Abstract

In this paper, a new methodology is presented to derive the aberration state of a lithographic projection system from wafer metrology data. For this purpose, new types of phase-shift gratings (PSGs) are introduced, with special features that give rise to a simple linear relation between the PSG image displacement and the phase aberration function of the imaging system. By using the PSGs as the top grating in a diffraction-based overlay stack, their displacement can be measured as an overlay error using a standard wafer metrology tool. In this way, the overlay error can be used as a measurand based on which the phase aberration function in the exit pupil of the lithographic system can be reconstructed. In practice, the overlay error is measured for a set of different PSG targets, after which this information serves as input to a least-squares optimization problem that, upon solving, provides estimates for the Zernike coefficients describing the aberration state of the lithographic system. In addition to a detailed method description, this paper also deals with the additional complications that arise when the method is implemented experimentally and this leads to a number of model refinements and a required calibration step. Finally, the overall performance of the method is assessed through a number of experiments in which the aberration state of the lithographic system is intentionally detuned and subsequently estimated by the new method. These experiments show a remarkably good agreement, with an error smaller than 5mλ, among the requested aberrations, the aberrations measured by the on-tool aberration sensor, and the results of the new wafer-based method.

© 2014 Optical Society of America

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References

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  1. H. Nomura, “New phase-shift gratings for measuring aberrations,” Proc. SPIE 4346, 25–35 (2001).
    [CrossRef]
  2. M. A. van de Kerkhof, W. de Boeij, H. Kok, M. Silova, J. Baselmans, and M. Hemerik, “Full optical column characterization of DUV lithographic projection tools,” Proc. SPIE 5377, 1960–1970 (2004).
    [CrossRef]
  3. C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, 1976).
  4. P. Vanoppen and T. Theeuwes, “Lithographic scanner stability improvements through advanced metrology and control,” Proc. SPIE 7640, 764010 (2010).
    [CrossRef]
  5. M. Ebert, H. Cramer, W. Tel, M. Kubis, and H. Megens, “Combined overlay, focus and CD metrology for leading edge lithography,” Proc. SPIE 7973, 797311 (2011).
    [CrossRef]
  6. T. Hastie, R. Tibshirani, and J. J. H. Friedman, The Elements of Statistical Learning (Springer, 2001), Vol. 1.
  7. M. Pisarenco and I. Setija, “Compact discrepancy and chi-squared principles for over-determined inverse problems” submitted to Inverse Methods.
  8. A. J. E. M. Janssen, “Computation of Hopkins’ 3-circle integrals using Zernike expansions,” J. Eur. Opt. Soc. Rapid Pub. 6, 11059 (2011).
    [CrossRef]
  9. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  10. A. J. E. M. Janssen, “A generalization of the Zernike circle polynomials for forward and inverse problems in diffraction theory,” arXiv:1110.2369v1 (2011).

2011 (2)

M. Ebert, H. Cramer, W. Tel, M. Kubis, and H. Megens, “Combined overlay, focus and CD metrology for leading edge lithography,” Proc. SPIE 7973, 797311 (2011).
[CrossRef]

A. J. E. M. Janssen, “Computation of Hopkins’ 3-circle integrals using Zernike expansions,” J. Eur. Opt. Soc. Rapid Pub. 6, 11059 (2011).
[CrossRef]

2010 (1)

P. Vanoppen and T. Theeuwes, “Lithographic scanner stability improvements through advanced metrology and control,” Proc. SPIE 7640, 764010 (2010).
[CrossRef]

2004 (1)

M. A. van de Kerkhof, W. de Boeij, H. Kok, M. Silova, J. Baselmans, and M. Hemerik, “Full optical column characterization of DUV lithographic projection tools,” Proc. SPIE 5377, 1960–1970 (2004).
[CrossRef]

2001 (1)

H. Nomura, “New phase-shift gratings for measuring aberrations,” Proc. SPIE 4346, 25–35 (2001).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Baselmans, J.

M. A. van de Kerkhof, W. de Boeij, H. Kok, M. Silova, J. Baselmans, and M. Hemerik, “Full optical column characterization of DUV lithographic projection tools,” Proc. SPIE 5377, 1960–1970 (2004).
[CrossRef]

Cramer, H.

M. Ebert, H. Cramer, W. Tel, M. Kubis, and H. Megens, “Combined overlay, focus and CD metrology for leading edge lithography,” Proc. SPIE 7973, 797311 (2011).
[CrossRef]

de Boeij, W.

