Abstract

Some simple measures of the difficulty of a variety of steps in asphere fabrication are defined by reference to fundamental geometric considerations. It is shown that effective approximations can then be exploited when an asphere’s shape is characterized by using a particular orthogonal basis. The efficiency of the results allows them to be used not only as quick manufacturability estimates at the production end, but more importantly as part of an efficient design process that can boost the resulting optical systems’ cost-effectiveness.

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References

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  1. A. Epple and H. Wang, “‘Design to manufacture’ from the perspective of optical design and fabrication,” in Optical Fabrication and Testing, OSA Technical Digest (Optical Society of America, 2008), OFB1.
  2. J. P. McGuire., “Manufacturable mobile phone optics: higher order aspheres are not always better,” Proc. SPIE 7652, 76521O, 76521O-8 (2010).
    [CrossRef]
  3. R. N. Youngworth and B. D. Stone, “Simple estimates for the effects of mid-spatial-frequency surface errors on image quality,” Appl. Opt. 39(13), 2198–2209 (2000).
    [CrossRef]
  4. J. W. Foreman., “Simple numerical measure of the manufacturability of aspheric optical surfaces,” Appl. Opt. 25(6), 826–827 (1986).
    [CrossRef] [PubMed]
  5. J. W. Foreman., “Mercier’s aspheric manufacturability index,” Appl. Opt. 26(22), 4711–4712 (1987).
    [CrossRef] [PubMed]
  6. J. Kumler, “Designing and specifying aspheres for manufacturability,” Proc. SPIE 5874, 121 (2005), doi:.
  7. C. du Jeu, “Criterion to appreciate difficulties of aspherical polishing,” Proc. SPIE 5494, 113–121 (2004).
    [CrossRef]
  8. G. W. Forbes and C. P. Brophy, “Designing cost-effective systems that incorporate high-precision aspheric optics,” SPIE Optifab (2009), paper TD06–25 (1), http://www.qedmrf.com .
  9. G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express 18(19), 19700–19712 (2010).
    [CrossRef] [PubMed]
  10. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007).
    [CrossRef] [PubMed]
  11. P. Murphy Jon Fleig, “Greg Forbes, Dragisha Miladinovic, Gary DeVries, and Stephen O'Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).

2010 (2)

J. P. McGuire., “Manufacturable mobile phone optics: higher order aspheres are not always better,” Proc. SPIE 7652, 76521O, 76521O-8 (2010).
[CrossRef]

G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express 18(19), 19700–19712 (2010).
[CrossRef] [PubMed]

2007 (1)

2006 (1)

P. Murphy Jon Fleig, “Greg Forbes, Dragisha Miladinovic, Gary DeVries, and Stephen O'Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).

2005 (1)

J. Kumler, “Designing and specifying aspheres for manufacturability,” Proc. SPIE 5874, 121 (2005), doi:.

2004 (1)

C. du Jeu, “Criterion to appreciate difficulties of aspherical polishing,” Proc. SPIE 5494, 113–121 (2004).
[CrossRef]

2000 (1)

1987 (1)

1986 (1)

du Jeu, C.

C. du Jeu, “Criterion to appreciate difficulties of aspherical polishing,” Proc. SPIE 5494, 113–121 (2004).
[CrossRef]

Forbes, G. W.

Foreman, J. W.

Kumler, J.

J. Kumler, “Designing and specifying aspheres for manufacturability,” Proc. SPIE 5874, 121 (2005), doi:.

McGuire, J. P.

J. P. McGuire., “Manufacturable mobile phone optics: higher order aspheres are not always better,” Proc. SPIE 7652, 76521O, 76521O-8 (2010).
[CrossRef]

Murphy Jon Fleig, P.

P. Murphy Jon Fleig, “Greg Forbes, Dragisha Miladinovic, Gary DeVries, and Stephen O'Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).

Stone, B. D.

Youngworth, R. N.

Appl. Opt. (3)

Opt. Express (2)

Proc. SPIE (4)

P. Murphy Jon Fleig, “Greg Forbes, Dragisha Miladinovic, Gary DeVries, and Stephen O'Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).

