Polarization Aberrations: A Comparison of Various Representations Greg McIntyre,a,b Jongwook Kyeb, Harry Levinsonb and Andrew Neureuthera a EECS Department, University of California- Berkeley, Berkeley, CA 94720 b Advanced Micro Devices, One AMD Place, Sunnyvale, CA 94088-3453 FLCC Seminar 31 October 2005

Purpose & Outline Purpose Outline What is polarization, why is it important Polarization aberrations: Various representations Physical properties Mueller matrix – pupil Jones matrix – pupil Pauli-spin matrix – pupil Others (Ein vs. Eout, coherence- & covariance - pupil) Preferred representation Proposed simulation flow & example Causality, reciprocity, differential Jones matrices Outline : to compare multiple representations and propose a common ‘language’ to describe polarization aberrations for optical lithography

What is polarization? Pure polarization states e- Ey,out ei Vector representation in x y plane Ex,out ei Ey,out ei y,out x,out Pure polarization states e- Partially polarized light = superposition of multiple pure states Polarization is an expression of the orientation of the lines of electric flux in an electromagnetic field. It can be constant or it can change either gradually or randomly. Linear Circular Elliptical Oscillating electron Propagating EM wave Polarization state

Why is polarization important in optical lithography? Z component of E-field introduced at High NA from radial pupil component decreases image contrast  z  Low NA High NA x y  mask TM TE = ETM NA Ez = ETM sin()  Z-component negligible  wafer Increasing NA

Scanner vendors are beginning to engineer polarization states in illuminator? Choice of illumination setting depends on features to be printed. ASML, Bernhard (Immersion symposium 2005) Polarization orientation TE Purpose: To increase exposure latitude (better contrast) by minimizing TM polarization

Polarization and immersion work together for improved imaging Immersion lithography can increase depth of focus Dry Wet l a liquid resist resist NA = .95 = sin(a) NA = .95 = nl sin(l) a = 71.8 l ~ 39.3 Depth of focus

Polarization and immersion work together for improved imaging Immersion lithography can also enable hyper-NA tools (thus smaller features) Total internal reflection prevents imaging NA = nl sin(l) > 1 l liquid resist Last lens element air Dry Wet Minimum feature

Immersion increases DOF and/or decreases minimum feature Polarization increases exposure latitude (better contrast) Wet Dry NA=0.95, Dipole 0.9/0.7, 60nm equal L/S (simulation) Polarization is needed to take full advantage of immersion benefits Dry, unpolarized Dry, polarized Wet, unpolarized Wet, polarized

Thus, polarization state is important Thus, polarization state is important. But there are many things that can impact polarization state as light propagates through optical system. Mask polarization effects Illuminator polarization design Polarization aberrations of projection optics Source polarization Wafer / Resist

Polarization Aberrations

Traditional scalar aberrations Scalar diffraction theory: Each pupil location characterized by a single number (OPD) defocus astigmatism coma Optical Path Difference Ein ei in Eout ei out a: illumination frequency Typically defined in Zernike’s

Polarization aberrations Subtle polarization-dependent wavefront distortions cause intricate (and often non-intuitive) coupling between complex electric field components Ex,in ei x,in Ex,out ei Ey,in ei y,in Ey,out ei y,out x,out Each pupil location no longer characterized by a single number

Changes in polarization state Diattenuation: Retardance: Degrees of Freedom: Magnitude Eigenpolarization orientation Eigenpolarization ellipticity E x y E' attenuates eigenpolarizations differently (partial polarizer) shifts the phase of eigenpolarizations differently (wave plate)

Sample pupil (physical properties) However, this format is inconvenient for understanding the impact on imaging inconvenient as an input format for simulation Apodization Scalar aberration Total representation has 8 degrees of freedom per pupil location diattenuation retardance

Mueller-pupil

Mueller Matrix - Pupil Consider time averaged intensities Sin Sout H V Sin Sout Stokes vector completely characterizes state of polarization PH = flux of light in H polarization Mueller matrix defines coupling between Sin and Sout

Mueller Matrix - Pupil Recast polarization aberration into Mueller pupil Mueller Pupil 16 degrees of freedom per pupil location m02,m20: 45-135 Linear diattenuation m01,m10: H-V Linear diattenuation m03,m30: Circular diattenuation m13,m31: 45-135 Linear retardance m12,m21: H-V Linear retardance m23,m32: Circular retardance

Polarization-dependent depolarization Mueller Matrix - Pupil Stokes vector represented as a unit vector on the Poincare Sphere Meuller Matrix maps any input Stokes vector (Sin) into output Stokes vector (Sout) Right Circular Left Circular 45 135 Linear S S’ The extra 8 degrees of freedom specify depolarization, how polarized light is coupled into unpolarized light Polarization-dependent depolarization Represented by warping of the Poincare’s sphere Chipman, Optics express, v.12, n.20, p.4941, Oct 2004 Uniform depolarization

Mueller Matrix - Pupil Advantages: Disadvantages: accounts for all polarization effects depolarization non-reciprocity intensity formalism measurement with slow detectors difficult to interpret loss of phase information not easily compatible with imaging equations hard to maintain physical realizability Generally inconvenient for partially coherent imaging

