Abstract

A novel design for an all-reflective unobscured optical-power zoom (OPZ) objective with a zoom factor of 3 is presented. In contrast to OPZ objectives based on liquid lenses, all-reflective objectives use only reflective elements and are therefore free of chromatic aberrations. Thus, they can be used for a wide spectral range or in combination with image sensors that differ in their spectral characteristics. To avoid a decrease in image contrast encountered in on-axis designs with central obscuration, an unobscured off-axis or “Schiefspiegler” approach is adopted. The effective focal length of the objective is changed by two deformable mirrors, each with one actuator only. The simulated final design shows adequate image quality over the whole zooming range. Before starting the complex and cost-intensive development of deformable mirrors with the size, curvature, and dynamic range needed, the optical design should be evaluated first with respect to the practical achievable optical performance. Therefore, optomechanical setups with ultraprecision-manufactured solid mirrors were realized for three different focal lengths.

© 2009 Optical Society of America

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References

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  1. H. S. Lavin and R. Winter, “Controlled focus mirror with rim controlled flexure,” U.S. patent 4,059,346 (22 November 1977).
  2. J. Evans, “Apparatus for deforming thin diaphragms,” U.S. patent 2,403,915 (16 July 1946).
  3. E. F. Flint, “Variable focus lens,” U.S. patent 2,300,251(27 October 1942).
  4. H. Gross, F. Blechinger, and B. Achtner, Handbook of Optical Systems, Vol. 4, Survey of Optical Instruments (Wiley-VCH, 2008).
  5. RMA Electronics Inc., http://www.rmassa.com/manu/ivacorp.htm.
  6. Varioptic SA, “Varioptic liquid lens in autofocus webcams,” http://www.varioptic.com/en/news/newsroom-press-releases.php?code=134.
  7. L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4′′ CCD and CMOS sensors,” Proc. SPIE 5249, 288-296 (2004).
    [CrossRef]
  8. F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
    [CrossRef]
  9. F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
    [CrossRef]
  10. S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
    [CrossRef]
  11. S. Reichelt and H. Zappe, “Design of spherically corrected, achromatic variable-focus liquid lenses,” Opt. Express 15, 14146-14154 (2007).
    [CrossRef] [PubMed]
  12. D. V. Wick, T. Martinez, D. M. Payne, W. C. Sweatt, and S. R. Restaino, “Active optical zoom system,” Proc. SPIE 5798, 151-157 (2005).
    [CrossRef]
  13. G. A. Michelet and G. G. Bret, “Objective having a variable focal length,” U.S. patent 4,407,567 (4 October 1983).
  14. W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
    [CrossRef]
  15. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, 1980).
  16. J. R. Rogers, “Aberrations of unobscured reflective optical systems,” Ph.D. Dissertation (University of Arizona, 1983).
  17. A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).
  18. E. Dietzsch, “Die entwicklungsgeschichte der retrofokusobjektive vom typ flektogon,” in Jenaer Jahrbuch zur Technik- und Industriegeschichte 2002 (Glaux Verlag Jena, 2002).
  19. M. Berek, Grundlagen der Praktischen Optik (de Gruyter, 1970).
    [CrossRef]
  20. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1991).
  21. L. G. Cook, “Fast folded wide angle large reflective unobscured system,” U.S. patent 5,331,470 (19 July 1994).
  22. K. L. Hallam, B. J. Howell, and M. E. Wilson, “Wide-angle flat field telescope,” U.S. patent 4,598,981 (8 July 1986).

2007 (3)

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[CrossRef]

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

S. Reichelt and H. Zappe, “Design of spherically corrected, achromatic variable-focus liquid lenses,” Opt. Express 15, 14146-14154 (2007).
[CrossRef] [PubMed]

2006 (2)

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
[CrossRef]

2005 (1)

D. V. Wick, T. Martinez, D. M. Payne, W. C. Sweatt, and S. R. Restaino, “Active optical zoom system,” Proc. SPIE 5798, 151-157 (2005).
[CrossRef]

2004 (1)

L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4′′ CCD and CMOS sensors,” Proc. SPIE 5249, 288-296 (2004).
[CrossRef]

Achtner, B.

H. Gross, F. Blechinger, and B. Achtner, Handbook of Optical Systems, Vol. 4, Survey of Optical Instruments (Wiley-VCH, 2008).

Berek, M.

M. Berek, Grundlagen der Praktischen Optik (de Gruyter, 1970).
[CrossRef]

Berge, B.

F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
[CrossRef]

L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4′′ CCD and CMOS sensors,” Proc. SPIE 5249, 288-296 (2004).
[CrossRef]

Blechinger, F.

