Abstract

We study the formation of the caustic surfaces formed in both convex-plane and plano-convex spherical lenses by considering a plane wave incident on the lens along the optical axis. Using the caustic formulas and a paraxial approximation we derive analytical expressions to evaluate the spherical aberration to third order. Furthermore, we apply the formulas to evaluate the circle of least confusion for a positive lens.

© 2010 Optical Society of America

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References

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  1. M. Avendaño-Alejo, R. Díaz-Uribe, and I. Moreno, “Caustics caused by refraction in the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 25, 1586–1593 (2008).
    [CrossRef]
  2. J. A. Lock, C. L. Adler, and E. A. Hovenac, “Exterior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder: semiclassical scattering theory analysis,” J. Opt. Soc. Am. A 17, 1846–1856 (2000).
    [CrossRef]
  3. A. Cordero-Dávila and J. Castro-Ramos, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror,” Appl. Opt. 37, 6774–6778 (1998).
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    [CrossRef] [PubMed]
  6. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-Function and its Ramifications (Wiley-VCH, 2006), Chap. 12, pp. 179–186.
    [CrossRef]
  7. O. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), Chap. 10, pp. 161–179.
  8. O. N. Stavroudis and R. C. Fronczek, “Caustic surfaces and the structure of the geometrical image,” J. Opt. Soc. Am. 66, 795–800 (1976).
    [CrossRef]
  9. O. N. Stavroudis, R. C. Fronczek, and R.-S. Chang, “Geometry of the half-symmetric image,” J. Opt. Soc. Am. 68, 739–742 (1978).
    [CrossRef]
  10. O. N. Stavroudis, “The k function in geometrical optics and its relationship to the archetypal wave front and the caustic surface,” J. Opt. Soc. Am. A 12, 1010–1016 (1995).
    [CrossRef]
  11. D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A 25, 2370–2382 (2008).
    [CrossRef]
  12. G. Silva-Ortigoza, J. Castro-Ramos, and A. Cordero-Dávila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021–1028 (2001).
    [CrossRef]
  13. G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A, Pure Appl. Opt. 4, 358–365 (2002).
    [CrossRef]
  14. J. Castro-Ramos, O. de Ita Prieto, and G. Silva-Ortigoza, “Computation of the disk of least confusion for conic mirrors,” Appl. Opt. 43, 6080–6089 (2004).
    [CrossRef] [PubMed]
  15. D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
    [CrossRef] [PubMed]
  16. A. V. Gitin, “Legendre transformation: Connection between transverse aberration of an optical system and its caustic,” Opt. Commun. 281, 3062–3066 (2008).
  17. R. W. Hosken, “Circle of least confusion of a spherical reflector,” Appl. Opt. 46, 3107–3117 (2007).
    [CrossRef] [PubMed]
  18. M. Avendaño-Alejo, D. M. González-Utrera, V. I. Moreno-Oliva, and R. Díaz-Uribe, “Null Hartmann screen for measuring the spherical aberration in a plane-convex lens,” in Proceedings of the 18th IMEKO TC 2 (2008), pp. 1–6.
  19. J. J. Stoker, Differential Geometry (Wiley-Interscience, 1969), Chap. 2, pp. 12–52.
  20. A. E. Conrady, Applied Optics & Optical Design Part 1 (Dover, 1957), Chap. II, pp. 72–125.
  21. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University Press, 1970), Chap. 4, pp. 35–82.
  22. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, 1974), Chap. 6, pp. 73–110.
  23. D. Malacara and Z. Malacara, Handbook of Lens Design (Marcel Dekker, 1994), Chap. 5, pp. 149–153.
  24. R. C. Spencer, “Focal region of a spherical reflector circle of least confusion,” Appl. Opt. 7, 1644–1645 (1968).
    [CrossRef] [PubMed]
  25. R. C. Spencer and G. Hyde, “Studies of the focal region of a spherical reflector: geometric optics,” IEEE Trans. Antennas Propag. 16, 317–324 (1968).
    [CrossRef]
  26. C. Y. Tsai, “A general calculation of the 3-D disk of least confusion using skew ray tracing,” Appl. Phys. B 96, 517–525 (2009).
    [CrossRef]
  27. O. Stravroudis, Handbook of Optical Engineering (Marcel Dekker, 2001), Chap. 1, pp. 1–38.
  28. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999), Chap. V, pp. 229–260.

