Paraxial analysis of three-component zoom lens with fixed distance between object and image points and fixed position of image-space focal point

Open Access

Abstract

This work performs an analysis of basic optical properties of zoom lenses with a fixed distance between object and image points and a fixed position of the image-space focal point. Formulas for the calculation of paraxial parameters of such optical systems are derived and the calculation is presented on examples.

© 2014 Optical Society of America

1. Introduction

Optical systems with variable optical parameters (zoom lenses) find applications in various areas and many papers [122] are dedicated to their optical design. The change of a focal length or a transverse magnification can be achieved by the change of the position of individual elements of the optical system. The position of the image plane is required to be fixed during the change of the focal length in order the image was sharp in the whole range of the focal length change. If it is required that the image should be located at a specific constant distance from the object for a given range of magnification, then the position of the image-space focal point of the classical zoom lens is not fixed and changes its position during the magnification change.

The aim of this work is to perform an analysis of paraxial optical properties of zoom lenses with a fixed distance between object and image points and a fixed position of the image-space focal point and derive formulas for the calculation of paraxial parameters of such optical systems. Zoom lenses with the fixed distance between object and image points and fixed position of the image-space focal point may find their applications in optical systems for information processing [23], where it is possible to affect the amplitude and phase using the spatial filter, which is positioned in the focal plane. As far as we know the analysis of such a type of the zoom lens was not published yet.

2. Paraxial imaging properties of optical system

It is well-known from the theory of geometrical optics that every optical system is characterized by its focal length $f′$, the position of the object focal point $sF$, and the position of the image focal point $s′F′$. The power $φ$ of the optical system is defined by the formula $φ=n′/f′$, where $n′$ is the refractive index of image space. Further, if we consider that the image and object media is air ($n=n′=1$) and the optical system is composed of N thin lenses, then we can define Gaussian brackets [24, 25]

$α=[dN−1,−φN−1,dN−2,−φN−2,dN−3,−φN−3,...,d1,−φ1],β=[dN−1,−φN−1,dN−2,−φN−2,dN−3,−φN−3,...,d1],γ=[−φN,dN−1,−φN−1,dN−2,−φN−2,dN−3,...,d1,−φ1],δ=[−φN,dN−1,−φN−1,dN−2,−φN−2,dN−3,...,d1],$
where $φi$ is the power of i-th lens, and $di$is the distance between (i + 1)-st a i-th lens. Then, it holds for basic paraxial parameters [24, 25]
$φ=−γ, sF=δ/γ, s′F′=−α/γ, s′=−β−α sδ−γ s, m=s′γ+α=1δ−sγ,$
where s is the distance of the object from the first element of the optical system, $s′$is the distance of the image from the last element of the optical system, and m is the transverse magnification of the optical system.

We focus on the analysis of paraxial properties of a three-component zoom lens (Fig. 1), where $α,β,γ,$, and $δ$ are given by Eq. (1), $φ1, φ2, φ3$ are values of the optical power of individual components, and d1, d2 are distances between components of the zoom lens. We can write for such zoom lens [24, 25] following formulas

$α=1−d2(φ1+φ2−φ1φ2d1)−φ1d1,$
$β=d1+d2−φ2d1d2,$
$γ=−φ=−(φ1+φ2+φ3)+φ1φ2d1+φ2φ3d2+φ1φ3(d1+d2)−φ1φ2φ3d1d2,$
$δ=1−d1(φ2+φ3)−d2φ3+d1d2φ2φ3.$
Consider an optical system presented in Fig. 1, where ξ is the object plane, ξ' is the image plane and m is the transverse magnification of the optical system. The mutual distance $L=AA′¯$ of planes ξ and ξ' (Fig. 1) can be expressed as
$L=AA′¯=−s+d1+d2+s′$
and the distance $D=AF′¯$ of points A and $F′$(image-space focal point) can be expressed as
$D=AF′¯=−s+d1+d2+s′F′.$
Let us demand now that the distances $L=AA′¯$ and $D=AF′¯$ remain constant during the change of magnification m. If we require the correction of field curvature of the third order of the optical system [2631], then it holds
$SIV≈0.62(φ1+φ2+φ3)=0,$
where $SIV$ is the Petzval sum.