Jones-pupil

Jones Matrix - Pupil Consider instantaneous fields: Ex,in ei Ey,in ei y,in Ex,out ei Ey,out ei y,out Elements are complex, thus 8 degrees of freedom Jones vector Jones matrix Mask diffracted fields High-NA & resist effects Lens effect Jones Pupil x,out a: illumination frequency Vector imaging equation:

Jones Matrix - Pupil Mask coordinate system (x,y) i.e. Jxy = coupling between input x and output y polarization fields Jxx(mag) Jxy(mag) Jyx(mag) Jyy(mag) Jxx(phase) Jxy(phase) Jyx(phase) Jyy(phase) Mask coordinate system (x,y) x y Jtete(mag) Jtetm(mag) Jtmte(mag) Jtmtm(mag) Jtete(phase) Jtetm(phase) Jtmte(phase) Jtmtm(phase) Pupil coordinate system (te,tm) TM TE

Zernike coefficients (An,m) Jones Matrix - Pupil Jxx (real) Jxx (imag) Jxy (real) Jxy (imag) Jyx (real) Jyx (imag) Jyy (real) Jyy (imag) Decomposition into Zernike polynomials Annular Zernike polynomials (or Zernikes weighted by radial function) might be more useful Lowest 16 zernikes => 128 degrees of freedom for pupil Zernike coefficients (An,m) real imaginary Similar to Totzeck, SPIE 05

Pauli-pupil

Pauli-spin Matrix - Pupil Decompose Jones Matrix into Pauli-spin matrix basis mag(a0) phase(a0) real(a1/a0) real(a2/a0) real(a3/a0) imag(a1/a0) imag(a2/a0) imag(a3/a0)

Meaning of the Pauli-Pupil mag(a0) phase(a0) real(a1/a0) real(a2/a0) real(a3/a0) imag(a1/a0) imag(a2/a0) imag(a3/a0) Scalar transmission (Apodization) & normalization constant for diattenuation & retardance Diattenuation along x & y axis Diattenuation along 45  & 135 axis Circular Diattenuation Scalar phase (Aberration) Retardance along x & y axis Retardance along 45  & 135 axis Circular Retardance

traditional scalar phase Diattenuation effects Usefulness of Pauli-Pupil to Lithography Pupil can be specified by only: a2 (complex) a1 traditional scalar phase Diattenuation effects Retardance effects |a0| calculated to ensure physically realizable pupil assuming: no scalar attenuation eigenpolarizations are linear

The advantage of Pauli-Pupils Jxx(mag) Jxy(mag) Jyx(mag) Jyy(mag) Jxx(phase) Jxy(phase) Jyx(phase) Jyy(phase) Jones Pauli 8 coupled pupil functions (easy to create unrealizable pupil) 128 Zernike coefficients not very intuitive fits imaging equations 4 independent pupil functions (scalar effects considered separately) 64 Zernike coefficients physically intuitive easily converted to Jones for a1 real a1 imag a2 real a2 imag imaging equations

Proposed simulation flow (to determine polarization aberration specifications and tolerances) Input: a1, a2, scalar aberration Convert to Jones Pupil Simulate Calculate a0

Simulation example Monte Carlo simulation done with Panoramic software and Matlab API to determine variation in image due to polarization aberrations Polarization monitor Resist image Intensity at center is polarization-dependent signal Simulate many randomly generated Pauli-pupils to determine how polarization aberrations affect signal Example: polarization monitor (McIntyre, SPIE 05) Center intensity change (%CF) iteration Signal variation

A word of caution… Jinstrument This analysis is based on the “Instrumental Jones Matrix” Ein Eout Jinstrument Apodization Aberration Magnitude Orientation Ellipticity of eignpolarization “Instrumental parameters”

Constraints of Causality & Reciprocity Ein Eout JA JC JD JF JB JE Reciprocity: time reversed symmetry (“parameters of element A”) Causality: polarization state can not depend on future states (order dependent) (except in presence of magnetic fields)

Differential Jones Matrix N = differential Jones N= generalized propagation vector (homogeneous media) Wave Equation: General solution Also: = dielectric tensor EM Theory: symmetric Anti-symmetric

Differential Jones Matrix Assumed real(ai) => dichroic property & imag(ai) => birefringent property Barakat (1996): Contradiction resolved for small values of polarization effects Jones' assumption was wrong

Other representations

E-field test representation X Y 45 rcp TE TM Output electric field, given input polarization state Color degree of circular polarization

Intensity test representation X Y 45 rcp TE TM Output intensity, given input polarization state

Covariance & Coherency Matrix Covariance Matrix (C) Coherency Matrix (T) Trace describes average power transmitted Kt1 (mag) Kt2 (mag) Kt3 (mag) Kt1 (phase) Kt2 (phase) Kt3 (phase) Assumes reciprocity (Jxy = Jyx) Power Convenient with partially polarized light (similar to Jones-pupil) (similar to Pauli-pupil)

Additional comments on polarization in lithography Different mathematics convenient with different aspects of imaging Source, mask Stokes vector Lenses Jones vector Each vendor uses different terminology Initially, source and mask polarization effects will be most likely source of error

Conclusion Polarization is becoming increasingly important in lithography Compared various representations of polarization aberrations & proposed Pauli-pupil as ‘language’ to describe them Proposed simulation flow and input format Multiple representations of same pupil help to understand complex and non-intuitive effects of polarization aberrations