2009

C. Y. Tsai, “A general calculation of the 3-D disk of least confusion using skew ray tracing,” Appl. Phys. B 96, 517–525 (2009).
[CrossRef]

2008

A. V. Gitin, “Legendre transformation: Connection between transverse aberration of an optical system and its caustic,” Opt. Commun. 281, 3062–3066 (2008).

M. Avendaño-Alejo, D. M. González-Utrera, V. I. Moreno-Oliva, and R. Díaz-Uribe, “Null Hartmann screen for measuring the spherical aberration in a plane-convex lens,” in Proceedings of the 18th IMEKO TC 2 (2008), pp. 1–6.

M. Avendaño-Alejo, R. Díaz-Uribe, and I. Moreno, “Caustics caused by refraction in the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 25, 1586–1593 (2008).
[CrossRef]

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A 25, 2370–2382 (2008).
[CrossRef]

2007

2006

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-Function and its Ramifications (Wiley-VCH, 2006), Chap. 12, pp. 179–186.
[CrossRef]

2004

2002

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A, Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

2001

2000

1999

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999), Chap. V, pp. 229–260.

1998

1995

1994

D. Malacara and Z. Malacara, Handbook of Lens Design (Marcel Dekker, 1994), Chap. 5, pp. 149–153.

1982

1981

1978

1977

1976

1974

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, 1974), Chap. 6, pp. 73–110.

1972

O. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), Chap. 10, pp. 161–179.

1970

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University Press, 1970), Chap. 4, pp. 35–82.

1969

J. J. Stoker, Differential Geometry (Wiley-Interscience, 1969), Chap. 2, pp. 12–52.

1968

R. C. Spencer and G. Hyde, “Studies of the focal region of a spherical reflector: geometric optics,” IEEE Trans. Antennas Propag. 16, 317–324 (1968).
[CrossRef]

R. C. Spencer, “Focal region of a spherical reflector circle of least confusion,” Appl. Opt. 7, 1644–1645 (1968).
[CrossRef] [PubMed]

1957

A. E. Conrady, Applied Optics & Optical Design Part 1 (Dover, 1957), Chap. II, pp. 72–125.

Adler, C. L.

Avendaño-Alejo, M.

M. Avendaño-Alejo, R. Díaz-Uribe, and I. Moreno, “Caustics caused by refraction in the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 25, 1586–1593 (2008).
[CrossRef]

M. Avendaño-Alejo, D. M. González-Utrera, V. I. Moreno-Oliva, and R. Díaz-Uribe, “Null Hartmann screen for measuring the spherical aberration in a plane-convex lens,” in Proceedings of the 18th IMEKO TC 2 (2008), pp. 1–6.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999), Chap. V, pp. 229–260.

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University Press, 1970), Chap. 4, pp. 35–82.

Burkhard, D. G.

Carvente-Muñoz, O.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A, Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

Castro-Ramos, J.

Chang, R. -S.

Conrady, A. E.

A. E. Conrady, Applied Optics & Optical Design Part 1 (Dover, 1957), Chap. II, pp. 72–125.

Cordero-Dávila, A.

de Ita Prieto, O.

Díaz-Uribe, R.

M. Avendaño-Alejo, R. Díaz-Uribe, and I. Moreno, “Caustics caused by refraction in the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 25, 1586–1593 (2008).
[CrossRef]

M. Avendaño-Alejo, D. M. González-Utrera, V. I. Moreno-Oliva, and R. Díaz-Uribe, “Null Hartmann screen for measuring the spherical aberration in a plane-convex lens,” in Proceedings of the 18th IMEKO TC 2 (2008), pp. 1–6.

Fronczek, R. C.

Gitin, A. V.

A. V. Gitin, “Legendre transformation: Connection between transverse aberration of an optical system and its caustic,” Opt. Commun. 281, 3062–3066 (2008).

