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 yxg )},({ ffG yx ),(  x < 5.0,1 π f )sin( xrect )( = x  > π  x 5.0,0 f x

From Fundamental  x > 0,1 1 i Principles of Optical xstep )( =  δ ()f − x < 0,0 x π + = + Lithography , by  22 fx Linearity: F ),({ yxbgyxaf )},( ffaF yx ),( ffbG yx ),( Chris A. Mack (Wiley & Sons, 2007) Delta function δ x)( 1 − π + − − = (2 yx bfafi ) Shift Theorem: F ({ , byaxg )} ),( effG ∞ ∞ yx () δ −= δ − xcomb ∑ jx )( ∑ x jf )( j −∞= j −∞=

1  1  1  1  π δ + δ −+  f  cos( x)  fx   fx  = 1  fx y  2  2  2  2  Similarity: F byaxg )},({ G ,  ab  a b      i δ + 1 i δ −− 1 sin(πx)  fx   fx  2   22  2 

 ∞    −π 2 −π f 2  g ηξ (),( xh − ξ, y − ) dd ηξη  = ffHffG ),(),( Gaussian e x e x Convolution: F ∫ ∫ yx yx    < πρ  ∞−  = r 1,1 J1 )2( rcirc )(  πρ  r > 1,0 ρ += 22 += yxr 22 ff yx © 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

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

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”

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 ,( ffP yx )  E.  ,0 otherwise =F −1{ } yxE ),( PT m

yxI ),( = yxE ),( 2

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

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