Fig. 1 Three-element optical system.

By substitution of Eqs. (2)(6) and Eq. (9) into Eqs. (7) and (8) we obtain after a tedious derivation for the calculation of the distance $d2$ the following nonlinear equation

$c4d24+c3d23+c2d22+c1d2+c0=0,$
where
$Q=L−D,c4=φ12φ23(φ1+φ2)3Q[φ1m+φ2Q(φ1+φ2)],c3=φ12φ22(φ1+φ2)2Qm[(Q(4mφ22+φ1φ2)−m2(2φ1−2φ2+φ1φ2D))(φ1+φ2)+2φ1φ2m2−4φ2mQ(φ1+φ2)2],c2=φ12φ2(φ1+φ2)m[φ2m3(φ1+φ2)+φ13m2Q+φ1φ2Q(φ1+φ2)(3φ1m2D−4m2+φ1Q(7m−3))],c1=−φ13φ2(φ1+φ2)m[2m3+3φ12(2m−1)Q2+φ1m2Q(3φ1D−2)],c0=φ14m2+φ16Qm[m2D+(2m−1)Q].$
The distance $d1$ can be calculated from the following equations
$a2d12+a1d1+a0=0,$
where
$a2=φ1φ2d2(φ1+φ2)−φ12,a1=φ12L−φ1(φ1+φ2)(2d2−φ2d22+φ2Ld2),a0=m+d2(L−d2)(φ1+φ2)2+1/m−2,$
and
$b2d12+b1d1+b0=0,$
where
$b2=φ1φ2d2(φ1+φ2)−φ12,b1=φ12D−φ1(φ1+φ2)(2d2−φ2d22+φ2Dd2),b0=d2(D−d2)(φ1+φ2)2+1/m−2.$
The values L, D, $φ1$ and $φ2$ are given as input parameters. Other parameters can be calculated by the following procedure. The distance $d2$ is calculated using Eq. (10) for a given value m of the transverse magnification. This distance is then substituted into Eq. (11) and Eq. (12). The distance $d1$ is calculated as the mutual solution of both Eqs. (11) and (12). One has to choose the solution for the distance $d1$, which is the same for both equations. In such case that Eq. (10) has complex roots or the value $d2$ is negative, then we must change the values of powers $φ1$ and $φ2$ in order the solution was real and positive ($d2>0$).

In case that the image-space focal point $F′$ or the image point $A′$ lies inside the optical system ($s′F′<0$,$s′<0$), then one needs to put another optical system with the fixed focal length behind the zoom system, which images points $F′$ and $A′$ as points $F″$ and $A″$ that are located behind the system as one can see from Fig. 2. It is possible to place the spatial filter in the plane, which pass through the point $F″$, and the image will lie in the plane, which passes through the point $A″$. The position of points $A″$ and $F″$ is fixed and does not change during the change of magnification.

Fig. 2 Combination of proposed zoom lens and fix focus lens.

3. Examples

We present an example of the calculation of basic parameters of the three-component zoom lens with the fixed distance between object and image points and the fixed position of the image-space focal point. We choose L = 150 mm and D = 110 mm. The results of the calculation for different values of focal length $f′1$ and $f′2$ and different values m of the transverse magnification are presented for three variants of the zoom lens in Tables 1, 2, and 3. All linear dimensions in tables are given in millimeters.

Table 1. Parameters of the First Variant of Zoom Lens (L = 150 mm, D = 110 mm)

Table 2. Parameters of the Second Variant of Zoom Lens (L = 150 mm, D = 110 mm)

Table 3. Parameters of the Third Variant of Zoom Lens (L = 150 mm, D = 110 mm)

4. Conclusion

A theoretical analysis of paraxial optical properties of the three-component zoom lens with the fixed distance between object and image points and the fixed position of the image-space focal point was presented in our work. Formulas for the calculation of paraxial parameters of such zoom lenses were derived. The procedure of the calculation of parameters of zoom lenses was presented on examples. Zoom lenses with the fixed distance between object and image points and the fixed position of image-space focal point may find their applications in optical systems for information processing, where it is possible to affect the amplitude and phase using the spatial filter, which is positioned in the fixed focal plane of the zoom lens. The advantage of the proposed solution, i.e. the zoom lens with the fixed distance between object and image points and the fixed position of the image-space focal point, is the fact that one can affect Fourier spectra of objects, which are characterized by the different spatial structure, with just one spatial filter. The position of the image-space focal point does not change with the change of focal length or the transverse magnification of the zoom lens and the position of the investigated object stays also fixed.