González-Utrera, D. M.

M. Avendaño-Alejo, D. M. González-Utrera, V. I. Moreno-Oliva, and R. Díaz-Uribe, “Null Hartmann screen for measuring the spherical aberration in a plane-convex lens,” in Proceedings of the 18th IMEKO TC 2 (2008), pp. 1–6.

Hoffnagle, J. A.

Hosken, R. W.

Hovenac, E. A.

Hyde, G.

R. C. Spencer and G. Hyde, “Studies of the focal region of a spherical reflector: geometric optics,” IEEE Trans. Antennas Propag. 16, 317–324 (1968).
[CrossRef]

Lock, J. A.

Malacara, D.

D. Malacara and Z. Malacara, Handbook of Lens Design (Marcel Dekker, 1994), Chap. 5, pp. 149–153.

Malacara, Z.

D. Malacara and Z. Malacara, Handbook of Lens Design (Marcel Dekker, 1994), Chap. 5, pp. 149–153.

Marciano-Melchor, M.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A, Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

Moreno, I.

Moreno-Oliva, V. I.

M. Avendaño-Alejo, D. M. González-Utrera, V. I. Moreno-Oliva, and R. Díaz-Uribe, “Null Hartmann screen for measuring the spherical aberration in a plane-convex lens,” in Proceedings of the 18th IMEKO TC 2 (2008), pp. 1–6.

Shealy, D. L.

Silva-Ortigoza, G.

Silva-Ortigoza, R.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A, Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

Spencer, R. C.

R. C. Spencer, “Focal region of a spherical reflector circle of least confusion,” Appl. Opt. 7, 1644–1645 (1968).
[CrossRef] [PubMed]

R. C. Spencer and G. Hyde, “Studies of the focal region of a spherical reflector: geometric optics,” IEEE Trans. Antennas Propag. 16, 317–324 (1968).
[CrossRef]

Stavroudis, O.

O. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), Chap. 10, pp. 161–179.

Stavroudis, O. N.

Stoker, J. J.

J. J. Stoker, Differential Geometry (Wiley-Interscience, 1969), Chap. 2, pp. 12–52.

Stravroudis, O.

O. Stravroudis, Handbook of Optical Engineering (Marcel Dekker, 2001), Chap. 1, pp. 1–38.

Theocaris, P. S.

Tsai, C. Y.

C. Y. Tsai, “A general calculation of the 3-D disk of least confusion using skew ray tracing,” Appl. Phys. B 96, 517–525 (2009).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, 1974), Chap. 6, pp. 73–110.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999), Chap. V, pp. 229–260.

Appl. Opt.

Appl. Phys. B

C. Y. Tsai, “A general calculation of the 3-D disk of least confusion using skew ray tracing,” Appl. Phys. B 96, 517–525 (2009).
[CrossRef]

IEEE Trans. Antennas Propag.

R. C. Spencer and G. Hyde, “Studies of the focal region of a spherical reflector: geometric optics,” IEEE Trans. Antennas Propag. 16, 317–324 (1968).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A, Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

A. V. Gitin, “Legendre transformation: Connection between transverse aberration of an optical system and its caustic,” Opt. Commun. 281, 3062–3066 (2008).

Other

M. Avendaño-Alejo, D. M. González-Utrera, V. I. Moreno-Oliva, and R. Díaz-Uribe, “Null Hartmann screen for measuring the spherical aberration in a plane-convex lens,” in Proceedings of the 18th IMEKO TC 2 (2008), pp. 1–6.

J. J. Stoker, Differential Geometry (Wiley-Interscience, 1969), Chap. 2, pp. 12–52.

A. E. Conrady, Applied Optics & Optical Design Part 1 (Dover, 1957), Chap. II, pp. 72–125.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University Press, 1970), Chap. 4, pp. 35–82.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, 1974), Chap. 6, pp. 73–110.

D. Malacara and Z. Malacara, Handbook of Lens Design (Marcel Dekker, 1994), Chap. 5, pp. 149–153.

O. Stravroudis, Handbook of Optical Engineering (Marcel Dekker, 2001), Chap. 1, pp. 1–38.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999), Chap. V, pp. 229–260.