M. A. van de Kerkhof, W. de Boeij, H. Kok, M. Silova, J. Baselmans, and M. Hemerik, “Full optical column characterization of DUV lithographic projection tools,” Proc. SPIE 5377, 1960–1970 (2004).
[CrossRef]

Ebert, M.

M. Ebert, H. Cramer, W. Tel, M. Kubis, and H. Megens, “Combined overlay, focus and CD metrology for leading edge lithography,” Proc. SPIE 7973, 797311 (2011).
[CrossRef]

Friedman, J. J. H.

T. Hastie, R. Tibshirani, and J. J. H. Friedman, The Elements of Statistical Learning (Springer, 2001), Vol. 1.

Hastie, T.

T. Hastie, R. Tibshirani, and J. J. H. Friedman, The Elements of Statistical Learning (Springer, 2001), Vol. 1.

Hemerik, M.

M. A. van de Kerkhof, W. de Boeij, H. Kok, M. Silova, J. Baselmans, and M. Hemerik, “Full optical column characterization of DUV lithographic projection tools,” Proc. SPIE 5377, 1960–1970 (2004).
[CrossRef]

Janssen, A. J. E. M.

A. J. E. M. Janssen, “Computation of Hopkins’ 3-circle integrals using Zernike expansions,” J. Eur. Opt. Soc. Rapid Pub. 6, 11059 (2011).
[CrossRef]

A. J. E. M. Janssen, “A generalization of the Zernike circle polynomials for forward and inverse problems in diffraction theory,” arXiv:1110.2369v1 (2011).

Kittel, C.

C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, 1976).

Kok, H.

M. A. van de Kerkhof, W. de Boeij, H. Kok, M. Silova, J. Baselmans, and M. Hemerik, “Full optical column characterization of DUV lithographic projection tools,” Proc. SPIE 5377, 1960–1970 (2004).
[CrossRef]

Kubis, M.

M. Ebert, H. Cramer, W. Tel, M. Kubis, and H. Megens, “Combined overlay, focus and CD metrology for leading edge lithography,” Proc. SPIE 7973, 797311 (2011).
[CrossRef]

Megens, H.

M. Ebert, H. Cramer, W. Tel, M. Kubis, and H. Megens, “Combined overlay, focus and CD metrology for leading edge lithography,” Proc. SPIE 7973, 797311 (2011).
[CrossRef]

Nomura, H.

H. Nomura, “New phase-shift gratings for measuring aberrations,” Proc. SPIE 4346, 25–35 (2001).
[CrossRef]

Pisarenco, M.

M. Pisarenco and I. Setija, “Compact discrepancy and chi-squared principles for over-determined inverse problems” submitted to Inverse Methods.

Setija, I.

M. Pisarenco and I. Setija, “Compact discrepancy and chi-squared principles for over-determined inverse problems” submitted to Inverse Methods.

Silova, M.

M. A. van de Kerkhof, W. de Boeij, H. Kok, M. Silova, J. Baselmans, and M. Hemerik, “Full optical column characterization of DUV lithographic projection tools,” Proc. SPIE 5377, 1960–1970 (2004).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Tel, W.

M. Ebert, H. Cramer, W. Tel, M. Kubis, and H. Megens, “Combined overlay, focus and CD metrology for leading edge lithography,” Proc. SPIE 7973, 797311 (2011).
[CrossRef]

Theeuwes, T.

P. Vanoppen and T. Theeuwes, “Lithographic scanner stability improvements through advanced metrology and control,” Proc. SPIE 7640, 764010 (2010).
[CrossRef]

Tibshirani, R.

T. Hastie, R. Tibshirani, and J. J. H. Friedman, The Elements of Statistical Learning (Springer, 2001), Vol. 1.

van de Kerkhof, M. A.

M. A. van de Kerkhof, W. de Boeij, H. Kok, M. Silova, J. Baselmans, and M. Hemerik, “Full optical column characterization of DUV lithographic projection tools,” Proc. SPIE 5377, 1960–1970 (2004).
[CrossRef]

Vanoppen, P.

