Abstract

This paper analyzes the influence of spherical aberration on the depth of focus of symmetrical optical systems for imaging of axial points. A calculation of a beam’s caustics is discussed using ray equations in the image plane and considering longitudinal spherical aberration as well. Concurrently, the influence of aberration coefficients on extremes of such a curve is presented. Afterwards, conditions for aberration coefficients are derived if the Strehl definition should be the same in two symmetrically placed planes with respect to the paraxial image plane. Such conditions for optical systems with large aberrations are derived with the use of geometric-optical approximation where the gyration diameter of the beam in given planes of the optical system is evaluated. Therefore, one can calculate aberration coefficients in such a way that the optical system generates a beam of rays that has the gyration radius in a given interval smaller than the defined limit value. Moreover, one can calculate the maximal depth of focus of the optical system respecting the aforementioned conditions.

© 2017 Optical Society of America

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References

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  1. A. Mikš, Applied Optics (CTU, 2009).
  2. S. F. Ray, Applied Photographic Optics (Focal, 2002).
  3. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, 1963).
  4. M. Born and E. Wolf, Principles of Optics (Oxford University, 1964).
  5. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).
  6. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
  7. O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, 1972).
  8. K. Miyamoto, “Wave optics and geometrical optics in optical design,” Prog. Opt. 1, 31–66 (1961).
    [Crossref]
  9. A. Miks, J. Novak, and P. Novak, “Dependence of Strehl ratio on f-number of optical system,” Appl. Opt. 51, 3804–3810 (2012).
    [Crossref]
  10. A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol. 44, 1043–1049 (2012).
    [Crossref]
  11. Z. Zalevsky, “Extended depth of focus imaging: a review,” SPIE Rev. 1, 018001 (2010).
    [Crossref]
  12. J. Ojeda-Castañeda and C. M. Gómez-Sarabia, “Tuning field depth at high resolution by pupil engineering,” Adv. Opt. Photon. 7, 814–880 (2015).
    [Crossref]
  13. A. Vázquez-Villa, J. A. Delgado-Atencio, S. Vázquez-Montiel, J. Castro-Ramos, and M. Cunill-Rodríguez, “Aspheric lens to increase the depth of focus,” Opt. Lett. 40, 2842–2845 (2015).
    [Crossref]
  14. S. Liu and H. Hua, “Extended depth-of-field microscopic imaging with a variable focus microscope objective,” Opt. Express 19, 353–362 (2011).
    [Crossref]
  15. T. Colomb, N. Pavillon, J. Kühn, E. Cuche, C. Depeursinge, and Y. Emery, “Extended depth-of-focus by digital holographic microscopy,” Opt. Lett. 35, 1840–1842 (2010).
    [Crossref]
  16. D. Elkind, Z. Zalevsky, U. Levy, and D. Mendlovic, “Optical transfer function shaping and depth of focus by using a phase only filter,” Appl. Opt. 42, 1925–1931 (2003).
    [Crossref]
  17. H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl. Opt. 40, 5658–5662 (2001).
    [Crossref]
  18. F. Diaz, F. Goudail, B. Loiseaux, and J. P. Huignard, “Design of a complex filter for depth of focus extension,” Opt. Lett. 34, 1171–1173 (2009).
    [Crossref]
  19. J. Ojeda-Castañeda and L. R. B. Valdós, “Arbitrarily high focal depth with finite apertures,” Opt. Lett. 13, 183–185 (1988).
    [Crossref]
  20. C. Rivolta, “Depth of focus of optical systems with a small amount of spherical aberration,” Appl. Opt. 29, 3249–3254 (1990).
    [Crossref]
  21. A. Mikš and J. Novák, “Dependence of depth of focus on spherical aberration of optical systems,” Appl. Opt. 55, 5931–5935 (2016).
    [Crossref]
  22. D. L. Shealy and D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
    [Crossref]
  23. D. G. Burkhard and D. L. Shealy, “Flux density for ray propagation in geometrical optics,” J. Opt. Soc. Am. 63, 299–304 (1973).
    [Crossref]
  24. D. L. Shealy and D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic paraboloid, and elliptic cone,” Appl. Opt. 12, 2955–2959 (1973).
    [Crossref]
  25. D. L. Shealy and D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
    [Crossref]
  26. D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
    [Crossref]
  27. T. B. Andersen, “Optical aberration functions: computation of caustic surfaces and illuminance in symmetrical systems,” Appl. Opt. 20, 3723–3728 (1981).
    [Crossref]
  28. K. Rektorys, Survey of Applicable Mathematics (MIT, 1969).
  29. Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, 1999).
  30. K. Miyamoto, “On a comparison between wave optics and geometrical optics by using Fourier analysis. I. General theory,” J. Opt. Soc. Am. 48, 57–63 (1958).
    [Crossref]
  31. K. Miyamoto, “On a comparison between wave optics and geometrical optics by using Fourier analysis. II. Astigmatism, coma, spherical aberration,” J. Opt. Soc. Am. 48, 567–575 (1958).
    [Crossref]
  32. K. Miyamoto, “On a comparison between wave optics and geometrical optics by using Fourier analysis. III. Image evaluation by spot diagram,” J. Opt. Soc. Am. 49, 35–40 (1959).
    [Crossref]

