Abstract

We develop a way to estimate the approximate analytic astigmatism with a high accuracy for any unit-magnification multipass system (UMS). The coaxial optical transmission model for UMS is simplified based on the system’s features. Furthermore, astigmatism is derived as a distinct form of vector addition and, thus, feasible analytic astigmatism can be obtained. The effectiveness of our method is verified by simulations for a Bernstein–Herzberg White cell. In our cases, the relative error of optimization for astigmatism correction by our method is smaller than 5‰, which is only one-tenth of that by Kohn’s method. Our method significantly improves the efficiency for astigmatism correction, and further benefits the optical design of a UMS.

© 2010 Optical Society of America

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References

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  1. K. Chen, H. Yang, L. Sun, and G. Jin, “Generalized method for calculating astigmatism of unit-magnification multipass system,” Appl. Opt.  49, 1964–1971 (2010).
    [CrossRef] [PubMed]
  2. R. P. Blickensderfer, G. E. Ewing, and R. Leonard, “A long path, low temperature cell,” Appl. Opt.  7, 2214–2217(1968).
    [CrossRef] [PubMed]
  3. H. M. Pickett, G. M. Bradley, and H. L. Strauss, “A new White type multiple pass absorption cell,” Appl. Opt.  9, 2397–2398 (1970).
    [CrossRef] [PubMed]
  4. D. Horn and G. C. Pimentel, “2.5 Km low-temperature multiple-reflection cell,” Appl. Opt.  10, 1892–1898 (1971).
    [CrossRef] [PubMed]
  5. J.-F. O. Doussin, R. Dominique, and C. Patrick, “Multiple-pass cell for very-long-path infrared spectrometry,” Appl. Opt.  38, 4145–4150 (1999).
    [CrossRef]
  6. S. M. Chernin, “Promising version of the three-objective multipass matrix system,” Opt. Express  10, 104–107(2002).
    [PubMed]
  7. D. R. Glowacki, A. Goddard, and P. W. Seakins, “Design and performance of a throughput-matched, zero-geometric-loss, modified three objective multipass matrix system for FTIR spectrometry,” Appl. Opt.  46, 7872–7883 (2007).
    [CrossRef] [PubMed]
  8. S. M. Chernin, S. B. Mikhailov, and E. G. Barskaya, “Aberrations of a multipass matrix system,” Appl. Opt.  31, 765–769(1992).
    [CrossRef] [PubMed]
  9. Y. G. Barskaya, “Aberrations of a multipass cell,” Opt. Technol.  38, 278–280 (1971).
  10. C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE  7156, 71560G (2008).
  11. T. R. Reesor, “The astigmatism of a multiple path absorption cell,” J. Opt. Soc. Am.  41, 1059–1060 (1951).
    [CrossRef]
  12. T. H. Edwards, “Multiple-traverse absorption cell design,” J. Opt. Soc. Am.  51, 98–102 (1961).
    [CrossRef]
  13. W. H. Kohn, “Astigmatism and White cells: theoretical considerations on the construction of an anastigmatic White cell,” Appl. Opt.  31, 6757–6764 (1992).
    [CrossRef] [PubMed]
  14. H. J. Bernstein and G. Herzberg, “Rotation-vibration spectra of diatomic and simple polyatomic molecules with long absorbing paths,” J. Chem. Phys.  16, 30–39 (1948).
    [CrossRef]
  15. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), pp. 158–161.
  16. R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE  251, 146–153 (1980).
  17. K. P. Thompson, “Practical methods for the optical design of systems without symmetry,” Proc. SPIE  2774, 2–12(1996).
    [CrossRef]
  18. J. R. Rogers, “Design techniques for systems containing tilted components,” Proc. SPIE  3737, 286–300 (1999).
    [CrossRef]
  19. J. R. Rogers, “Techniques and tools for obtaining symmetrical performance from tilted-component systems,” Opt. Eng.  39, 1776–1787 (2000).
    [CrossRef]
  20. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A  22, 1389–1401 (2005).
    [CrossRef]
  21. L. B. Moore, A. M. Hvisc, and J. Sasian, “Aberration fields of a combination of plane symmetric systems,” Opt. Express  16, 15655–15670 (2008).
    [CrossRef] [PubMed]

2010 (1)

2008 (2)

L. B. Moore, A. M. Hvisc, and J. Sasian, “Aberration fields of a combination of plane symmetric systems,” Opt. Express  16, 15655–15670 (2008).
[CrossRef] [PubMed]

C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE  7156, 71560G (2008).