H. Gross, F. Blechinger, and B. Achtner, Handbook of Optical Systems, Vol. 4, Survey of Optical Instruments (Wiley-VCH, 2008).

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1991).

Bräuer, A.

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[CrossRef]

F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
[CrossRef]

Bret, G. G.

G. A. Michelet and G. G. Bret, “Objective having a variable focal length,” U.S. patent 4,407,567 (4 October 1983).

Burch, J. M.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

Cook, L. G.

L. G. Cook, “Fast folded wide angle large reflective unobscured system,” U.S. patent 5,331,470 (19 July 1994).

Craen, P.

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[CrossRef]

Deladi, S.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

Dietzsch, E.

E. Dietzsch, “Die entwicklungsgeschichte der retrofokusobjektive vom typ flektogon,” in Jenaer Jahrbuch zur Technik- und Industriegeschichte 2002 (Glaux Verlag Jena, 2002).

Evans, J.

J. Evans, “Apparatus for deforming thin diaphragms,” U.S. patent 2,403,915 (16 July 1946).

Fellner, T.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Flint, E. F.

E. F. Flint, “Variable focus lens,” U.S. patent 2,300,251(27 October 1942).

Gerrard, A.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

Greger, W.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Gross, H.

H. Gross, F. Blechinger, and B. Achtner, Handbook of Optical Systems, Vol. 4, Survey of Optical Instruments (Wiley-VCH, 2008).

Hallam, K. L.

K. L. Hallam, B. J. Howell, and M. E. Wilson, “Wide-angle flat field telescope,” U.S. patent 4,598,981 (8 July 1986).

Helwegen, I.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

Hendriks, B. H. W.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

Hösel, T.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Howell, B. J.

K. L. Hallam, B. J. Howell, and M. E. Wilson, “Wide-angle flat field telescope,” U.S. patent 4,598,981 (8 July 1986).

Kuiper, S.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

Lavin, H. S.

H. S. Lavin and R. Winter, “Controlled focus mirror with rim controlled flexure,” U.S. patent 4,059,346 (22 November 1977).

Martinez, T.

D. V. Wick, T. Martinez, D. M. Payne, W. C. Sweatt, and S. R. Restaino, “Active optical zoom system,” Proc. SPIE 5798, 151-157 (2005).
[CrossRef]

Mathieu, G.

L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4′′ CCD and CMOS sensors,” Proc. SPIE 5249, 288-296 (2004).
[CrossRef]

Michelet, G. A.

G. A. Michelet and G. G. Bret, “Objective having a variable focal length,” U.S. patent 4,407,567 (4 October 1983).

Mueller, C.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Payne, D. M.

D. V. Wick, T. Martinez, D. M. Payne, W. C. Sweatt, and S. R. Restaino, “Active optical zoom system,” Proc. SPIE 5798, 151-157 (2005).
[CrossRef]

Reichelt, S.

Reinicke, H.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Restaino, S. R.

D. V. Wick, T. Martinez, D. M. Payne, W. C. Sweatt, and S. R. Restaino, “Active optical zoom system,” Proc. SPIE 5798, 151-157 (2005).
[CrossRef]

Rogers, J. R.

J. R. Rogers, “Aberrations of unobscured reflective optical systems,” Ph.D. Dissertation (University of Arizona, 1983).

Saurei, L.

L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4′′ CCD and CMOS sensors,” Proc. SPIE 5249, 288-296 (2004).
[CrossRef]

Schoth, A.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Schreiber, P.

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[CrossRef]

F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
[CrossRef]

Suijver, J. F.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

Sweatt, W. C.

D. V. Wick, T. Martinez, D. M. Payne, W. C. Sweatt, and S. R. Restaino, “Active optical zoom system,” Proc. SPIE 5798, 151-157 (2005).
[CrossRef]

Thompson, K. P.

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, 1980).

Wick, D. V.

D. V. Wick, T. Martinez, D. M. Payne, W. C. Sweatt, and S. R. Restaino, “Active optical zoom system,” Proc. SPIE 5798, 151-157 (2005).
[CrossRef]

Wilde, J.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Wilson, M. E.

K. L. Hallam, B. J. Howell, and M. E. Wilson, “Wide-angle flat field telescope,” U.S. patent 4,598,981 (8 July 1986).

Winter, R.

H. S. Lavin and R. Winter, “Controlled focus mirror with rim controlled flexure,” U.S. patent 4,059,346 (22 November 1977).

Wippermann, F. C.

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[CrossRef]

F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1991).

Zappe, H.

