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Page 1 Magnification/Reduction Numerical Aperture CHE323/CHE384 Chemical Processes for Micro- and Nanofabrication www.lithoguru.com/scientist/CHE323 Diffraction Review Lecture 43 • Diffraction is the propagation of light in the presence of Lithography: boundaries Projection Imaging, part 1 • In lithography, diffraction can be described by Fraunhofer diffraction - the diffraction pattern is the Fourier Transform Chris A. Mack of the mask transmittance function • Small patterns diffract more: frequency components of Adjunct Associate Professor the diffraction pattern are inversely proportional to dimensions on the mask Reading : Chapter 7, Fabrication Engineering at the Micro- and Nanoscale , 4 th edition, Campbell • For repeating patterns (such as a line/space array), the diffraction pattern becomes discrete diffracted orders • Information about the pitch is contained in the positions of the diffracted orders, and the amplitude of the orders determines the duty cycle (w/p) © 2013 by Chris A. Mack www.lithoguru.com © Chris Mack 2 Fourier Transform Examples Fourier Transform Properties g(x) Graph of g(x) G(f ) = x F{g ( x , y )} G(,) fx f y 1,x < 0.5 sin(π f ) rect() x = x > π 0,x 0.5 f x From Fundamental 1,x > 0 1 i Principles of Optical step() x = δ ()f − 0,x < 0 x π + = + Lithography , by 2 2 fx Linearity: F{(,)af x y bg( x , y )} aF(,) fx f y bG(,) fx f y Chris A. Mack (Wiley & Sons, 2007) Delta function δ ()x 1 − π + − − = i2 ( fx a f y b) Shift Theorem: F{(g x a, y b)} G(,) f f e ∞ ∞ x y ()=δ − δ − comb x ∑ ()x j ∑ ()fx j j =−∞ j =−∞ 1 1 1 1 π δ + +δ − f cos( x) fx fx = 1 fx y 2 2 2 2 Similarity: F{g ( ax , by )} G , ab a b i δ + 1 −i δ − 1 sin(πx) fx fx 2 2 2 2 ∞ −π 2 −π f 2 g(,)(ξ η h x − ξ, y −η)d ξ d η = G(,)(,) f f H f f Gaussian e x e x Convolution: F ∫ ∫ x y x y 1,r < 1 J (2πρ ) −∞ = 1 circ() r πρ 0,r > 1 ρ =2 + 2 r= x2 + y 2 fx f y © Chris Mack 3 © Chris Mack 4 Numerical Aperture F - Number Aperture Object Plane α a The F/# or F-Stop is used by cameras as an alternative to the Numerical Aperture NA = n sin ααα b b 1 n = index of refraction of media (air in this case) F/# = = ααα = maximum half-angle of light making it through the lens a 2NA © Chris Mack 5 © Chris Mack 6 Page 1 Magnification/Reduction Numerical Aperture • Given the reduction ratio R: • Graphing the diffraction pattern with the aperture: n sin θ = nR sin θ w w m m Amplitude Refractive index on wafer side = nw θθθ θθθ m w − 3 3 p p Refractive index 2 1 1 2 f on mask side = nm − − x Entrance Aperture Exit p p 0 p p Pupil Stop Pupil • Entrance Pupil - the image of the aperture stop as seen from the front of the lens Lens Opening (Aperture) Radius = NA/ λ • Exit Pupil - the image of the aperture stop as seen from the back of the lens © Chris Mack 7 © Chris Mack 8 Numerical Aperture Numerical Aperture Mask Mask Objective Lens Aperture Top down view: -1st 0th +1 st 0th Order 0th Order ±1st Order Lens ±1st Order Lens © Chris Mack 9 © Chris Mack 10 Forming an Image Fourier Optics • The objective lens focuses the diffraction orders onto the • The objective lens produces an image equal to wafer the Fourier transform of the light entering the +1 st lens. • The combination of diffraction followed by the imaging lens produces a Fourier transform of a Oth Fourier transform, which gives back the original pattern. -1st • However, the objective lens cuts off the high spatial frequencies of the diffraction pattern. • The image plane is defined as the plane where all diffraction orders arrive “in phase” © Chris Mack 11 © Chris Mack 12 Page 2 Fourier Optics Fourier Optics • The objective lens is described by its pupil • Given a diffraction pattern Tm and a pupil function function , P(f x,f y). For an ideal lens, this function P, the light which makes it into the lens is just just describes the size of the aperture. PT m. • The lens then takes the inverse Fourier transform ,1 when f 2 + f 2 ≤ NA / λ of this light to give the electric field of the image, = x y P(, fx f y ) E. ,0 otherwise =F −1{ } E(,) x y PT m I(,) x y = E(,) x y 2 © Chris Mack 13 © Chris Mack 14 Lecture 43: What have we Learned? • Define “numerical aperture” • What are the entrance and exit pupils of an imaging lens? • How can one determine which diffraction orders pass through an imaging lens? • What is the pupil function of a lens? • Explain the concept of Fourier Optics © Chris Mack 15 Page 3.
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