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-Function and its Ramifications (Wiley-VCH, 2006), Chap. 12, pp. 179–186.
[CrossRef]

O. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), Chap. 10, pp. 161–179.

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Figures (8)

Fig. 1
Fig. 1

Process of refraction produced by a plano-convex lens and its associated parameters by considering that the point source is located at infinity.

Fig. 2
Fig. 2

Caustic produced by a plano-convex lens when the point source is located at infinity. Also shown is the process used to obtain the CLC.

Fig. 3
Fig. 3

(a) Comparison between exact and paraxial caustics produced by plano-convex lenses considering three numbers F / # . The aperture for all the cases is 25   mm h 25   mm . (b) Zoom exclusively of the caustics.

Fig. 4
Fig. 4

Graphical method to obtain the maximum value which gives the radius of the CLC and the plane where it is placed by using n i = 1.517 , n a = 1 , R = 38.76   mm for a plano-convex lens as a function of the height.

Fig. 5
Fig. 5

Process of refraction produced by a convex-plane lens and its associated parameters considering that the point source is located at infinity.

Fig. 6
Fig. 6

Caustic produced by a convex-plane lens when the point source is located at infinity and the principal surface formed by it.

Fig. 7
Fig. 7

(a) Comparison between exact and paraxial caustics produced by convex-plane lenses considering three numbers F / # for an aperture of 25   mm h 25   mm . (b) Zoom exclusively of the caustics.

Fig. 8
Fig. 8

Systems of references: x, y, and z are the exit pupil coordinates; ξ, η, and ζ are the coordinates of the image point. The aberrations are measured taking the Gaussian image as the ideal point.

Equations (38)