J. P. McGuire., “Manufacturable mobile phone optics: higher order aspheres are not always better,” Proc. SPIE 7652, 76521O, 76521O-8 (2010).
[CrossRef]

J. Kumler, “Designing and specifying aspheres for manufacturability,” Proc. SPIE 5874, 121 (2005), doi:.

C. du Jeu, “Criterion to appreciate difficulties of aspherical polishing,” Proc. SPIE 5494, 113–121 (2004).
[CrossRef]

Other (2)

G. W. Forbes and C. P. Brophy, “Designing cost-effective systems that incorporate high-precision aspheric optics,” SPIE Optifab (2009), paper TD06–25 (1), http://www.qedmrf.com .

A. Epple and H. Wang, “‘Design to manufacture’ from the perspective of optical design and fabrication,” in Optical Fabrication and Testing, OSA Technical Digest (Optical Society of America, 2008), OFB1.

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Figures (6)

Fig. 1
Fig. 1

Either annular or off-axis sections of conicoids can be tested at null in double-pass reflection tests involving a flat or a sphere as a retro. Two such options for testing a paraboloid are sketched above where the interferometer sits to the left in both cases.

Fig. 2
Fig. 2

A selection of surface normals on a rotationally symmetric asphere serve to indicate the principal radii of curvature. The three red normal vectors all fall in the same plane of symmetry whereas the green normals lie on a cone.

Fig. 3
Fig. 3

Sketch of a lattice of subapertures that provides total coverage of an asphere. Schematic fringes are represented for one of the subaperture tests.

Fig. 4
Fig. 4

An asphere’s cross section is shown in the upper panel with its best-fit sphere drawn as the dotted purple curve. Plots of its principal curvatures (in-plane in red and out-of-plane in blue) are overlaid by the first-order approximations of Eq. (5.8) as dashed curves in the lower panel. The dashed gray line indicates the best-fit curvature of 0.03864mm–1. Notice that the principal curvatures reach more than +/–10 times the best-fit curvature.

Fig. 5
Fig. 5

A comparison of the domains defined by the constraints associated with Eqs. (2.9) and (6.2) for the case M = 1, γmax = 0.25, N = 1024, λ = 633nm, and χmax = 4. The domain of full-aperture testability is shown as the shaded circle in the center of the plot. The ellipses represent the “stitchable” domain boundaries for a selection of values of the part NA, i.e. of η. In the inset, this selection of best-fit spheres is drawn with equal CA to compare their raw shapes.

Fig. 6
Fig. 6

Plots of CAmin for the same parameter values used in Fig. 5. Each cone-like surface has its apex at the origin. When viewed along the vertical axis, the rims of these surfaces coincide with the ellipses drawn on the ceiling, and these are precisely the testable domains of Fig. 5. The largest value attained by each cone is marked on the right-hand end wall.

Equations (83)