P. Vanoppen and T. Theeuwes, “Lithographic scanner stability improvements through advanced metrology and control,” Proc. SPIE 7640, 764010 (2010).
[CrossRef]

J. Eur. Opt. Soc. Rapid Pub. (1)

A. J. E. M. Janssen, “Computation of Hopkins’ 3-circle integrals using Zernike expansions,” J. Eur. Opt. Soc. Rapid Pub. 6, 11059 (2011).
[CrossRef]

Proc. SPIE (4)

H. Nomura, “New phase-shift gratings for measuring aberrations,” Proc. SPIE 4346, 25–35 (2001).
[CrossRef]

M. A. van de Kerkhof, W. de Boeij, H. Kok, M. Silova, J. Baselmans, and M. Hemerik, “Full optical column characterization of DUV lithographic projection tools,” Proc. SPIE 5377, 1960–1970 (2004).
[CrossRef]

P. Vanoppen and T. Theeuwes, “Lithographic scanner stability improvements through advanced metrology and control,” Proc. SPIE 7640, 764010 (2010).
[CrossRef]

M. Ebert, H. Cramer, W. Tel, M. Kubis, and H. Megens, “Combined overlay, focus and CD metrology for leading edge lithography,” Proc. SPIE 7973, 797311 (2011).
[CrossRef]

Other (5)

T. Hastie, R. Tibshirani, and J. J. H. Friedman, The Elements of Statistical Learning (Springer, 2001), Vol. 1.

M. Pisarenco and I. Setija, “Compact discrepancy and chi-squared principles for over-determined inverse problems” submitted to Inverse Methods.

C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, 1976).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

A. J. E. M. Janssen, “A generalization of the Zernike circle polynomials for forward and inverse problems in diffraction theory,” arXiv:1110.2369v1 (2011).

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Figures (22)

Fig. 1.
Fig. 1.

PSG-based image formation. A schematic representation of image formation for a 1D PSG, having a unit cell with N=4 different taps and period P, is shown. The PSG at the reticle level is illuminated by a normal incidence plane wave. Each tap in the PSG unit cell has an effective scattering factor, fj, with j=0,,N1. The scattering factors shown correspond to the tap design values given in Eq. (11), where a complex valued fj indicates a relative phase shift for the light passing through that particular area of the unit cell. Interaction between the incident light and the PSG will generate a number of diffraction orders in the entrance pupil of the projection optics. Due to the complex scattering factors, destructive interference can take place and a number of diffraction orders are forbidden (shown in gray). We basically end up with only two non-zero orders within the NA, the zeroth and 1st diffraction orders, and only these will propagate through the projection optics to contribute to the image. Note that the reciprocal space coordinates, (h1,2,g1,2), as used in Eq. (1), are given by the coordinates of the two allowed diffraction orders in the pupil and that ρ is the distance between the first diffraction order and the pupil center in normalized pupil coordinates.

Fig. 2.
Fig. 2.

Two-beam interference image formation. Beams 1 and 2 have equal wavelengths with dashed and solid lines representing tops and valleys of the plane wave, respectively. Where solid (or dashed) lines intersect, both beams are in phase and constructive interference takes place to form the image (interference fringes indicated by the thick black lines). A shift of the image plane in the z direction (defocus) and/or a phase change of either beam will only shift the complete interference fringe in the image plane (z=0) in the lateral (x) direction.

Fig. 3.
Fig. 3.

PSG-based diffraction pattern control. Dots and crosses represent allowed and forbidden diffraction orders, respectively. Variation of the PSG pitch scales the diffraction pattern and changes the radial sampling position of the 1st diffraction order. A rotation, θ, of the PSG grating in the xy plane gives rise to an identical rotation of its diffraction pattern in pupil space.

Fig. 4.
Fig. 4.

Basic principle of DBO. A focused spot, with an annular pupil distribution, is used to illuminate a stack with two gratings. The backreflected light, containing diffraction orders, is collected and imaged on a CCD camera (top row images). In the case that both gratings are perfectly aligned, the image on the CCD will be symmetrical (center column), while if a displacement is present, the recorded CCD image is asymmetrical (left and right columns). Then using two such grating stacks, with known biases in overlay, the relative displacement between the top and bottom grating layers can be determined from the measured asymmetries in the free first diffraction orders reflected by both stacks.

Fig. 5.
Fig. 5.