2007 (1)

2005 (1)

2002 (1)

2000 (1)

J. R. Rogers, “Techniques and tools for obtaining symmetrical performance from tilted-component systems,” Opt. Eng.  39, 1776–1787 (2000).
[CrossRef]

1999 (2)

J. R. Rogers, “Design techniques for systems containing tilted components,” Proc. SPIE  3737, 286–300 (1999).
[CrossRef]

J.-F. O. Doussin, R. Dominique, and C. Patrick, “Multiple-pass cell for very-long-path infrared spectrometry,” Appl. Opt.  38, 4145–4150 (1999).
[CrossRef]

1996 (1)

K. P. Thompson, “Practical methods for the optical design of systems without symmetry,” Proc. SPIE  2774, 2–12(1996).
[CrossRef]

1992 (2)

1980 (1)

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE  251, 146–153 (1980).

1971 (2)

1970 (1)

1968 (1)

1961 (1)

1951 (1)

1948 (1)

H. J. Bernstein and G. Herzberg, “Rotation-vibration spectra of diatomic and simple polyatomic molecules with long absorbing paths,” J. Chem. Phys.  16, 30–39 (1948).
[CrossRef]

Barskaya, E. G.

Barskaya, Y. G.

Y. G. Barskaya, “Aberrations of a multipass cell,” Opt. Technol.  38, 278–280 (1971).

Bernstein, H. J.

H. J. Bernstein and G. Herzberg, “Rotation-vibration spectra of diatomic and simple polyatomic molecules with long absorbing paths,” J. Chem. Phys.  16, 30–39 (1948).
[CrossRef]

Blickensderfer, R. P.

Bradley, G. M.

Chen, K.

Chernin, S. M.

Dominique, R.

Doussin, J.-F. O.

Edwards, T. H.

Ewing, G. E.

Glowacki, D. R.

Goddard, A.

Guofan, J.

C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE  7156, 71560G (2008).

Herzberg, G.

H. J. Bernstein and G. Herzberg, “Rotation-vibration spectra of diatomic and simple polyatomic molecules with long absorbing paths,” J. Chem. Phys.  16, 30–39 (1948).
[CrossRef]

Horn, D.

Huaidong, Y.

C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE  7156, 71560G (2008).

Hvisc, A. M.

Jin, G.

Kexin, C.

C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE  7156, 71560G (2008).

Kohn, W. H.

Leonard, R.

Liqun, S.

C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE  7156, 71560G (2008).

Mikhailov, S. B.

Moore, L. B.

Patrick, C.

Pickett, H. M.

Pimentel, G. C.

Reesor, T. R.

Rogers, J. R.

J. R. Rogers, “Techniques and tools for obtaining symmetrical performance from tilted-component systems,” Opt. Eng.  39, 1776–1787 (2000).
[CrossRef]

J. R. Rogers, “Design techniques for systems containing tilted components,” Proc. SPIE  3737, 286–300 (1999).
[CrossRef]

Sasian, J.

Seakins, P. W.

Shack, R. V.

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE  251, 146–153 (1980).

Strauss, H. L.

Sun, L.

Thompson, K.

K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A  22, 1389–1401 (2005).
[CrossRef]

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE  251, 146–153 (1980).

Thompson, K. P.

K. P. Thompson, “Practical methods for the optical design of systems without symmetry,” Proc. SPIE  2774, 2–12(1996).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), pp. 158–161.

Yang, H.

Appl. Opt. (8)

J. Chem. Phys. (1)

H. J. Bernstein and G. Herzberg, “Rotation-vibration spectra of diatomic and simple polyatomic molecules with long absorbing paths,” J. Chem. Phys.  16, 30–39 (1948).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

J. R. Rogers, “Techniques and tools for obtaining symmetrical performance from tilted-component systems,” Opt. Eng.  39, 1776–1787 (2000).
[CrossRef]

Opt. Express (2)

Opt. Technol. (1)

Y. G. Barskaya, “Aberrations of a multipass cell,” Opt. Technol.  38, 278–280 (1971).

Proc. SPIE (4)

C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE  7156, 71560G (2008).