Acknowledgment

This work has been supported by the Czech Science Foundation grant 13-31765S.

1. A. D. Clark, Zoom Lenses (Adam Hilger, 1973).

2. K. Yamaji, “Design of zoom lenses,” in Progress in Optics, ed. E.Wolf (North-Holland, 1967), vol. VI, pp. 105–170.

3. A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. 47(32), 6088–6098 (2008). [CrossRef]   [PubMed]

4. T. Kryszczyński and J. Mikucki, “Structural optical design of the complex multi-group zoom systems by means of matrix optics,” Opt. Express 21(17), 19634–19647 (2013). [CrossRef]   [PubMed]

5. A. Walther, “Angle eikonals for a perfect zoom system,” J. Opt. Soc. Am. A 18(8), 1968–1971 (2001). [CrossRef]   [PubMed]

6. G. Wooters and E. W. Silvertooth, “Optically compensated zoom lens,” J. Opt. Soc. Am. 55(4), 347–351 (1965). [CrossRef]

7. D. F. Kienholz, “The design of a zoom lens with a large computer,” Appl. Opt. 9(6), 1443–1452 (1970). [CrossRef]   [PubMed]

8. A. V. Grinkevich, “Version of an objective with variable focal length,” J. Opt. Technol. 73(5), 343–345 (2006). [CrossRef]

9. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component type,” Appl. Opt. 21(12), 2174–2183 (1982). [CrossRef]   [PubMed]

10. G. H. Matter and E. T. Luszcz, “A family of optically compensated zoom lenses,” Appl. Opt. 9(4), 844–848 (1970). [CrossRef]   [PubMed]

11. K. Tanaka, “Erratum: Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component type,” Appl. Opt. 21(21), 3805 (1982). [CrossRef]   [PubMed]

12. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 2: Generalization of Yamaji type V,” Appl. Opt. 21(22), 4045–4053 (1982). [CrossRef]   [PubMed]

13. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 3: Five-component type,” Appl. Opt. 22(4), 541–553 (1983). [CrossRef]   [PubMed]

14. A. Miks and J. Novak, “Paraxial analysis of four-component zoom lens with fixed distance between focal points,” Appl. Opt. 51(21), 5231–5235 (2012). [CrossRef]   [PubMed]

15. A. Mikš, J. Novák, and P. Novák, “Three-element zoom lens with fixed distance between focal points,” Opt. Lett. 37(12), 2187–2189 (2012). [CrossRef]   [PubMed]

16. A. Mikš and J. Novák, “Design of a double-sided telecentric zoom lens,” Appl. Opt. 51(24), 5928–5935 (2012). [CrossRef]   [PubMed]

17. T. Kryszczyński, “Development of the double-sided telecentric three-component zoom systems by means of matrix optics,” Proc. SPIE 7141, 71411Y (2008). [CrossRef]

18. T. Kryszczyński, M. Leśniewski, and J. Mikucki, “Use of matrix optics to analyze the complex multi-group zoom systems,” Proc. SPIE 8697, 86970I (2012). [CrossRef]

19. S. Pal and L. Hazra, “Ab initio synthesis of linearly compensated zoom lenses by evolutionary programming,” Appl. Opt. 50(10), 1434–1441 (2011). [CrossRef]   [PubMed]

20. L. Hazra and S. Pal, “A novel approach for structural synthesis of zoom systems,” Proc. SPIE 7786, 778607 (2010). [CrossRef]

21. S. Pal and L. Hazra, “Stabilization of pupils in a zoom lens with two independent movements,” Appl. Opt. 52(23), 5611–5618 (2013). [CrossRef]   [PubMed]

22. S. Pal, “Aberration correction of zoom lenses using evolutionary programming,” Appl. Opt. 52(23), 5724–5732 (2013). [CrossRef]   [PubMed]

23. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

24. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

25. M. Herzberger, “Gaussian optics and Gaussian brackets,” J. Opt. Soc. Am. 33(12), 651–652 (1943). [CrossRef]

26. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

27. A. Mikš, “Modification of the formulas for third-order aberration coefficients,” J. Opt. Soc. Am. A 19(9), 1867–1871 (2002). [CrossRef]   [PubMed]

28. A. Miks, Applied Optics (Czech Technical University, 2009).

29. H. Haferkorn, Bewertung Optischer Systeme (VEB Deutscher Verlag der Wissenschaften, 1986).

30. W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).

31. A. Mikš and P. Novák, “Theoretical and experimental analysis of basic parameters of two-element optical systems,” Appl. Opt. 51(30), 7286–7294 (2012). [CrossRef]   [PubMed]

References

• View by:

1. A. D. Clark, Zoom Lenses (Adam Hilger, 1973).
2. K. Yamaji, “Design of zoom lenses,” in Progress in Optics, ed. E.Wolf (North-Holland, 1967), vol. VI, pp. 105–170.
3. A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. 47(32), 6088–6098 (2008).
[Crossref] [PubMed]
4. T. Kryszczyński and J. Mikucki, “Structural optical design of the complex multi-group zoom systems by means of matrix optics,” Opt. Express 21(17), 19634–19647 (2013).
[Crossref] [PubMed]
5. A. Walther, “Angle eikonals for a perfect zoom system,” J. Opt. Soc. Am. A 18(8), 1968–1971 (2001).
[Crossref] [PubMed]
6. G. Wooters and E. W. Silvertooth, “Optically compensated zoom lens,” J. Opt. Soc. Am. 55(4), 347–351 (1965).
[Crossref]
7. D. F. Kienholz, “The design of a zoom lens with a large computer,” Appl. Opt. 9(6), 1443–1452 (1970).
[Crossref] [PubMed]
8. A. V. Grinkevich, “Version of an objective with variable focal length,” J. Opt. Technol. 73(5), 343–345 (2006).
[Crossref]
9. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component type,” Appl. Opt. 21(12), 2174–2183 (1982).
[Crossref] [PubMed]
10. G. H. Matter and E. T. Luszcz, “A family of optically compensated zoom lenses,” Appl. Opt. 9(4), 844–848 (1970).
[Crossref] [PubMed]
11. K. Tanaka, “Erratum: Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component type,” Appl. Opt. 21(21), 3805 (1982).
[Crossref] [PubMed]
12. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 2: Generalization of Yamaji type V,” Appl. Opt. 21(22), 4045–4053 (1982).
[Crossref] [PubMed]
13. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 3: Five-component type,” Appl. Opt. 22(4), 541–553 (1983).
[Crossref] [PubMed]
14. A. Miks and J. Novak, “Paraxial analysis of four-component zoom lens with fixed distance between focal points,” Appl. Opt. 51(21), 5231–5235 (2012).
[Crossref] [PubMed]
15. A. Mikš, J. Novák, and P. Novák, “Three-element zoom lens with fixed distance between focal points,” Opt. Lett. 37(12), 2187–2189 (2012).
[Crossref] [PubMed]
16. A. Mikš and J. Novák, “Design of a double-sided telecentric zoom lens,” Appl. Opt. 51(24), 5928–5935 (2012).
[Crossref] [PubMed]
17. T. Kryszczyński, “Development of the double-sided telecentric three-component zoom systems by means of matrix optics,” Proc. SPIE 7141, 71411Y (2008).
[Crossref]
18. T. Kryszczyński, M. Leśniewski, and J. Mikucki, “Use of matrix optics to analyze the complex multi-group zoom systems,” Proc. SPIE 8697, 86970I (2012).
[Crossref]
19. S. Pal and L. Hazra, “Ab initio synthesis of linearly compensated zoom lenses by evolutionary programming,” Appl. Opt. 50(10), 1434–1441 (2011).
[Crossref] [PubMed]
20. L. Hazra and S. Pal, “A novel approach for structural synthesis of zoom systems,” Proc. SPIE 7786, 778607 (2010).
[Crossref]
21. S. Pal and L. Hazra, “Stabilization of pupils in a zoom lens with two independent movements,” Appl. Opt. 52(23), 5611–5618 (2013).
[Crossref] [PubMed]
22. S. Pal, “Aberration correction of zoom lenses using evolutionary programming,” Appl. Opt. 52(23), 5724–5732 (2013).
[Crossref] [PubMed]
23. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
24. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
25. M. Herzberger, “Gaussian optics and Gaussian brackets,” J. Opt. Soc. Am. 33(12), 651–652 (1943).
[Crossref]
26. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
27. A. Mikš, “Modification of the formulas for third-order aberration coefficients,” J. Opt. Soc. Am. A 19(9), 1867–1871 (2002).
[Crossref] [PubMed]
28. A. Miks, Applied Optics (Czech Technical University, 2009).
29. H. Haferkorn, Bewertung Optischer Systeme (VEB Deutscher Verlag der Wissenschaften, 1986).
30. W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).
31. A. Mikš and P. Novák, “Theoretical and experimental analysis of basic parameters of two-element optical systems,” Appl. Opt. 51(30), 7286–7294 (2012).
[Crossref] [PubMed]

