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Polarization Aberrations: A Comparison of Various Representations

Polarization Aberrations: A Comparison of Various Representations. Greg McIntyre , a,b Jongwook Kye b , Harry Levinson b and Andrew Neureuther a. a EECS Department, University of California- Berkeley, Berkeley, CA 94720 b Advanced Micro Devices, One AMD Place, Sunnyvale, CA 94088-3453.

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Polarization Aberrations: A Comparison of Various Representations

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  1. 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

  2. : to compare multiple representations and propose a common ‘language’ to describe polarization aberrations for optical lithography 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 Purpose & Outline Purpose

  3. What is polarization? 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. • Pure polarization states Oscillating electron Polarization state Propagating EM wave e- Ey,out ei y,out Vector representation in x y plane Ex,out ei x,out Linear Circular Elliptical • Partially polarized light = superposition of multiple pure states

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

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

  6. Polarization and immersion work together for improved imaging Depth of focus 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

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

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

  9. 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

  10. Polarization Aberrations

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

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

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

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

  15. Mueller-pupil

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

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

  18. Right Circular S • Stokes vector represented as a unit vector on the Poincare Sphere Linear 45 135 0 • Meuller Matrix maps any input Stokes vector (Sin) into output Stokes vector (Sout) S’ Left Circular • The extra 8 degrees of freedom specify depolarization, how polarized light is coupled into unpolarized light Represented by warping of the Poincare’s sphere Polarization-dependent depolarization Uniform depolarization Chipman, Optics express, v.12, n.20, p.4941, Oct 2004 Mueller Matrix - Pupil

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

  20. Jones-pupil

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

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

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

  24. Pauli-pupil

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

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

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

  28. Jones Pauli Jxx(mag) Jxy(mag) Jxx(phase) Jxy(phase) • 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 Jyx(mag) Jyy(mag) Jyx(phase) Jyy(phase) imaging equations a1 imag a1 real a2 imag a2 real The advantage of Pauli-Pupils

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

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

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

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

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

  34. Jones (1947): Assumed real(ai) => dichroic property & imag(ai) => birefringent property Barakat (1996): Jones' assumption was wrong Contradiction resolved for small values of polarization effects Differential Jones Matrix

  35. Other representations

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

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

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

  39. 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 Additional comments on polarization in lithography

  40. 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

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