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE  251, 146–153 (1980).

K. P. Thompson, “Practical methods for the optical design of systems without symmetry,” Proc. SPIE  2774, 2–12(1996).
[CrossRef]

J. R. Rogers, “Design techniques for systems containing tilted components,” Proc. SPIE  3737, 286–300 (1999).
[CrossRef]

Other (1)

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), pp. 158–161.

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Figures (11)

Fig. 1
Fig. 1

Simplified model for astigmatism of UMS. The kth reflection on the objective mirrors is equivalent to the kth GATL.

Fig. 2
Fig. 2

Parameters for GATL. (a) Reflections in UMS. (b) Corresponding part of model for astigmatism of UMS. X Y Z , global coordinate in the model whose optical axis is Z; X k Y k Z , local coordinate of kth GATL whose meridional plane is Y k Z .

Fig. 3
Fig. 3

Vector addition of astigmatism.

Fig. 4
Fig. 4

Overview of astigmatism calculation procedure: (a) our method for approximate analytic astigmatism and (b) our previous generalized method for accurate numerical astigmatism.

Fig. 5
Fig. 5

System layout of the 16-pass BHWC. F, the field mirror with the center of curvature C F ; M 1 and M 2 , two spherical objective mirrors with centers of curvature C 1 and C 2 , respectively; In, the entrance aperture; Out, the exit aperture; tags 1 to 7, the images of the entrance aperture in sequence.

Fig. 6
Fig. 6

Reflections in BHWC. (a)  ( k 1 ) and kth reflections on the objective mirrors. (b) Side view against the Y r axis. O k , incident point of the kth reflection on the objective mirrors; F k , incident point of the kth reflection on the field mirror.

Fig. 7
Fig. 7

Astigmatism of a 40-pass BHWC. Solid curve, accurate numerical results by our previous generalized method; Dashed curve, analytical results by our method; short-dashed curve, analytical results by Reesor’s theory; dotted curve, analytical results by Kohn’s method.

Fig. 8
Fig. 8

Deviation of analytical results from accurate astigmatism by two methods: dashed curve, our method; dotted curve, Kohn’s method.

Fig. 9
Fig. 9

Astigmatism of BHWC versus p, Δ, and n. (a) Astigmatism varying with p in a 40-pass PBWC with R = 625 mm , h = 20 mm , and Δ = 50 mm . (b) Astigmatism varying with Δ in a 40-pass PBWC with R = 625 mm , p = 40 mm , and h = 20 mm . (c) Astigmatism in the varying n-pass PBWC with R = 625 mm , p = 40 mm , h = 20 mm , and Δ = 50 mm . Solid curve, accurate numerical results; dashed curve, analytical results by our method; dotted curve, analytical results by Kohn’s method.

Fig. 10
Fig. 10

Optimizing h for astigmatism correction versus p in a 40-pass BHWC with Δ = 0.08 R . (a) Optimum h for astigmatism correction by two methods. (b) Relative error of optimum h for astigmatism correction by two methods. Solid curve, accurate numerical results by our previous generalized method; dashed curves, analytic results by our method; dotted curves, analytic results by Kohn’s method.

Fig. 11
Fig. 11

Reflection on the spherical mirror M 2 . The symbols follow the convention in Fig. 6.

Equations (65)