2012 (5)

T. Kryszczyński, M. Leśniewski, and J. Mikucki, “Use of matrix optics to analyze the complex multi-group zoom systems,” Proc. SPIE 8697, 86970I (2012).
[Crossref]

2010 (1)

L. Hazra and S. Pal, “A novel approach for structural synthesis of zoom systems,” Proc. SPIE 7786, 778607 (2010).
[Crossref]

2008 (2)

T. Kryszczyński, “Development of the double-sided telecentric three-component zoom systems by means of matrix optics,” Proc. SPIE 7141, 71411Y (2008).
[Crossref]

Hazra, L.

L. Hazra and S. Pal, “A novel approach for structural synthesis of zoom systems,” Proc. SPIE 7786, 778607 (2010).
[Crossref]

Kryszczynski, T.

T. Kryszczyński, M. Leśniewski, and J. Mikucki, “Use of matrix optics to analyze the complex multi-group zoom systems,” Proc. SPIE 8697, 86970I (2012).
[Crossref]

T. Kryszczyński, “Development of the double-sided telecentric three-component zoom systems by means of matrix optics,” Proc. SPIE 7141, 71411Y (2008).
[Crossref]

Lesniewski, M.

T. Kryszczyński, M. Leśniewski, and J. Mikucki, “Use of matrix optics to analyze the complex multi-group zoom systems,” Proc. SPIE 8697, 86970I (2012).
[Crossref]

Mikucki, J.

T. Kryszczyński, M. Leśniewski, and J. Mikucki, “Use of matrix optics to analyze the complex multi-group zoom systems,” Proc. SPIE 8697, 86970I (2012).
[Crossref]

Pal, S.

L. Hazra and S. Pal, “A novel approach for structural synthesis of zoom systems,” Proc. SPIE 7786, 778607 (2010).
[Crossref]

Proc. SPIE (3)

L. Hazra and S. Pal, “A novel approach for structural synthesis of zoom systems,” Proc. SPIE 7786, 778607 (2010).
[Crossref]

T. Kryszczyński, “Development of the double-sided telecentric three-component zoom systems by means of matrix optics,” Proc. SPIE 7141, 71411Y (2008).
[Crossref]

T. Kryszczyński, M. Leśniewski, and J. Mikucki, “Use of matrix optics to analyze the complex multi-group zoom systems,” Proc. SPIE 8697, 86970I (2012).
[Crossref]

Other (8)

A. D. Clark, Zoom Lenses (Adam Hilger, 1973).

K. Yamaji, “Design of zoom lenses,” in Progress in Optics, ed. E.Wolf (North-Holland, 1967), vol. VI, pp. 105–170.

A. Miks, Applied Optics (Czech Technical University, 2009).

H. Haferkorn, Bewertung Optischer Systeme (VEB Deutscher Verlag der Wissenschaften, 1986).

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Figures (2)

Fig. 1 Three-element optical system.
Fig. 2 Combination of proposed zoom lens and fix focus lens.