Pupil accessibility of 1D-PSGs in combination with DBO metrology. The concentric circle at pupil radius 0.57 represents the maximum radial sampling position allowed by the constraint in Eq. (13) (for DBO with a YieldStar S/T-200; NA=0.95, λ=425nm). The circles at pupil radii 0.45 and 0.36 represent the minimum radial pupil position allowed by the constraint in Eq. (12) for the PSG designs given in Eqs. (10) and (11), respectively. Consequently, phase information from the hatched Pupil area 1 can be obtained using the PSG design in Eq. (11) while the design in Eq. (10) can only be used to sample the smaller pupil subset labeled Pupil area 2.

Fig. 6.
Fig. 6.

Two 2D-PSG unit cell designs satisfying the conditions in Eq. (15). Both designs contain four scattering objects and their relative scattering factors are color coded according to the legend on the right; i and i indicate relative phase offsets of +90 and 90 deg, respectively. Note that the correct positioning and relative scattering factor of the objects in the unit cell are essential in achieving destructive interference for certain diffraction orders; the actual shape of the scattering objects is, in this respect, of minor importance.

Fig. 7.
Fig. 7.

2D-PSG unit cells and their corresponding diffraction patterns (dots and crosses represent allowed and forbidden orders, respectively). The first row pertains to the nominal 2D-PSG design, which is equivalent to the minimum unit cell given on the right-hand side of Fig 6. In the second row, the unit cell is transformed such that AB, resulting in a compression of the diffraction pattern in one direction. In the third row, the obliquity transformation is applied, resulting in the generation of the orders (1, 0) and (0, 1) at different distances from the pupil center. Note that the diffraction orders (1, 0) and (0, 1) are the two truly diffracted beams. The orders that are forbidden because of the 2D-PSG are (1, 0), (0, 1), (1,1), and (1, 1). Some higher orders, for example (2,2) and (2, 2), are not forbidden, but remain outside of the NA of the optical system. Note that the nomenclature used here, and throughout this paper, to indicate a diffraction order pertains to the minimum unit cell as defined in the right-hand side of Fig. 6.

Fig. 8.
Fig. 8.

Pupil accessibility of 1D- and 2D-PSGs in combination with DBO.

Fig. 9.
Fig. 9.

Numerical proof-of-principle. A set of 60 Zernikes (Z5 to Z64) is defined with random values between 50 and +50mλ (top row). For this Zernike set, the aberration- induced shift of all 420 PSG targets is computed and the resulting simulated values serve as input to the minimization problem defined in Eq. (21). The bottom row presents the error in the estimated Zernike coefficients, which is better than 5mλ.

Fig. 10.
Fig. 10.

Schematic representation of the projection optics exit pupil for the case of a 1D-PSG in combination with an on-axis finite monopole effective source. The monopole source, the size of which is defined by the radius σ relative to the NA of the projection system, is convoluted with the PSG diffraction orders at locations (ρ1,θ1) and (ρ2,θ2). As the source is assumed spatially incoherent, each point on the resulting diffraction order disks can interfere only with its corresponding point on the other disk and each of these point pairs effectively generates a single coherent contribution.

Fig. 11.
Fig. 11.

2D-PSG diffraction orders partly outside the pupil. A point on one diffraction order disk can only interfere, and thus contribute to the overall image, if its corresponding point on the other disk also lies within the pupil. As a result, the effective diffraction order phase is obtained by integration over the common area of both orders, which will be an irregular shaped area when one, or both, diffraction orders lie partly outside the pupil. Systematically computing the average phase for such areas is dealt with in Appendix C.

Fig. 12.
Fig. 12.

Finite source impact. Assume a phase distribution in the pupil, defined by Zernike function Z36. On the left, we plot Z36 averaged over a disk as a function of the averaging disk radius, σ, for four different radial positions in the pupil. On the right, the disk average is plotted as a function of its radial position to illustrate the impact of a finite monopole source (σ=0.122) compared to the coherent case (σ=0.0).

Fig. 13.
Fig. 13.

Cross-section scanning electron microscopy (SEM) images. On the left a cross-section SEM image of a 1D-PSG wafer stack is shown and one can observe a significant difference in side-wall angle between the sides of the grating lines. On the right, a cross-section SEM image of a 2D-PSG wafer stack is shown. It is clearly observed that, although the line appears to be built up from tilted pillars, the 2D-PSG generates effectively a 1D grating with a pitch matched to the bottom grating. The fine structure in the grating line is subresolution for the DBO measurement, but can possibly contribute to the asymmetry signal generated by the whole stack.

Fig. 14.
Fig. 14.