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α k = α 1 + j = 1 l 1 ε j ,
P ( z ) = [ I z I 0 I ] ,
Λ g ( θ , α ) = [ I 0 L g ( θ , α ) I ] = [ S ( α ) 0 0 S ( α ) ] [ I 0 L ( θ ) I ] [ S ( α ) 0 0 S ( α ) ] , S ( α ) = [ cos α sin α sin α cos α ] , L g ( θ , α ) = S ( α ) L ( θ ) S ( α ) , L ( θ ) = [ f s 1 0 0 f m 1 ] = [ 2 cos θ / R 0 0 2 / R / cos θ ] ,
M = P ( R ) { k = 2 n / 2 Λ g ( θ k , α k ) P ( R ) Λ g ( 0 , 0 ) P ( R ) } , Λ g ( θ 1 , α 1 ) P ( R ) P ( R / 2 ) Λ g ( 0 , 0 ) .
M = [ A B C D ] = [ I E 0 I ] [ 0 F F * 0 ] [ I G 0 I ] , E = A C 1 = [ e 11 e 12 e 12 e 22 ] = S ( β ) [ λ s 0 0 λ m ] S ( β ) , l s m = λ m λ s = ( e 11 e 22 ) 2 + 4 e 12 2 ,
M 0 = P ( R / 2 ) Λ g ( 0 , 0 ) , M 1 = Λ g ( θ 1 , α 1 ) P ( R ) P ( R / 2 ) Λ g ( 0 , 0 ) , M k = { k = 2 n / 2 Λ g ( θ k , α k ) P ( R ) Λ g ( 0 , 0 ) P ( R ) } · Λ g ( θ 1 , α 1 ) P ( R ) P ( R / 2 ) Λ g ( 0 , 0 ) ( k = 2 , ... , n / 2 ) ,
M = P ( z ) M = [ A + z C B + z D C D ] .
E = ( A + z C ) C 1 = E + z I .
M = Λ g M = [ I 0 L g I ] [ A B C D ] = [ A B L g A + C L g B + D ] .
E = A ( L g A + C ) 1 = ( E 1 + L g ) 1 .
E k 1 = S ( β k 1 ) [ ( λ s ) k 1 0 0 ( λ m ) k 1 ] S ( β k 1 ) ,
E k = [ ( { [ E k 1 + R · I ] 1 + L g ( 0 , 0 ) } 1 + R · I ) 1 + L g ( θ k , α k ) ] 1 .
E k 1 = S ( β k 1 ) [ ( e 11 ) k ( e 12 ) k ( e 12 ) k ( e 22 ) k ] S ( β k 1 ) , ( e 11 ) k = s k cos 2 γ k + m k sin 2 γ k + { [ 2 R 1 + ( R + ( λ s ) k 1 ) 1 ] 1 + R } 1 , ( e 12 ) k = ( s k m k ) sin γ k cos γ k , ( e 22 ) k = s k sin 2 γ k + m k cos 2 γ k + { [ 2 R 1 + ( R + ( λ m ) k 1 ) 1 ] 1 + R } 1 ,
( λ s ( m ) ) k R ,
| R + ( λ s ( m ) ) k 1 | R ,
( λ s ) k ( λ m ) k R 2 .
[ 2 R + ( λ s ( m ) ) k 1 ] 1 = [ R + ( R + ( λ s ( m ) ) k 1 ) ] 1 R 1 [ 1 ( R + ( λ s ( m ) ) k 1 ) R 1 ] ( λ s ( m ) ) k 1 / R 2 .
[ 2 R 1 + ( R + ( λ s ( m ) ) k 1 ) 1 ] 1 R + ( λ s ( m ) ) k 1 .
( e 11 ) k = s k cos 2 γ k + m k sin 2 γ k ( λ s ) k 1 / R 2 , ( e 12 ) k = ( s k m k ) sin γ k cos γ k , ( e 22 ) k = s k sin 2 γ k + m k cos 2 γ k ( λ m ) k 1 / R 2 .
[ ( e 11 ) k ( e 12 ) k ( e 12 ) k ( e 22 ) k ] 1 = ( λ s ) k ( λ m ) k [ ( e 22 ) k ( e 12 ) k ( e 12 ) k ( e 11 ) k ] .
