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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 FLCC Purpose & Outline Purpose: 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 2 McIntyre, FLCC, 10/31/05 FLCC 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 eVector representation in x y plane Ey,out eiy,out Linear Circular Elliptical Ex,out ei x,out • Partially polarized light = superposition of multiple pure states 3 McIntyre, FLCC, 10/31/05 FLCC Why is polarization important in optical lithography? z x High NA Low NA Z component of E-field introduced at High NA from radial pupil component decreases image contrast mask TM y TE Z-component negligible Ez = ETM sin() = ETM NA wafer 4 Increasing NA McIntyre, FLCC, 10/31/05 FLCC 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 5 McIntyre, FLCC, 10/31/05 FLCC 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) a = 71.8 Depth of focus NA = .95 = nl sin(l) l ~ 39.3 2n(1 - cos( )) 6 McIntyre, FLCC, 10/31/05 FLCC Polarization and immersion work together for improved imaging Immersion lithography can also enable hyper-NA tools (thus smaller features) Dry Wet Last lens element Last lens element l liquid air resist resist Total internal reflection prevents imaging NA = nl sin(l) > 1 Minimum feature k 1 NA 7 McIntyre, FLCC, 10/31/05 FLCC 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) 8 McIntyre, FLCC, 10/31/05 FLCC 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 9 McIntyre, FLCC, 10/31/05 FLCC Polarization Aberrations 10 McIntyre, FLCC, 10/31/05 FLCC Traditional scalar aberrations Scalar diffraction theory: Each pupil location characterized by a single number (OPD) EWafer ( x' , y' , a ) 1 2 1 ik cos x ' sin y ' ik ( , ) E ( , , a ) e e dd Diff 0 0 Optical Path Difference Typically defined in Zernike’s n Ein eiin ( , ) Eout eiout A n , m Z n, m ( , ) n 1 m 0 defocus astigmatism coma a: illumination frequency 11 McIntyre, FLCC, 10/31/05 FLCC Polarization aberrations Subtle polarization-dependent wavefront distortions cause intricate (and often non-intuitive) coupling between complex electric field components Ey,in ei y,in Ey,out eiy,out Ex,out ei x,out Ex,in eix,in Each pupil location no longer characterized by a single number 12 McIntyre, FLCC, 10/31/05 FLCC Changes in polarization state Diattenuation: attenuates Retardance: shifts the phase of eigenpolarizations differently (partial polarizer) eigenpolarizations differently (wave plate) Ey Ey E'y Degrees of Freedom: Degrees of Freedom: • Magnitude • Eigenpolarization orientation E'x Ex E'x Ex E'y • Magnitude • Eigenpolarization orientation •Eigenpolarization ellipticity 13 McIntyre, FLCC, 10/31/05 •Eigenpolarization ellipticity FLCC Sample pupil (physical properties) Total representation has 8 degrees of freedom per pupil location Apodization diattenuation Scalar aberration retardance However, this format is • inconvenient for understanding the impact on imaging • inconvenient as an input format for simulation 14 McIntyre, FLCC, 10/31/05 FLCC Mueller-pupil 15 McIntyre, FLCC, 10/31/05 FLCC Mueller Matrix - Pupil Consider time averaged intensities Stokes vector completely characterizes state of polarization V V H H Sin Sout s0 PH PV P P s H V 1 S s2 P45 P135 s3 PR PL PH = flux of light in H polarization Mueller matrix defines coupling between Sin and Sout Sout MS in m00 m M 10 m20 m30 m01 m02 m11 m12 m21 m 22 m31 m32 m03 m13 m23 m33 16 McIntyre, FLCC, 