2016 (1)

2015 (2)

2012 (2)

A. Miks, J. Novak, and P. Novak, “Dependence of Strehl ratio on f-number of optical system,” Appl. Opt. 51, 3804–3810 (2012).
[Crossref]

A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol. 44, 1043–1049 (2012).
[Crossref]

2011 (1)

2010 (2)

2009 (1)

2003 (1)

2001 (1)

1990 (1)

1988 (1)

1981 (2)

1975 (1)

D. L. Shealy and D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[Crossref]

1973 (3)

1961 (1)

K. Miyamoto, “Wave optics and geometrical optics in optical design,” Prog. Opt. 1, 31–66 (1961).
[Crossref]

1959 (1)

1958 (2)

Andersen, T. B.

Born, M.

M. Born and E. Wolf, Principles of Optics (Oxford University, 1964).

Burkhard, D. G.

D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
[Crossref]

D. L. Shealy and D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[Crossref]

D. G. Burkhard and D. L. Shealy, “Flux density for ray propagation in geometrical optics,” J. Opt. Soc. Am. 63, 299–304 (1973).
[Crossref]

D. L. Shealy and D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
[Crossref]

D. L. Shealy and D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic paraboloid, and elliptic cone,” Appl. Opt. 12, 2955–2959 (1973).
[Crossref]

Castro-Ramos, J.

Colomb, T.

Cuche, E.

Cunill-Rodríguez, M.

Delgado-Atencio, J. A.

Depeursinge, C.

Diaz, F.

Elkind, D.

Emery, Y.

Gan, F.

Gómez-Sarabia, C. M.

Goudail, F.

Herzberger, M.

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

Hua, H.

Huignard, J. P.

Kravtsov, Y. A.

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, 1999).

Kühn, J.

Levy, U.

Liu, S.

Loiseaux, B.

Mendlovic, D.

Miks, A.

A. Miks, J. Novak, and P. Novak, “Dependence of Strehl ratio on f-number of optical system,” Appl. Opt. 51, 3804–3810 (2012).
[Crossref]

A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol. 44, 1043–1049 (2012).
[Crossref]

Mikš, A.

Miyamoto, K.

Novak, J.

A. Miks, J. Novak, and P. Novak, “Dependence of Strehl ratio on f-number of optical system,” Appl. Opt. 51, 3804–3810 (2012).
[Crossref]

A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol. 44, 1043–1049 (2012).
[Crossref]

Novak, P.

Novák, J.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, 1963).

Ojeda-Castañeda, J.

Orlov, Y. I.

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, 1999).

Pavillon, N.

Ray, S. F.

S. F. Ray, Applied Photographic Optics (Focal, 2002).

Rektorys, K.

K. Rektorys, Survey of Applicable Mathematics (MIT, 1969).

Rivolta, C.

Shealy, D. L.