Opt. Express (1)

Proc. SPIE (6)

L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4′′ CCD and CMOS sensors,” Proc. SPIE 5249, 288-296 (2004).
[CrossRef]

F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
[CrossRef]

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[CrossRef]

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

D. V. Wick, T. Martinez, D. M. Payne, W. C. Sweatt, and S. R. Restaino, “Active optical zoom system,” Proc. SPIE 5798, 151-157 (2005).
[CrossRef]

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Other (15)

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, 1980).

J. R. Rogers, “Aberrations of unobscured reflective optical systems,” Ph.D. Dissertation (University of Arizona, 1983).

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

E. Dietzsch, “Die entwicklungsgeschichte der retrofokusobjektive vom typ flektogon,” in Jenaer Jahrbuch zur Technik- und Industriegeschichte 2002 (Glaux Verlag Jena, 2002).

M. Berek, Grundlagen der Praktischen Optik (de Gruyter, 1970).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1991).

L. G. Cook, “Fast folded wide angle large reflective unobscured system,” U.S. patent 5,331,470 (19 July 1994).

K. L. Hallam, B. J. Howell, and M. E. Wilson, “Wide-angle flat field telescope,” U.S. patent 4,598,981 (8 July 1986).

G. A. Michelet and G. G. Bret, “Objective having a variable focal length,” U.S. patent 4,407,567 (4 October 1983).

H. S. Lavin and R. Winter, “Controlled focus mirror with rim controlled flexure,” U.S. patent 4,059,346 (22 November 1977).

J. Evans, “Apparatus for deforming thin diaphragms,” U.S. patent 2,403,915 (16 July 1946).

E. F. Flint, “Variable focus lens,” U.S. patent 2,300,251(27 October 1942).

H. Gross, F. Blechinger, and B. Achtner, Handbook of Optical Systems, Vol. 4, Survey of Optical Instruments (Wiley-VCH, 2008).

RMA Electronics Inc., http://www.rmassa.com/manu/ivacorp.htm.

Varioptic SA, “Varioptic liquid lens in autofocus webcams,” http://www.varioptic.com/en/news/newsroom-press-releases.php?code=134.

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Figures (9)

Fig. 1
Fig. 1

Collinear model of a simple optical-power zoom system in the wide-angle position with two variable optical-power elements at position 1 and 3, respectively.

Fig. 2
Fig. 2

Standard deviation of the Petzval sum versus separation d ^ 2 .

Fig. 3
Fig. 3

Optical power of the primary P ^ 1 , secondary P ^ 2 , and tertiary mirror P ^ 3 , as well as the Petzval sum versus the normalized system focal length with separation d ^ 2 = 11.8 .

Fig. 4
Fig. 4

Geometric estimation of the circle of confusion, which is caused by a curved image field.

Fig. 5
Fig. 5

Enhanced first-order design—OPZ objective shown in three different zooming positions: (a) wide-angle; (b) midzooming position; (c) tele.

Fig. 6
Fig. 6

Final design of a Schiefspiegler OPZ objective: (a)  f S = 5.2 mm ; (b)  f S = 7.8 mm ; (c)  f S = 15.6 mm ; 1, DM; 2, off-axis biconic concave mirror; 3, biconic convex mirror (acts as aperture stop); 4, second DM; 5, cover glass; 6, image sensor.

Fig. 7
Fig. 7

Spot diagrams for (a)  f S = 5.2 mm , (b)  f S = 7.8 mm , and (c)  f S = 15.6 mm for different field angles utilizing the horizontal symmetry of the system (scale: 40 μm ).

Fig. 8
Fig. 8

(a) Sectional 3D-CAD-model. (b) Opto-mechanical setup.

Fig. 9
Fig. 9

Images taken with OPZ setups: (a)  f S = 5.2 mm , (b)  f S = 7.8 mm , (c)  f S = 15.6 mm .

Tables (2)

Tables Icon

Table 1 Design Parameters for a Given Back Focal Length Resulting from the Enhanced First-Order Design

Tables Icon

Table 2 Design Parameters of the OPZ’s Final Design

Equations (26)