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z     = f con ( ρ , c , κ ) : =     c ρ 2 / [ 1 + ϕ ( ρ ) ] ,
ϕ ( ρ )     : =     1 ( 1 + κ ) c 2 ρ 2 .
σ ( ρ ) : =     [ 1 + f ( ρ ) 2 ] 1 / 2     =     ϕ ( ρ ) / 1 κ c 2 ρ 2 .
z = f con ( ρ , c , κ ) + 1 σ ( ρ ) u 2 ( 1 u 2 ) m = 0 M a m Q m ( u 2 ) ,
δ ( u 2 )     : =     u 2 ( 1 u 2 ) m = 0 M a m Q m ( u 2 ) .
[ d d ρ δ ( u 2 ) ] 2     =     [ 2 u ρ max δ ( u 2 ) ] 2     =     1 ρ max 2 m = 0 M a m 2 ,
g ( u )     : =     0 1 g ( u ) [ 1 u 2 ] 1 / 2 d u / 0 1 [ 1 u 2 ] 1 / 2 d u .
γ ¯     =     8 ρ max N λ [ d d ρ δ ( u 2 ) ] 2     =     8 N λ m = 0 M a m 2 .
m = 0 M a m 2     <     ( γ ¯ max N λ / 8 ) 2 .
f ( ρ )     =     c ρ 2 / ( 1 + ϕ ) + δ / σ ,
f ( ρ )     =     c ρ / ϕ + c 2 ρ σ δ / ϕ 4 + 2 ρ δ / ( σ ρ max 2 ) ,
f ( ρ )     =     c / ϕ 3 + c 2 σ ( 3 3 ϕ 2 + σ 2 ) δ / ϕ 6         + 2 ( 2 + ϕ 2 2 σ 2 ) δ / ( ϕ 2 σ ρ max 2 ) + 4 ρ 2 δ / ( σ ρ max 4 ) .
z     =     c ρ 2 / [ 1 + ϕ ( ρ ) ]     + δ ε ( u 2 ) / σ ( ρ ) ,
δ ε ( u 2 )     : =     1 ( 1 ε 2 ) 1 + ε ( u 2 ε 2 ) ( 1 u 2 ) m = 0 M a m Q m ( u 2 ε 2 1 ε 2 ) .
c IP ( ρ ) : =     f ( ρ ) [ 1 + f ( ρ ) 2 ] 3 / 2 .
c OOP ( ρ ) : =     f ( ρ ) ρ [ 1 + f ( ρ ) 2 ] 1 / 2 .
c IP ( ρ )     =     c OOP ( ρ ) + ρ c OOP ( ρ ) .
c OOP ( ρ )     =     j = 0 c j ρ 2 j .
μ ¯ : =     [ c IP ( ρ ) c ] 2 T 2 / 8.
φ ( x , y )     2 π c 2 test λ [ ( c IP c test ) x 2 + ( c OOP c test ) y 2 ] ,
( x , y )     2 η TS N ( j , k ) ,
c test = ( c IP + c OOP ) / 2 ,
φ j k     16 π λ [ η TS ( c IP + c OOP ) N ] 2 ( c IP c OOP ) ( j 2 k 2 ) .
16 π λ [ η TS ( c IP + c OOP ) N ] 2 | c IP c OOP | 2 j         γ π ,
j         γ λ 32 | c IP c OOP | [ ( c IP + c OOP ) N / η TS ] 2 .
η TS 2         γ N λ 16 ( c IP + c OOP ) 2 | c IP c OOP | .
η TS 2 = γ N λ 16 / [ c IP c OOP ( c IP + c OOP ) 2 ] 2 .
c OOP ( u ρ max )     =     c ( 1 η 2 u 2 ) + ε OOP ( u 2 ) ( 1 η 2 u 2 ) 2 + η 2 u 2 ε OOP ( u 2 ) [ 2 ( 1 η 2 u 2 ) + ε OOP ( u 2 ) ] ,
ε OOP ( u 2 ) : =     1 η ρ max { 2 δ ( u 2 ) + η 2 [ δ ( u 2 ) 2 u 2 δ ( u 2 ) ] } .
c OOP ( u ρ max )     =     c [ 1 + ε OOP ( u 2 ) 3 2 η 2 u 2 1 η 2 u 2 ε OOP ( u 2 ) 2 1 2 η 2 u 2 ( 1 5 η 2 u 2 ) ( 1 η 2 u 2 ) 2 ε OOP ( u 2 ) 3 + O ( 4 ) ] .
| ε OOP ( u 2 ) |     <     | 1 + i [ ( 1 η 2 u 2 ) / ( u η ) ] 2 1 |     =     ( 1 η 2 u 2 ) / | u η | .
c IP ( u ρ max )     =     c { [ 1 + ε IP ( u 2 ) ] ( 1 η 2 u 2 ) + 3 η 2 u 2 ε OOP ( u 2 ) } ( 1 η 2 u 2 ) 2 { ( 1 η 2 u 2 ) 2 + η 2 u 2 ε OOP ( u 2 ) [ 2 ( 1 η 2 u 2 ) + ε OOP ( u 2 ) ] } 3 / 2 ,