E k = S ( β k 1 ) [ ( e 11 ) k ( e 12 ) k ; ( e 12 ) k ( e 22 ) k ] S ( β k 1 ) = S ( β k 1 ) [ ( e 11 ) k ( e 12 ) k ; ( e 12 ) k ( e 22 ) k ] S ( β k 1 ) ,
( e 11 ) k = ( λ s ) k ( λ m ) k ( e 22 ) k = R 2 [ s k sin 2 γ k + m k cos 2 γ k ] ( λ m ) k 1 , ( e 22 ) k = ( λ s ) k ( λ m ) k ( e 11 ) k = R 2 [ s k cos 2 γ k + m k sin 2 γ k ] ( λ s ) k 1 , ( e 12 ) k = ( λ s ) k ( λ m ) k ( e 12 ) k = R 2 [ ( s k m k ) sin γ k cos γ k ] .
σ k = [ ρ k 2 + σ k 1 2 + 2 ρ k σ k 1 cos 2 γ k ] 1 2 ,
2 ( β k β k 1 ) = { arctan ( ρ k sin ( 2 γ k ) ρ k cos ( 2 γ k ) + σ k 1 ) + π , for     ρ k cos ( 2 γ k ) + σ k 1 0 arctan ( ρ k sin ( 2 γ k ) ρ k cos ( 2 γ k ) + σ k 1 ) , for     ρ k cos ( 2 γ k ) + σ k 1 > 0 .
ρ k = 2 R sin 2 θ k .
σ k = σ k 1 + ρ k .
l s m = σ n / 2 = | k = 1 n / 2 ρ k | .
δ k = ( k 1 ) h Δ 2 2 R 2 .
l k = ( l k 1 c ) ( 1 + r k ) , h k = h k 1 ( 1 + r k ) ,
l k = l k 0 + ( p c / 2 ) i = 1 k r i c i = 1 k ( i 1 ) r i , h k = h 2 + h 2 i = 1 k r i .
l k = p ( k 1 2 ) c + Δ k R 2 [ 2 p ( p k c ) c 2 6 ( k 2 4 ) ] , h k = h 2 + h 2 Δ k R 2 ( 2 p k c ) .
sin θ k tan θ k l k 2 + h k 2 R ( 1 + l k Δ R 2 ) .
sin 2 θ k l k 2 + h k 2 R 2 ( 1 + 2 l k Δ R 2 ) = ( p ( k 1 / 2 ) c ) 2 + h 2 / 4 R 2 + τ R 4 + O ( R 6 ) ,
α k = α 1 + i = 1 k 1 ( ε k 1 + ε k 1 ) ,
cos ε k 1 l k l k 2 + h k 2 ( 1 + h k δ k l k Δ ) .
cos ε k 1 l k 1 ( 2 Δ + c ) + 2 h k 1 δ k 1 l k 1 2 + h k 1 2 ( 2 Δ + c ) l k 1 l k 1 2 + h k 1 2 ( 1 + h k 1 δ k 1 l k 1 Δ ) .
cos α k l k l k 2 + h k 2 ( 1 + h k δ k l k Δ ) .
l s m = | i = 1 n / 2 ρ k ( 2 R sin 2 θ k , 2 α k ) | .
l s m = i = 1 n / 2 2 R sin 2 θ k ( 2 cos 2 α k 1 ) .
l s m = n R ( p 2 3 h 2 4 ) n 2 p Δ 30 R 3 ( 5 h 2 + 2 p 2 ) .
l s m = n R ( p 2 3 ( 1 4 n 2 ) + h 2 4 ) n 2 p Δ 30 R 3 ( ( 10 n 2 5 2 ) h 2 + ( 2 32 n 4 ) p 2 ) .
Q O k O F k 1 2 P C F R + P C F .
δ k δ k 1 h Δ 2 2 R 2 .
δ k = ( k 1 ) h Δ 2 2 R 2 .
l k = l k 1 c , h k = h k 1 = h / 2 ,
cos θ k + 1 = [ R 2 + ( F k O k + 1 ) 2 ( C 2 F k ) 2 ] / ( 2 R · F k O k + 1 ) .
C 2 F k = ( ( l k c ) 2 + h k 2 ) 1 / 2 , F k O k + 1 = [ ( h k δ k + 1 ) 2 + ( l k c Δ ) 2 + ( R 2 Δ 2 δ k + 1 2 ) ] 1 / 2 ,