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y   cos ( θ a θ i ) + z   sin ( θ a θ i ) = ( t R ) sin ( θ a θ i ) + R   sin   θ a .
θ a = arcsin [ n i n a sin   θ i ] ,     θ a θ i = n i   cos   θ i n a 2 n i 2 sin 2 θ i .
y   sin ( θ a θ i ) + z   cos ( θ a θ i ) = ( t R ) cos ( θ a θ i ) + ( θ a / θ i ) R   cos   θ a θ a / θ i 1 ,
z ( θ i ) = t R + ( n i R n a 2 ( n i 2 n a 2 ) ) [ n i n a 2 cos 3 θ i + ( n a 2 n i 2 sin 2 θ i ) 3 / 2 ] ,
y ( θ i ) = R n i 2 sin 3 θ i n a 2 .
z ( h ) = t R + n i [ n i n a 2 ( R 2 h 2 ) 3 / 2 + ( R 2 n a 2 h 2 n i 2 ) 3 / 2 ] n a 2 ( n i 2 n a 2 ) R 2 ,
y ( h ) = h 3 n i 2 R 2 n a 2 .
z p ( h ) t + n a R n i n a [ 1 3 2 ( h n i R n a ) 2 ] ,
y p ( h ) = R n a n i ( h n i R n a ) 3 ,
Y p = K p c 1 / 2 Z p 3 / 2 ,
y = H ( n i 2 n a 2 ) ( z ( R 2 H 2 + t R ) ) n a 2 R 2 H 2 + n i n a 2 R 2 n i 2 H 2 H .
y = h ( n i 2 n a 2 ) ( z ( R 2 h 2 + t R ) ) n a 2 R 2 h 2 + n i n a 2 R 2 n i 2 h 2 + h .
Z i ( h ) = t R + n i [ h ( n a 2 U + n i V ) ( n i u + v ) + H ( n a 2 u + n i v ) ( n i U + V ) ] ( n i 2 n a 2 ) ( n a 2 ( H u + h U ) + n i ( H v + h V ) ) ,
Y i ( h ) = h H n i ( v V + n i ( u U ) ) n a 2 ( H u + h U ) + n i ( H v + h V ) ,
h 2 n i H R 2 n a 2 = n a 2 R 2 n i 2 h 2 n a 2 R 2 n i 2 H 2 + n i ( R 2 h 2 R 2 H 2 ) n a 2 ( H R 2 h 2 + h R 2 H 2 ) + n i ( H n a 2 R 2 n i 2 h 2 + h n a 2 R 2 n i 2 H 2 ) ,
h 2 n i [ n a 2 ( H u + h U ) + n i ( H v + h V ) ] H R 2 n a 2 [ v V + n i ( u U ) ] = 0.
2 n a R ( ( n a + n i ) R ( n i R 2 H 2 + n a 2 R 2 n i 2 H 2 ) ) 3 n i ( n a + n i ) h 2 = 0.
h CLC = 2 n a R ( ( n a + n i ) R ( n i R 2 H 2 + n a 2 R 2 n i 2 H 2 ) ) 3 n i ( n a + n i ) .
h c a = 2 n a 3 n i ( 1 n i n a n i + n a ) R .
y R   sin   θ a = tan ( θ a θ i ) [ z R ( 1 cos   θ a ) ] ,
P i = ( t , R   sin   θ a tan ( θ a θ i ) [ t R ( 1 cos   θ a ) ] ) ,
y = R   sin   θ a tan   θ t [ t R ( 1 cos   θ a ) ] tan   θ r [ z t ] ,
y   cos   θ r + z   sin   θ r = R [ cos   θ r   sin   θ i cos   θ t ] + t   sin   θ r + ( R t ) cos   θ r   tan   θ t .
y   sin   θ r + z   cos   θ r = R [ cos   θ r   cos   θ i cos   θ t ] θ i / θ a θ r / θ a R   sin   θ r ( sin   θ i + sin   θ t ) cos   θ t + t   cos ( θ r θ t ) cos   θ t + ( R t + R   sin   θ i   sin   θ t ) [ cos   θ r cos 2 θ t ] θ t / θ a θ r / θ a .
z ( θ a ) = Q cos 2 θ r cos   θ t + t ,
y ( θ a ) = Q   sin   θ r   cos   θ r cos   θ t + ( R t ) tan   θ t + R   sin   θ i cos   θ t ,
Q = R   cos   θ i ( θ i / θ a θ r / θ a ) + ( R t cos   θ t + R   tan   θ t   sin   θ i ) ( θ t / θ a θ r / θ a ) ,
P S = ( t + ( R t R 2 h 2 ) n a 2 R 4 h 2 ( n i 2 R 2 n a 2 h 2 n a R 2 h 2 ) 2 n a h 2 + n i 2 R 2 n a 2 h 2 R 2 h 2 , h ) .
P P P = ( n i n a ) t / n i .
z p ( h ) F 3 h 2 { n i 2 R [ n i 3 2 n a ( n i 2 n a 2 ) ] ( n i n a ) 4 ( n a + n i ) t } 2 n a ( n i n a ) n i 3 R 2 ,
y p ( h ) h 3 { n i 2 R [ n i 3 2 n a ( n i 2 n a 2 ) ] ( n i n a ) 4 ( n a + n i ) t } n a 2 n i 3 R 3 .
Y p = K c p 1 / 2 Z p 3 / 2 ,
K c p = 8 ( n i n a ) 3 n i 3 27 n a ( n i 2 [ n i 3 2 n a ( n i 2 n a 2 ) ] R ( n i n a ) 4 ( n a + n i ) t ) ,
W ( x 2 + y 2 , y η , η 2 ) = b 1 ( x 2 + y 2 ) 2 + b 2 y η ( x 2 + y 2 ) + b 3 y 2 η 2 + b 4 η 2 ( x 2 + y 2 ) + b 5 y η 3 + third-   and   higher - order   terms + ,
η = ζ 3 / 2 3 R 2 n 3 b 1 ,
Y p = Z p 3 / 2 3 ( n i n a n a R ) 2 n a 3 b 1 .
b 1 p c = n i 2 ( n i n a ) 8 n a 2 R 3 .
b 1 c p = ( n i n a ) ( n i 2 [ n i 3 + 2 n a ( n a 2 n i 2 ) ] R + ( n a n i ) 4 ( n a + n i ) t 8 n a 2 n i 3 R 4 ) .

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