D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
[Crossref]

D. L. Shealy and D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[Crossref]

D. G. Burkhard and D. L. Shealy, “Flux density for ray propagation in geometrical optics,” J. Opt. Soc. Am. 63, 299–304 (1973).
[Crossref]

D. L. Shealy and D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic paraboloid, and elliptic cone,” Appl. Opt. 12, 2955–2959 (1973).
[Crossref]

D. L. Shealy and D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
[Crossref]

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, 1972).

Valdós, L. R. B.

Vázquez-Montiel, S.

Vázquez-Villa, A.

Wang, H.

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Oxford University, 1964).

Zalevsky, Z.

Adv. Opt. Photon. (1)

Appl. Opt. (8)

J. Opt. Soc. Am. (4)

Opt. Acta (2)

D. L. Shealy and D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
[Crossref]

D. L. Shealy and D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[Crossref]

Opt. Express (1)

Opt. Laser Technol. (1)

A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol. 44, 1043–1049 (2012).
[Crossref]

Opt. Lett. (4)

Prog. Opt. (1)

K. Miyamoto, “Wave optics and geometrical optics in optical design,” Prog. Opt. 1, 31–66 (1961).
[Crossref]

SPIE Rev. (1)

Z. Zalevsky, “Extended depth of focus imaging: a review,” SPIE Rev. 1, 018001 (2010).
[Crossref]

Other (9)

K. Rektorys, Survey of Applicable Mathematics (MIT, 1969).

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, 1999).

A. Mikš, Applied Optics (CTU, 2009).

S. F. Ray, Applied Photographic Optics (Focal, 2002).

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, 1963).

M. Born and E. Wolf, Principles of Optics (Oxford University, 1964).

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).

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

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, 1972).

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Figures (6)

Fig. 1.
Fig. 1. Transverse spherical aberration.
Fig. 2.
Fig. 2. Calculation of caustics of a beam of rays.
Fig. 3.
Fig. 3. Transverse spherical aberration.
Fig. 4.
Fig. 4. Dependency of gyration diameter D g on defocus s 0 .
Fig. 5.
Fig. 5. Dependency of caustics (red line) of a beam of rays (blue lines) with respect to the gyration radius (green line) on defocus s 0 for a constant step in ray height h in the aperture plane.
Fig. 6.
Fig. 6. Energetic study of a beam with division of aperture into equal areas (blue, rays coming from centroids of the subapertures; green, gyration radius).