B 2 C 2 = R F k O k + 1 cos θ k + 1 = [ ( l k c ) Δ + h k δ k + 1 ] / R ( l k c ) Δ / R .
B B 3 C 2 O k + 1 B B 3 = B 3 F k + 1 C 2 B 3 ,
B 3 F k + 1 2 B 2 C 2 · F k C 2 C 2 O k + 1 = 2 ( l k c ) Δ R 2 F k C 2 .
h k + 1 = ( 1 + 2 ( l k c ) Δ R 2 ) h k , l k + 1 = ( 1 + 2 ( l k c ) Δ R 2 ) ( l k c ) .
h k + 1 = h k ( 1 + r k + 1 ) , l k + 1 = ( l k c ) ( 1 + r k + 1 ) ,
r k + 1 = 2 ( l k 0 c ) Δ R 2 = 2 ( l ( k + 1 ) 0 c ) Δ R 2 ,
tan θ k + 1 = B 2 F k R B 2 C 2 C 2 F k + 1 B 3 F k + 1 R ( 1 B 2 C 2 / R ) .
tan θ k + 1 C 2 F k + 1 R ( 1 + B 2 C 2 R ) = l k + 1 2 + h k + 1 2 R ( 1 + Δ l k + 1 R 2 ( 1 + r k + 1 ) 1 ) .
tan θ k + 1 l k + 1 2 + h k + 1 2 R ( 1 + l k + 1 Δ R 2 ) .
cos ( C 1 F k 1 H 1 ) = ( C 1 F k 1 ) 2 + ( F k 1 O k ) 2 ( C 1 O k ) 2 2 C 1 F k 1 · F k 1 O k .
C 1 H 1 = C 1 F k 1 sin ( C 1 F k 1 H 1 ) , F k 1 H 1 = C 1 F k 1 cos ( C 1 F k 1 H 1 ) .
cos ( G 1 F k 1 H 1 ) = ( O k 1 F k 1 ) 2 + ( F k 1 O k ) 2 ( O k 1 O k ) 2 2 O k 1 F k 1 · F k 1 O k , cos ( C 1 F k 1 G 1 ) = ( C 1 F k 1 ) 2 + ( F k 1 O k 1 ) 2 ( C 1 O k 1 ) 2 2 C 1 F k 1 · F k 1 O k 1 .
F k 1 G 1 = F k 1 H 1 / cos ( G 1 F k 1 H 1 ) , G 1 H 1 = F k 1 H 1 tan ( G 1 F k 1 H 1 ) .
( C 1 G 1 ) 2 = ( C 1 F k 1 ) 2 + ( F k 1 G 1 ) 2 2 C 1 F k 1 · F k 1 G 1 · cos ( C 1 F k 1 G 1 ) ,
cos ε k 1 = ( C 1 H 1 ) 2 + ( G 1 H 1 ) 2 ( C 1 G 1 ) 2 2 C 1 H 1 · G 1 H 1 .
C 1 P 1 = ( R 2 Δ 2 δ k 2 ) 1 / 2 , C 1 O k = R , C 1 F k 1 = ( l k 2 + h k 2 ) 1 / 2 / ( 1 + r k ) , F k 1 O k = [ ( C 1 P 1 ) 2 + ( h k / ( 1 + r k ) δ k ) 2 + ( l k / ( 1 + r k ) Δ ) 2 ] 1 / 2 , O k 1 O k = [ ( 2 Δ + c ) 2 + ( δ k + δ k 1 ) 2 ] 1 / 2 , F k 1 O k 1 = [ ( Δ + c + l k / ( 1 + r k ) ) 2 + ( C 1 P 1 ) 2 + ( h k / ( 1 + r k ) + δ k 1 ) 2 ] 1 / 2 , C 1 O k 1 = [ ( C 1 P 1 ) 2 + ( Δ + c ) 2 + δ k 1 2 ] 1 / 2 .
cos ε k 1 = [ ( 2 Δ + c ) l k + 2 h k δ k ] R 2 + ξ 1 R 0 { [ ( 2 Δ + c ) 2 + δ k 2 ] R 2 + ξ 2 R 0 } 1 / 2 · [ ( l k 2 + h k 2 ) R 2 + ξ 3 R 0 ] 1 / 2 .
cos ε k 1 = l k l k 2 + h k 2 ( 1 + 2 h k δ k ( 2 Δ + c ) l k ) l k l k 2 + h k 2 ( 1 + h k δ k l k Δ ) .

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