Computed diffraction patterns for the nominal 1D-PSG defined in Eq. (11). The top image shows the diffraction pattern computed according to Kirchhoff, with, as expected, only two non-zero orders within the NA (indicated by the white circle). The middle image shows the diffraction pattern for the same target when correctly accounting for the reticle 3D effects. In this case, three non-zero orders are predicted within the NA, destroying the pure two-beam process. In the bottom image, the rigorously computed diffraction pattern is shown for an optimized 1D-PSG design, showing that the pure two-beam process is restored.

Fig. 15.
Fig. 15.

Illustration and impact of reticle manufacturing defects. The top row shows SEM images of typical 1D- and 2D-PSGs on the reticle. The 1D SEM image was taken before the phase etch step, while the 2D SEM image was taken afterward. One can observe significant corner rounding and additional deformations due to the phase etch into the substrate. In the bottom row, typical diffraction patterns for 1D- and 2D-PSGs are shown as measured by an AIMS. Due to the writing and etching defects, the intended forbidden orders (indicated by dashed circles) are non-zero. Note that the monopole source of the AIMS is relatively large, resulting in overlapping orders and higher diffraction orders showing up at the pupil edge that are of no concern in the actual experiment in the lithographic system where a much smaller source is used.

Fig. 16.
Fig. 16.

Zernikes estimated with the PSG-OVL method for the case that a single non-zero coefficient (Z7=3nm) was dialed in during exposure of the PSG layer onto the wafer. On the top, estimates for the Zernikes (Z5Z36), obtained from the raw overlay measurements, are shown. In the middle, the corresponding retrieval result is shown for the case that the measured overlay values are first calibrated with respect to an aberration-free reference system. The bottom row shows the calibrated result when additional regularization is applied in the fitting procedure.

Fig. 17.
Fig. 17.

PSG-OVL reconstructed Zernikes for a requested cocktail of three non-zero coefficients (Z8=2nm, Z10=2nm, and Z27=2nm).

Fig. 18.
Fig. 18.

Zernikes estimated with the PSG-OVL method for the case that a cocktail of three even Zernikes (Z5=2nm, Z13=2nm, and Z17=2nm) is dialed in. The top axis shows the Zernike estimates obtained using both calibration and regularization. The results in the bottom panel are obtained using an additional correction based on the AIMS-measured residual forbidden-order intensities generated by the PSG reticle.

Fig. 19.
Fig. 19.

Comparison between the aberration coefficients obtained with the on-tool aberration sensors of the exposure tool and the PSG-OVL method (identical experimental settings as in Fig. 17).

Fig. 20.
Fig. 20.

Definition of the relevant distances in reciprocal (pupil) space.

Fig. 21.
Fig. 21.

Definition of the real-space variables and relation between the minimum unit cell (dotted box), computational unit cell (dashed box) and the unit cell as defined in Fig. 7.

Fig. 22.
Fig. 22.

Integration range S in Eq. (C1) consisting of the intersection of two pupil disks centered at ν̲1 and ν̲2 and a third disk centered at o̲ defined by the scanner’s illumination monopole, with choice of origin and radii facilitating solving the mathematical problem.

Tables (3)

Tables Icon

Table 1. Seven Possible Sequences of Diffraction Orders for N=4, with “O” and “X” Indicating Nonforbidden and Forbidden Diffraction Orders, Respectively, and the Top Row Giving the Indices k of the Diffraction Order

Tables Icon

Table 2. Schematic Representation of the Diffraction Pattern Defined by the Conditions in Eq. (15), with “O” and “X” Indicating Nonforbidden and Forbidden Orders, Respectively

Tables Icon

Table 3. Parameters Defining the Left and Right Unit Cells Shown in Fig. 6, Respectively, with the Four Scattering Objects Listed for Each Case

Equations (61)