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

y = k x + h ,
k = ( h δ y ) / p ,
δ y = m = 2 M a 2 m 1 h 2 m 1 ,
k = h + m = 2 M a 2 m 1 h 2 m 1 p .
ϕ ( x , y , h ) = 0 , ϕ ( x , y , h ) / h = 0 ,
ϕ ( x , y , h ) = y k x h = 0 .
x c = p g ( h ) , y c = h f ( h ) g ( h ) ,
f ( h ) = h m = 2 M a 2 m 1 h 2 m 1 , g ( h ) = 1 m = 2 M ( 2 m 1 ) a 2 m 1 h 2 ( m 1 ) .
x = δ s + t cos α , y = t sin α ,
δ s = n = 1 N a 2 n ( tan α ) 2 n ,
J = | x α x t y α y t | = 0 .
t = 2 cos α n = 1 N n s 2 n ( tan α ) 2 n .
x c = n = 1 N ( 2 n + 1 ) s 2 n ( tan α ) 2 n , y c = 2 n = 1 N n s 2 n ( tan α ) ( 2 n + 1 ) .
f ( h ) e ( h ) = 0 ,
e ( h ) = m = 2 m = M 2 ( 2 m 1 ) ( m 1 ) a 2 m 1 h 2 m 3 .
10 a 5 2 h 8 + 13 a 3 a 5 h 6 + ( 3 a 3 2 10 a 5 ) h 4 3 a 3 h 2 = 0 .
h 1 = 0 , h 2 = 0 , h 3 = 30 10 a 3 a 5 a 5 , h 4 = h 3 , h 5 = a 3 + a 3 2 + 4 a 5 2 a 5 , h 6 = h 5 , h 7 = a 3 a 3 2 + 4 a 5 2 a 5 , h 8 = h 7 .
δ y = a 3 h 3 + a 5 h 5 + a 7 h 7 .
h 01 = a 5 + a 5 2 4 a 3 a 7 2 a 7 , h 02 = a 5 a 5 2 4 a 3 a 7 2 a 7 .
h 0 = ± a 3 a 5 .
W = W 20 r 2 + W 40 r 4 + W 60 r 6 ,
S.D. 1 ( 2 π / λ ) 2 E 0 ,
E 0 = W 2 ¯ W ¯ 2 , W ¯ = 2 0 1 W ( r ) r d r , W 2 ¯ = 2 0 1 W 2 ( r ) r d r .
E 0 = W 20 2 12 + W 20 ( W 40 6 + 3 W 60 20 ) + ( 4 W 40 2 45 + W 40 W 60 6 + 9 W 60 2 112 ) = e 2 W 20 2 + e 1 W 20 + e 0 ,
W 20 = s 0 8 F 2 , W 40 = 3 2 q 0 W 60 , W 60 = Δ s k 24 F 2 ( 1 q 0 ) = Δ s ext 6 F 2 q 0 2 ,
W 40 = 9 10 W 60 .
W 40 = 3 2 q 0 W 60 = 9 10 W 60 .
q 0 = r 0 2 = 3 / 5 .
W 40 = ± 5.351 12 ( E 0 ) p ( W 20 ) p 2 ,
( W 20 ) p = ( s 0 ) p 8 F 2 .
W 60 = 10 9 W 40 = 5.946 12 ( E 0 ) p ( W 20 ) p 2 .
| ( s 0 ) p | 16 F 2 3 ( E 0 ) p .
W 40 = H 4 4 R a 3 , W 60 = H 6 6 R a 5 ,
δ y = a 3 h 3 + a 5 h 5 .
r g 2 = 1 π H 2 0 2 π 0 H ( δ y 0 ) 2 h d h d φ = 2 H 2 0 H ( δ y 0 ) 2 h d h = 2 H 2 0 H ( δ y s 0 tan α ) 2 h d h ,
I g = β 1 + β 2 ( r g 2 ) ,
r g 2 = g 2 s 0 2 + g 1 s 0 + g 0 ,
g 2 = H 2 / ( 2 p 2 ) , g 1 = H 4 ( 3 H 2 a 5 + 4 a 3 ) / ( 6 p ) , g 0 = H 6 ( 10 H 4 a 5 2 + 24 H 2 a 3 a 5 + 15 a 3 2 ) / 60 .
g 2 = H 2 / ( 2 p 2 ) , g 1 = 8 W 40 / 3 3 W 60 , g 0 = 2 p 2 ( 10 W 40 2 + 24 W 40 W 60 + 15 W 60 2 ) / ( 5 H 2 ) .
a 3 = 3 4 H 2 a 5 .
a 5 = ± 4 210 2 p 2 ( r g 2 ) p H 2 ( s 0 ) p 2 7 H 5 p .
| ( s 0 ) lim | 2 p H ( r g ) p .
( r g ) min = g 0 .
( D g ) min = 2 ( r g ) min = 2 g 0 .
q 0 = r 0 2 = 3 / 4 .
a 5 = 9 H ( s 01 + s 02 ) ± 96 D 7 H 2 p , a 3 = 3 4 H 2 a 5 + 3 ( s 01 + s 02 ) 4 H 2 p ,
D = 9 H 2 ( s 01 2 + s 02 2 ) + 17 H 2 s 01 s 02 + 70 r g 2 p 2 .
r Ti = 4 3 Δ φ ( r i 3 r i 1 3 ) ( r i 2 r i 1 2 ) sin Δ φ 2 ,
r i + 1 = R 2 N + Δ ϕ · r i 2 , i = 0 , , N , r 0 = 0 .
r Ti 0 = 2 3 ( r i 3 r i 1 3 ) ( r i 2 r i 1 2 ) .

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