ε IP ( u 2 ) : =     1 η ρ max { [ 2 δ ( u 2 ) + 4 u 2 δ ( u 2 ) ] + η 2 [ δ ( u 2 ) 4 u 2 δ ( u 2 ) 4 u 4 δ ( u 2 ) ] } .
c IP ( u ρ max )     =     c { 1 + ε IP ( u 2 ) η 2 u 2 1 η 2 u 2 ε OOP ( u 2 ) [ 3 ε IP ( u 2 ) + 3 2 1 + η 2 u 2 1 η 2 u 2 ε OOP ( u 2 ) ]               η 2 u 2 ( 1 η 2 u 2 ) 2 ε OOP ( u 2 ) 2 [ 3 ( 1 5 η 2 u 2 ) 2 ε IP ( u 2 ) η 2 u 2 ( 3 + 5 η 2 u 2 ) 1 η 2 u 2 ε OOP ( u 2 ) ] + O ( 4 ) } ,
c IP ( u ρ max )         c [ 1 + ε IP ( u 2 ) ]     and     c OOP ( u ρ max )         c [ 1 + ε OOP ( u 2 ) ]
χ 2         4 γ N λ { ( η 2 / c ) [ ε IP ( u 2 ) ε OOP ( u 2 ) ] } 2 + O ( 3 )           4 γ N λ { 4 u 2 δ ( u 2 ) 2 η 2 u 2 [ δ ( u 2 ) + 2 u 2 δ ( u 2 ) ] } 2 .
( 1 4 γ N λ χ 2 ) 2         a T ( A 0 η 2 A 1 + η 4 A 2 ) a .
a T A n a     =     ( U n a ) 2 .
U 0 = ( 4.898979 12.36293 15.2826 21.35537 24.80396 30.47022 34.17734 39.6322 0. 10.05249 14.62284 18.54807 22.72841 26.87057 30.94862 35.15502 0. 0. 13.83908 18.51539 22.02259 26.59522 30.18392 34.68144 0. 0. 0. 17.98119 22.29843 26.13997 30.28506 34.24423 0. 0. 0. 0. 21.91791 26.28912 29.97733 34.31793 0. 0. 0. 0. 0. 25.97587 30.20157 34.05626 0. 0. 0. 0. 0. 0. 29.94555 34.20084 0. 0. 0. 0. 0. 0. 0. 33.9772 ) ,
U 1 = ( 6.928203 19.27195 24.69144 33.75769 39.67753 48.3265 54.52582 62.93859 0. 9.892182 21.01649 27.18734 32.94823 39.26074 44.97279 51.3028 0. 0. 12.13989 26.24577 31.74023 37.87812 43.34389 49.48468 0. 0. 0. 14.93647 31.54287 37.3337 42.96587 48.83548 0. 0. 0. 0. 17.62935 37.15725 42.72747 48.59571 0. 0. 0. 0. 0. 20.45256 42.68451 48.40322 0. 0. 0. 0. 0. 0. 23.22303 48.33439 0. 0. 0. 0. 0. 0. 0. 26.05161 ) ,
U 2 = ( 5.086747 13.9192 18.80456 25.47325 30.09193 36.51867 41.30643 47.58664 0. 5.697364 13.8236 19.94909 24.12012 28.7889 32.93957 37.60946 0. 0. 6.146558 16.35225 22.92788 27.27238 31.27854 35.64707 0. 0. 0. 7.01248 19.0356 26.73161 30.74467 34.96211 0. 0. 0. 0. 7.87676 21.949 30.48723 34.62888 0. 0. 0. 0. 0. 8.82526 24.8344 34.43803 0. 0. 0. 0. 0. 0. 9.764812 27.80165 0. 0. 0. 0. 0. 0. 0. 10.73794 ) .
| ε IP ( u 2 ) |     <     1 ,
| ε OOP ( u 2 ) |     <     1 4 ( 1 η 2 u 2 ) / | u η | .
δ ( u 2 )     : =     u 2 ( 1 u 2 ) m = 0 M a m Q m ( u 2 )     =     u 2 ( 1 u 2 ) m = 0 M b m P m ( u 2 ) ,
P m ( cos 2 θ )     =     ( 1 ) m 2 cos [ ( 2 m + 1 ) θ ] cos θ ,
g ( u )     =     2 π 0 π / 2 g ( cos θ ) d θ     =     1 π π / 2 π / 2 g ( cos θ ) d θ .
δ ( cos 2 θ )     =     2 cos θ sin 2 θ m = 0 M ( 1 ) m b m cos [ ( 2 m + 1 ) θ ] .
