pattern onto the photoresist coated on the wafer. This projection system is the heart of optical lithography, and is conventionally composed of many lenses, spanning well over a meter in length. The design and manufacturing of this projection system will largely determine the quality and performance of the lithography process, and its attainable resolution.

The designing of a lithography tool, portrayed in many textbooks as the pinnacle of imaging optical system design, the sheer number of optical elements required, coupled with the stringentness of the required imaging quality, all contributes to the overall design difficulty. Currently, many publications detailing such a design tend to focus mainly on the result of the design rather than the process of the design. While the design result of the lithography tool and its performance are undeniably important, the other equally important question is how to reach that design form in the first place, in keeping with a given set of optical constraints and requirements.

Unlike optical lithography tools in the previous deep ultraviolet (DUV) generation, the shorter wavelength of extreme ultra violet (EUV) exhibits high absorption in almost all propagation mediums. This absorption forces optical projection systems consisting mainly of lenses to be abandoned in favor of the reflective projection systems consisting entirely of mirrors. [35] As such, the optics undergoes a paradigm shift from mainly refractive systems to entirely reflective systems, and designers are forced outside the past comfort of the familiar refractive lens design. To address this, the hope of this study therefore, is to propose a method to analyze and design a completely reflective EUV lithography (EUVL) tool, using the mathematics of the Generalized Gaussian Constants (GGC) [36], based on the Gaussian Bracket [37].