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I˜h,g=C12+C22+2C1C2cos[2π(h1h2)x+2π(g1g2)y+φ1φ2+ΔΦ1,2],
ΔXAI=ΔΦ1,2P2π,
FN(k)=j=0N1fjexp{2πijkN}.
1+if1f2if3=0.
1+if1f2if3=0,
1f1+f2f3=0.
1+if1f2if3=0,
1f1+f2f3=0,
1if1f2+if3=0,
{1,i,0,0}OXOOO,
{1,1+i,i,0}OXXOO.
PkminλNA,
P>λDBODNADBO,
FN(h,k)=j=0M1fjexp{2πi(xjh+yjk)},
FN(1,0)=0,FN(0,1)=0,FN(1,1)=0,FN(1,1)=0.
(0,0)(1,0)=(1,0),(0,0)(0,1)=(0,1),(1,0)(0,1)=(1,1).
Φ(ρ,θ)=n,mαnmZnm(ρ,θ),
Znm(ρ,θ)=Rn|m|(ρ){cos(|m|θ)form0sin(|m|θ)form<0.
ΔΦ1,2=Φ(ρ2,θ2)Φ(ρ1,θ1).
OVL=ΔXAI=ΔΦ1,2P2π=P2πn,mαnm[Znm(ρ2,θ2)Znm(ρ1,θ1)],
αmin=argminαEαOVLmeas2,
ΔXAI=P2π(Φ¯σ(ρ2,θ2)Φ¯σ(ρ1,θ1)),
OVL=ΔXAI=P2π(Φ¯σ(ρ2,θ2)Φ¯σ(ρ1,θ1))=P2πn,mαnm[Z¯n;σm(ρ2,θ2)Z¯n;σm(ρ1,θ1)],
α˜min=argminαQ(Eα(OVLmeasOVLref))2,
K=Kmeas+Kref,
K1=QQ*.
α˜min=(E*K1E)1E*K1(OVLmeasOVLref),
α˜min(ζ)=argminαQ(Eα(OVLmeasOVLref))2+ζ2D(ααp)2.
q=t1,p=tq212,
n=q+p,m=qp;
Zt(ρ,θ)=Rnm(ρ){(tq22p)cosmθ+[1(tq22p)]sinmθ}=Znm(ρ,θ).
Rnm(ρ)cosmθ=Z(n+m2)2+nm+1,
Rnm(ρ)sinmθ=Z(n+m2)2+nm+2,
ρ0=λNAP,
ρ1=ρ1,
ρ2=(1ξ)2ρ02+ρ12ξ2ρ02,
ρ3=ξρ0,
ρA=(1ξ)ρ0
ρB=(ρ12ξ2ρ02),
ω=arccos[ρ12+ρ22ρ022ρ1ρ2],
Ψ=arccos[ρ02+(2sin(ω2)ρ2)2(ρ1ρ2)24sin(ω2)ρ0ρ2],
Amin=λ2NAρ1(1cos(ω2))2+(1sin(ω2))2,
Bmin=ρ1ρ2Amin,
AC=ΞλNAξρ0,
BC=λNAρ12ξ2ρ02,
2λNA(1+σ)PλDBOCNADBO,
(1+σ)2(44ξ)(λNAP)2ρ11,
1((1+σ)2ρ12)(NAP)24λ2ξ12.
SZnm(ν̲ν̲1,2r)dν̲,
Znm(ν̲)Znm(ρeiθ)=Rn|m|(ρ)eimθ,0ρ1,0θ2π,
SZnm(ν̲ν̲1r)dν̲=ν̲1Znm(ν̲ν̲1r)Z00(ν̲)(Z00(ν̲ν̲2r)Z00(ν̲))*dν̲=n,mπn+1βn,1m(βn,2m)*,
βn,1m=n+1πZnm(ν̲ν̲1r)(Znm(ν̲))*dν̲,
βn,2m=n+1πZ00(ν̲ν̲2r)(Znm(ν̲))*dν̲,
βn,1m=n+1πnCnn,nmmZnmm(ν̲1r+1),|ν̲1|r+1,
Cnn,nmm=(rr+1)2(1)n(n+1)π(n+1)(n+1)×[Snnn+1Sn+2,nn+1Sn,n+2n+1+Sn+2,n+2n+1],
Sijk+1=(12(k+i+j))!(12(kij))!(12(k+ij))!(12(ki+j))!×ri(r+1)i+j(P12(kij)(i,j)(1r1+r))2,
ri(r+1)i+j(P12(kij)(i,j)(1r1+r))2=(1ρ2)i2Rkij,i(ρ),
(q+αq)(1ρ2)αRn|m|,α(ρ)=1Nk=0N1Cnα+1(ρcos2πkN)e2πikmN.
ak=(12(k+i+j))!(12(kij))!(12(k+ij))!(12(ki+j))!
ak=i+j=(i+ji),
ak+2=(12(k+i+j)+1)(12(kij)+1)(12(k+ij)+1)(12(ki+j)+1)ak.

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