d d u 2     =     1 2 sin θ cos θ d d θ .
4 cos 2 θ δ ( cos 2 θ )     =     m = 0 M ( 1 ) m b m [ m E m 1 ( θ ) + ( m 3 ) E m ( θ )                             ( m + 4 ) E m + 1 ( θ ) ( m + 1 ) E m + 2 ( θ ) ] ,
E m ( θ ) : =     2 m sin ( 2 m θ ) / sin ( 2 θ ) .
2 η 2 cos 2 θ [ δ ( cos 2 θ ) + 2 cos 2 θ δ ( cos 2 θ ) ]                   =     η 2 m = 0 M ( 1 ) m b m [ ( 2 m 1 ) F m 1 ( θ ) + ( 2 m 5 ) F m ( θ )                               ( 2 m + 7 ) F m + 1 ( θ ) ( 2 m + 3 ) F m + 2 ( θ ) ] ,
F m ( θ ) : =     1 2 m cot θ sin ( 2 m θ )     =     1 2 cos 2 θ E m ( θ ) .
g ( u )     =     1 2 π 0 2 π g ( cos θ ) d θ .
I m n : = 1 2 π 0 2 π E m ( θ ) E n ( θ ) d θ ,
J m n : = 1 2 π 0 2 π E m ( θ ) F n ( θ ) d θ ,
K m n : = 1 2 π 0 2 π F m ( θ ) F n ( θ ) d θ .
I m n =     2 m n π 0 2 π sin ( 2 m θ ) sin ( 2 n θ ) sin 2 ( 2 θ ) d θ     =     2 m n π 0 2 π sin ( m τ ) sin ( n τ ) sin 2 ( τ ) d τ     =     4 H m n ,
H m n : =     m n 2 π 0 2 π sin ( m τ ) sin ( n τ ) sin 2 ( τ ) d τ .
J m n = m n 4 π 0 2 π sin ( 2 m θ ) sin ( 2 n θ ) sin 2 θ d θ     =     1 8 H 2 m 2 n ,
K m n = m n / 4 2 π 0 2 π sin ( 2 m θ ) sin ( 2 n θ ) sin 2 θ cos 2 θ   d θ     =     1 16 H 2 m 2 n 1 8 | m n | δ | m | | n | ,
H m n =     m n 2 π i ( z m z m ) ( z n z n ) ( z z 1 ) 2 z d z     =     m n 2 π i ( z 2 m 1 ) ( z 2 n 1 ) ( z 2 1 ) 2 z m + n 1 d z .
H m n =     1 2 [ 1 + ( 1 ) | m | + | n | ]   Min( | m | , | n | )     | m n | .
J m n = | m n | Min( | m | , | n | ) , K m n = 1 2 | m n | [ Min( | m | , | n | ) 1 4 δ m n ] .
{ 4 u 2 δ ( u 2 ) 2 η 2 u 2 [ δ ( u 2 ) + 2 u 2 δ ( u 2 ) ] } 2 = b T ( B 0 η 2 B 1 + η 4 B 2 ) b
b m n 0     =     Y m ( θ ) Y n ( θ ) ,
b m n 1     =     Y m ( θ ) Z n ( θ ) + Y n ( θ ) Z m ( θ ) ,
b m n 2     =     Z m ( θ ) Z n ( θ ) ,
Y m ( θ )     =     ( 1 ) m [ m E m 1 ( θ ) + ( m 3 ) E m ( θ ) ( m + 4 ) E m + 1 ( θ ) ( m + 1 ) E m + 2 ( θ ) ] ,
Z m ( θ )     = ( 1 ) m [ ( 2 m 1 ) F m 1 ( θ ) + ( 2 m 5 ) F m ( θ )               ( 2 m + 7 ) F m + 1 ( θ ) ( 2 m + 3 ) F m + 2 ( θ ) ] .
c 0 ( j , k ) : =     ( 1 ) j + k 96 [ 1 + 2 j ( 1 + j ) ] ( 1 + 2 k ) ,
b m n 0     =     c 0 [ v m n ,   Max ( m , n ) ]     { 16 m ( m + 1 ) [ 5 m ( m + 1 ) ] ,       | m n | = 0 , 8 v m n ( v m n + 1 ) 2 ( v m n + 2 ) ,         | m n | = 1 , 0 ,                       | m n | > 1.
c 1 ( j , k ) : =     ( 1 ) j + k 24 [ 9 + 16 j ( 1 + j ) ] ( 1 + 2 k ) ,