10/31/05 FLCC Mueller Matrix - Pupil Recast polarization aberration into Mueller pupil Mueller Pupil m00 m M 10 m20 m30 m01 m02 m11 m12 m21 m 22 m31 m32 m03 m13 m23 m33 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 17 McIntyre, FLCC, 10/31/05 FLCC Right Circular Mueller Matrix - Pupil S • Stokes vector represented as a unit vector on the Poincare Sphere 135 45 0 • Meuller Matrix maps any input Stokes vector (Sin) into output Stokes vector (Sout) Sout MS in Linear 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 18 McIntyre, FLCC, 10/31/05 FLCC Mueller Matrix - Pupil 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 19 McIntyre, FLCC, 10/31/05 FLCC Jones-pupil 20 McIntyre, FLCC, 10/31/05 FLCC Jones Matrix - Pupil Consider instantaneous fields: Ey,in ei y,in Ex,in eix,in Ey,out eiy,out Ex,out eix,out E x ,out e i x ,out J xx i y ,out E y ,out e J yx Jones vector J xy E x ,ine i x ,in J yy E y ,ine i y ,in Jones matrix Elements are complex, thus 8 degrees of freedom Vector imaging equation: Fxx E x 2 1 1 E ( x ' , y ' , a , Pol ) F y 0 0 yx Fzx E z Wafer a: illumination frequency Fxy J xx F yy J yx Fzy High-NA & resist effects J xy E x J yy E y Jones Pupil e ik cos x ' sin y ' dd Diff Mask diffracted fields 21 McIntyre, FLCC, 10/31/05 Lens effect FLCC Jones Matrix - Pupil Jxx(mag) Jxy(mag) Jxx(phase) Jxy(phase) Jyx(mag) Jyy(mag) Jyx(phase) Jyy(phase) Mask coordinate system (x,y) y x 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) TE TM Jtmte(mag) Jtmtm(mag) Jtmte(phase) Jtmtm(phase) 22 McIntyre, FLCC, 10/31/05 FLCC Jones Matrix - Pupil Decomposition into Zernike polynomials • Lowest 16 zernikes => 128 degrees of freedom for pupil Zernike coefficients (An,m) Jxx (real) Jxx (imag) real imaginary Jxy (real) Jxy (imag) n ( , ) An ,m Z n ,m ( , ) n 1 m 0 •Annular Zernike polynomials (or Zernikes weighted by radial function) might be more useful Jyx (real) Jyx (imag) Jyy (real) Jyy (imag) 23 Similar to Totzeck, SPIE 05 McIntyre, FLCC, 10/31/05 FLCC Pauli-pupil 24 McIntyre, FLCC, 10/31/05 FLCC Pauli-spin Matrix - Pupil Decompose Jones Matrix into Pauli-spin matrix basis J ( H , , ) a0 0 a1 1 a2 2 a3 3 1 0 0 0 1 1 0 1 0 1 0 1 2 1 0 0 i 3 i 0 a0 a1 a2 a3 J xx J yy mag(a0) phase(a0) real(a1/a0) imag(a1/a0) real(a2/a0) imag(a2/a0) real(a3/a0) imag(a3/a0) 2 J xx J yy 2 J xy J yx 2 J xy J yx 2i 25 McIntyre, FLCC, 10/31/05 FLCC Meaning of the Pauli-Pupil Scalar transmission (Apodization) & normalization constant for diattenuation & retardance mag(a0) phase(a0) Scalar phase (Aberration) imag(a1/a0) real(a1/a0) Diattenuation along x & y axis Retardance along x & y axis real(a2/a0) imag(a2/a0) Diattenuation along 45 & 135 axis Retardance along 45 & 135 axis imag(a3/a0) real(a3/a0) Circular Diattenuation Circular Retardance 26 McIntyre, FLCC, 10/31/05 FLCC Usefulness of Pauli-Pupil to Lithography Pupil can be specified by only: traditional scalar phase a1 Diattenuation effects (complex) a2 Retardance effects (complex) |a0| calculated to ensure physically realizable pupil assuming: • no scalar attenuation • eigenpolarizations are linear 27 McIntyre, FLCC, 10/31/05 FLCC The advantage of Pauli-Pupils Jones Pauli • 8 coupled pupil functions • 4 independent pupil functions (scalar effects considered separately) (easy to create unrealizable pupil) • 64 Zernike coefficients • physically intuitive • easily converted to Jones for imaging equations • 128 Zernike coefficients • not very intuitive • fits imaging equations Jxx(mag) Jxy(mag) Jxx(phase)Jxy(phase) Jyx(mag) Jyy(mag) Jyx(phase)Jyy(phase) a1 real a1 imag a2 real a2 imag 28 McIntyre, FLCC, 10/31/05 FLCC Proposed simulation flow (to determine polarization aberration specifications and tolerances) Input: a1, a2, scalar aberration Calculate a0 Simulate Convert to Jones Pupil J ( H , , ) a0 0 a1 1 a2 2 a3 3 29 McIntyre, FLCC, 10/31/05 FLCC Simulation example 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) Polarization monitor Resist image Intensity at center is polarization-dependent signal Center intensity change (%CF) Simulate many randomly generated Pauli-pupils to determine how polarization aberrations affect signal 0.05 Signal variation 0.04 0.03 0.02 0.01 0 -0.01 0 50 100 -0.02 -0.03 -0.04 30 McIntyre, FLCC, 10/31/05 iteration FLCC 150 A word of caution… This analysis is based on the “Instrumental Jones Matrix” Ein Jinstrument Eout J a0 0 a1 1 a2 2 a3 3 a0 [1 0 J scalar real (a3 ) imag (a3 ) real (a1 ) real (a2 ) imag (a1 ) imag (a2 ) 1 2 3 ] [1 0 i 1 i 2 i 3] a0 a0 a0 a0 a0 a0 •Apodization •Aberration J diattenuation • Magnitude • Orientation • Ellipticity of eignpolarization J retardance • Magnitude • Orientation • Ellipticity of eignpolarization “Instrumental parameters” 31 McIntyre, FLCC, 10/31/05 FLCC Constraints of Causality & Reciprocity JA JC JB Ein Eout JE JD JF polarization state can not depend on future states (order dependent) Causality: J J F J E J D JC J B J A J A a0, A 0 a1, A 1 a2, A 2 a3, A 3 (“parameters of element A”) Reciprocity: time reversed symmetry (except in presence of magnetic fields) 32 McIntyre, FLCC, 10/31/05 FLCC Differential Jones Matrix J 0 , z J z ,z ' J z ', z '' Wave Equation: E KE 0 2 z 2 z z' Ez' J z ,z' Ez J J z 1 N lim z ' Jz z z' z' z J NJ z N = differential Jones J e Nz General solution Also: NE E z K 2 K N2 N= generalized propagation vector (homogeneous media) EM Theory: D E QE (G) E symmetric = dielectric tensor Anti-symmetric xx yx zx xy xz yy yz zy zz N K Q, G 33 McIntyre, FLCC, 10/31/05 FLCC Differential Jones Matrix Jones (1947): N a0 0 a1 1 a2 2 a3 3 Assumed real(ai) => dichroic property & imag(ai) => birefringent property Barakat (1996): Jones' assumption was wrong J e Nz N e0 0 e1 1 e2 2 e3 3 real (ei ) dichroic imag (ei ) diattenuation Contradiction resolved for small values of polarization effects e x 1 x x 2 ... e i a i , e0 a 0 1 34 McIntyre, FLCC, 10/31/05 FLCC Other representations 35 McIntyre, FLCC, 10/31/05 FLCC E-field test representation Output electric field, given input polarization state X Y 45 rcp TE TM Color degree of circular polarization 36 McIntyre, FLCC, 10/31/05 FLCC Intensity test representation Output intensity, given input polarization state X Y 45 rcp TE TM 37 McIntyre, FLCC, 10/31/05 FLCC Covariance & Coherency Matrix Covariance Matrix (C) C kC kC J xx kC 2 J xy J yy J xx J yy 1 kt J xx J yy 2 2 J xy Coherency Matrix (T) T kt kt Kt1 (mag) Kt2 (mag) Kt3 (mag) Kt1 (mag) Kt2 (mag) Kt3 (mag) Kt1 (phase) Kt2 (phase) Kt3 (phase) Kt1 (phase) Kt2 (phase) Kt3 (phase) (similar to Pauli-pupil) (similar to Jones-pupil) Power • Assumes reciprocity (Jxy = Jyx) • Convenient with partially polarized light • Trace describes average power transmitted 38 McIntyre, FLCC, 10/31/05 FLCC 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 39 McIntyre, FLCC, 10/31/05 FLCC 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 40 McIntyre, FLCC, 10/31/05 FLCC