Tables (3)

Table 1 Parameters of the First Variant of Zoom Lens (L = 150 mm, D = 110 mm)

Table 2 Parameters of the Second Variant of Zoom Lens (L = 150 mm, D = 110 mm)

Table 3 Parameters of the Third Variant of Zoom Lens (L = 150 mm, D = 110 mm)

Equations (15)

$α=[ d N−1 ,− φ N−1 , d N−2 ,− φ N−2 , d N−3 ,− φ N−3 ,..., d 1 ,− φ 1 ], β=[ d N−1 ,− φ N−1 , d N−2 ,− φ N−2 , d N−3 ,− φ N−3 ,..., d 1 ], γ=[ − φ N , d N−1 ,− φ N−1 , d N−2 ,− φ N−2 , d N−3 ,..., d 1 ,− φ 1 ], δ=[ − φ N , d N−1 ,− φ N−1 , d N−2 ,− φ N−2 , d N−3 ,..., d 1 ],$
$φ=−γ, s F =δ/γ, s ′ F ′ =−α/γ, s ′ =− β−α s δ−γ s , m= s ′ γ+α= 1 δ−sγ ,$
$α = 1 − d 2 ( φ 1 + φ 2 − φ 1 φ 2 d 1 ) − φ 1 d 1 ,$
$β = d 1 + d 2 − φ 2 d 1 d 2 ,$
$γ = − φ = − ( φ 1 + φ 2 + φ 3 ) + φ 1 φ 2 d 1 + φ 2 φ 3 d 2 + φ 1 φ 3 ( d 1 + d 2 ) − φ 1 φ 2 φ 3 d 1 d 2 ,$
$δ = 1 − d 1 ( φ 2 + φ 3 ) − d 2 φ 3 + d 1 d 2 φ 2 φ 3 .$
$L = A A ′ ¯ = − s + d 1 + d 2 + s ′$
$D = A F ′ ¯ = − s + d 1 + d 2 + s ′ F ′ .$
$S I V ≈ 0.62 ( φ 1 + φ 2 + φ 3 ) = 0 ,$
$c 4 d 2 4 + c 3 d 2 3 + c 2 d 2 2 + c 1 d 2 + c 0 =0,$
$Q=L−D, c 4 = φ 1 2 φ 2 3 ( φ 1 + φ 2 ) 3 Q[ φ 1 m+ φ 2 Q( φ 1 + φ 2 )], c 3 = φ 1 2 φ 2 2 ( φ 1 + φ 2 ) 2 Q m [(Q(4m φ 2 2 + φ 1 φ 2 )− m 2 (2 φ 1 −2 φ 2 + φ 1 φ 2 D))( φ 1 + φ 2 ) +2 φ 1 φ 2 m 2 −4 φ 2 mQ ( φ 1 + φ 2 ) 2 ], c 2 = φ 1 2 φ 2 ( φ 1 + φ 2 ) m [ φ 2 m 3 ( φ 1 + φ 2 )+ φ 1 3 m 2 Q+ φ 1 φ 2 Q( φ 1 + φ 2 )(3 φ 1 m 2 D−4 m 2 + φ 1 Q(7m−3))], c 1 =− φ 1 3 φ 2 ( φ 1 + φ 2 ) m [2 m 3 +3 φ 1 2 (2m−1) Q 2 + φ 1 m 2 Q(3 φ 1 D−2)], c 0 = φ 1 4 m 2 + φ 1 6 Q m [ m 2 D+(2m−1)Q].$
$a 2 d 1 2 + a 1 d 1 + a 0 =0,$
$a 2 = φ 1 φ 2 d 2 ( φ 1 + φ 2 )− φ 1 2 , a 1 = φ 1 2 L− φ 1 ( φ 1 + φ 2 )(2 d 2 − φ 2 d 2 2 + φ 2 L d 2 ), a 0 =m+ d 2 (L− d 2 ) ( φ 1 + φ 2 ) 2 +1/m−2,$
$b 2 d 1 2 + b 1 d 1 + b 0 =0,$
$b 2 = φ 1 φ 2 d 2 ( φ 1 + φ 2 )− φ 1 2 , b 1 = φ 1 2 D− φ 1 ( φ 1 + φ 2 )(2 d 2 − φ 2 d 2 2 + φ 2 D d 2 ), b 0 = d 2 (D− d 2 ) ( φ 1 + φ 2 ) 2 +1/m−2.$

Metrics

Select as filters

Cancel