b m n 1     =     c 1 [ v m n ,   Max ( m , n ) ]     { 6 { 4 m ( m + 1 ) [ 4 m ( m + 1 ) 41 ] } , | m n | = 0 , 2 ( v m n + 1 ) 2 [ 8 v m n ( v m n + 2 ) 9 ] ,       | m n | = 1 , ( v m n + 1 ) ( v m n + 2 ) ( 2 v m n + 3 ) 2 ,     | m n | = 2 , 0 ,                             | m n | > 2.
c 2 ( j , k ) : =     ( 1 ) j + k 24 [ 5 + 8 j ( 1 + j ) ] ( 1 + 2 k )     { 3 4 ,     j = 0     and     k = 0 , 3 8 ,     j = 0     and     1 k 2 , 0 ,     otherwise ,
b m n 2     =     c 2 [ v m n ,   Max ( m , n ) ]     { 1 4 { 63 2 m ( m + 1 ) [ 20 m ( m + 1 ) 301 ] } ,         | m n | = 0 , 3 8 { 5 v m n ( v m n + 2 ) [ 4 v m n ( v m n + 2 ) 9 ] 67 } ,       | m n | = 1 , 3 8 ( 2 v m n + 3 ) 2 [ 2 v m n ( v m n + 3 ) + 3 ] ,             | m n | = 2 , 1 8 ( v m n + 2 ) 2 ( 2 v m n + 3 ) ( 2 v m n + 5 ) ,           | m n | = 3 , 0 ,                                 | m n | > 3.
B 0 = ( 96 288 480 672 864 1056 1248 1440 288 1344 2496 3360 4320 5280 6240 7200 480 2496 6336 9312 11232 13728 16224 18720 672 3360 9312 18144 23520 26400 31200 36000 864 4320 11232 23520 40224 48096 51168 59040 1056 5280 13728 26400 48096 76416 86208 87840 1248 6240 16224 31200 51168 86208 130944 141216 1440 7200 18720 36000 59040 87840 141216 208416 ) ,
B 1 = ( 192 630 1098 1512 1944 2376 2808 630 2532 5040 7038 8856 10824 12792 1098 5040 11964 18630 23268 27720 32760 1512 7038 18630 34248 46968 54684 62712 1944 8856 23268 46968 75720 96006 106278 2376 10824 27720 54684 96006 143292 172080 2808 12792 32760 62712 106278 172080 244452 ) ,
B 2 = 1 8 ( 828 2676 4884 6780 8640 10560 12480 14400 2676 9882 20094 29049 36603 44352 52416 60480 4884 20094 46410 73791 94965 112944 132288 152640 6780 29049 73791 132690 185802 222789 254571 290880 8640 36603 94965 185802 292914 379599 433257 480348 10560 44352 112944 222789 379599 553194 680094 747681 12480 52416 132288 254571 433257 680094 941562 1113639 14400 60480 152640 290880 480348 747681 1113639 1487970 ) .
b T ( B 0 η 2 B 1 + η 4 B 2 ) b = a T ( A 0 η 2 A 1 + η 4 A 2 ) a ,
A n : = L 1 B n ( L 1 ) T .
μ ¯         [ c ε IP ( ρ ) ] 2 T 2 / 8     =     T 2 8 ρ max 2 { [ 2 δ ( u 2 ) + 4 u 2 δ ( u 2 ) ] + η 2 [ δ ( u 2 ) 4 u 2 δ ( u 2 ) 4 u 4 δ ( u 2 ) ] } 2 .
{ [ 2 δ ( u 2 ) + 4 u 2 δ ( u 2 ) ] + η 2 [ δ ( u 2 ) 4 u 2 δ ( u 2 ) 4 u 4 δ ( u 2 ) ] }       =     m = 0 M ( 1 ) m b m { [ ( m 1 ) G m ( θ ) G m + 1 ( θ ) ( m + 2 ) G m + 2 ( θ ) ]                 1 2 cos 2 θ η 2 [ ( 2 m 3 ) G m ( θ ) ( 2 m + 5 ) G m + 2 ( θ ) ] ,
G m ( θ )     : =     ( 2 m 1 ) sin [ ( 2 m 1 ) θ ] / sin θ .

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