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1 r ^ p = i = 1 N P ^ i = i = 1 N 1 f ^ i ,
P ^ 1 ( f ^ S ) = f ^ S d ^ 2 f ^ S P ^ 2 + s ^ f ^ S ( d ^ 1 d ^ 2 + d ^ 1 d ^ 2 P ^ 2 ) ,
P ^ 3 ( f ^ S ) = d ^ 2 f ^ S + s ^ d ^ 1 [ 1 + P ^ 2 ( d ^ 2 + s ^ ) ] s ^ ( d ^ 1 + d ^ 2 d ^ 1 d ^ 2 P ^ 2 ) .
d ^ 1 = 1 P ^ 1   mid + 1 P ^ 2 .
f ^ S = 1 P ^ 1 + P ^ 2 + P ^ 3 d ^ 2 P ^ 2 P ^ 3 d ^ 1 P ^ 1 P ^ 2 d ^ 1 P ^ 1 P ^ 3 d ^ 2 P ^ 1 P ^ 3 + d ^ 1 d ^ 2 P ^ 1 P ^ 2 P ^ 3 ,
s ^ = f ^ S ( 1 d ^ 2 P ^ 2 + d ^ 1 d ^ 2 P ^ 1 P ^ 2 d ^ 1 P ^ 1 d ^ 2 P ^ 1 ) ,
1 f ^ S   mid = P ^ 3   mid P ^ 1   mid P ^ 2 ,
s ^ = ( P ^ 3   mid ) 1 .
1 r ^ P   mid = P ^ 1   mid + P ^ 2 + P ^ 3   mid = 0.
P ^ 1   mid = ( f ^ S   mid s ^ ) 1 ,
P ^ 2 = [ s ^ ( s ^ f ^ S   mid 1 ) ] 1 .
d ^ 1 = 1 f ^ S   mid ( f ^ S   mid s ^ ) 2 .
1 r ^ P σ = [ 1 f ^ S max f ^ S min f ^ S min f ^ S max ( P ^ 1 + P ^ 2 + P ^ 3 ) 2 d f ^ S ] 1 / 2 .
d ^ 2 = L ^ d ^ 1 s ^ .
Δ z = r P [ r P 2 ( d 2 ) 2 ] 1 / 2 = 126 μm ,
u = Δ z F # = 126 μm 4.5 = 28 μm ,
z = r 2 R { 1 + [ 1 ( 1 + κ ) ( r R ) 2 ] 1 / 2 } ,
E 1 = ( 1 0 P ^ 1 1 ) , E 2 = ( 1 0 P ^ 2 1 ) , E 3 = ( 1 0 P ^ 3 1 ) .
S 1 = ( 1 d ^ 1 0 1 ) , S 2 = ( 1 d ^ 2 0 1 ) .
M = ( A B C D ) = E 3 · S 2 · E 2 · S 1 · E 1 ,
A = 1 d ^ 2 P ^ 2 + d ^ 1 d ^ 2 P ^ 1 P ^ 2 d ^ 1 P ^ 1 d ^ 2 P ^ 1 , B = d ^ 1 d ^ 1 d ^ 2 P ^ 2 + d ^ 2 , C = P ^ 1 P ^ 2 P ^ 3 + d ^ 2 P ^ 2 P ^ 3 + d ^ 1 P ^ 1 P ^ 2 + d ^ 1 P ^ 1 P ^ 3 + d ^ 2 P ^ 1 P ^ 3 d ^ 1 d ^ 2 P ^ 1 P ^ 2 P ^ 3 , D = 1 d ^ 1 P ^ 2 d ^ 1 P ^ 3 d ^ 2 P ^ 3 + d ^ 1 d ^ 2 P ^ 2 P ^ 3 .
s ^ = A C = f ^ S A = f ^ S ( 1 d ^ 2 P ^ 2 + d ^ 1 d ^ 2 P ^ 1 P ^ 2 d ^ 1 P ^ 1 d ^ 2 P ^ 1 ) ,
P ^ 1 = f ^ S d ^ 2 f ^ S P ^ 2 + s ^ f ^ S ( d ^ 1 d ^ 2 + d ^ 1 d ^ 2 P ^ 2 ) .
f ^ S = 1 C = 1 P ^ 1 + P ^ 2 + P ^ 3 d ^ 2 P ^ 2 P ^ 3 d ^ 1 P ^ 1 P ^ 2 d ^ 1 P ^ 1 P ^ 3 d ^ 2 P ^ 1 P ^ 3 + d ^ 1 d ^ 2 P ^ 1 P ^ 2 P ^ 3 .
P ^ 3 = 1 f ^ S ( 1 + f ^ S ( P ^ 1 + P ^ 2 d ^ 1 P ^ 1 P ^ 2 ) 1 + d ^ 1 P ^ 1 + d ^ 2 ( P ^ 1 + P ^ 2 d ^ 1 P ^ 1 P ^ 2 ) ) .
P ^ 3 = d ^ 2 f ^ S + s ^ d ^ 1 ( 1 + P ^ 2 ( d ^ 2 + s ^ ) ) s ^ ( d ^ 1 + d ^ 2 d ^ 1 d ^ 2 P ^ 2 ) .

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