Abstract

Alvarez and Lohmann lenses are variable focus optical devices based on lateral shifts of two lenses with cubic-type surfaces. I analyzed the optical performance of these types of lenses computing the first order optical properties (applying wavefront refraction and propagation) without the restriction of the thin lens approximation, and the spot diagram using a ray tracing algorithm. I proposed an analytic and numerical method to select the most optimum coefficients and the specific configuration of these lenses. The results show that Lohmann composite lens is slightly superior to Alvarez one because the overall thickness and optical aberrations are smaller.

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References

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  1. L. W. Alvarez, “Development of variable- focus lenses and a new refractor,” J. Am. Optom. Assoc. 49(1), 24–29 (1978).
    [PubMed]
  2. L. W. Alvarez, and W. E. Humphrey, “Variable power lens and system,” US Patent 3,507,565 (1970).
  3. A. W. Lohmann, “A New Class Of Varifocal Lenses,” Appl. Opt. 9(7), 1669–1671 (1970).
    [CrossRef] [PubMed]
  4. W. E. Humphrey, “Remote subjective refractor employing continuously variable sphere-cylinder corrections,” Opt. Eng. 15, 286–291 (1976).
  5. W. E. Humphrey, “Apparatus for opthalmological prescription readout,” US Patent 3,927,933 (1974).
  6. S. S. Rege, T. S. Tkaczyk, and M. R. Descour, “Application of the Alvarez-Humphrey concept to the design of a miniaturized scanning microscope,” Opt. Express 12(12), 2574–2588 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-12-2574 .
    [CrossRef] [PubMed]
  7. G. L. Van Der Heijde, “Artificial intraocular lens e.g. Alvarez-type lens for implantation in eye, comprises two lens elements with optical thickness such that power of the lens changes by transversal displacement of one lens element relative to the other element,” Int Patent 2006/025726 (2006).
  8. A. N. Simonov, G. Vdovin, and M. C. Rombach, “Cubic optical elements for an accommodative intraocular lens,” Opt. Express 14(17), 7757–7775 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7757 .
    [CrossRef] [PubMed]
  9. H. Mukaiyama, K. Kato, and A. Komatsu, “Variable focus visual power correction device - has optical lenses superposed so that main meridians are coincident with one another,” Int Patent 93/15432 (1994).
  10. G. L. van Der Heijde, “Universal spectacles for children in developing countries,” in Mopane: Conference on Visual Optics (Kruger National Park, South Africa, 2006).
  11. B. Spivey, “Lens for spectacles, has thickness designed so that by adjusting relative positions of two lenses in direction perpendicular to viewing direction, combined focus of two lenses is changed,” US Patent 151184 (2008).
  12. C. F. Cheung, and W. B. Lee, Surface generation in ultra-precision diamond turning: modelling and practices (Professional Engineering Publ., London 2003).
  13. I. M. Barton, S. N. Dixit, L. J. Summers, K. Avicola, J. Wilhelmsen, and J. Wilhelmsen, “Diffractive Alvarez lens,” Opt. Lett. 25(1), 1–3 (2000), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-25-1-1 .
    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, San Francisco, 1968).
  15. J. C. Maxwell, “On the Focal Lines of a Refracted Pencil,” Proc. London Math. Soc. s1–4, 337–343 (1873).
  16. A. Walther, The ray and wave theory of lenses (Cambridge University Press, 1995).
  17. J. Rubinstein and G. Wolansky, “Differential relations for the imaging coefficients of asymmetric systems,” J. Opt. Soc. Am. A 20(12), 2365–2369 (2003), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-20-12-2365 .
    [CrossRef]
  18. B. D. Stone and G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9(1), 96–109 (1992), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-9-5-832 .
    [CrossRef]
  19. A. Gullstrand, “Die Reelle Optische Abbildung,” Sven. Vetensk. Handl 41, 1–119 (1906).
  20. J. B. Keller and H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Opt. Soc. Am. A 40(1), 48–52 (1950), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-40-1-48 .
    [CrossRef]
  21. J. A. Kneisly, “Local Curvature Of Wavefronts In Optical System,” J. Opt. Soc. Am. A 54(2), 229–235 (1964), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-54-2-229 .
    [CrossRef]
  22. O. N. Stavroudis, The mathematics of geometrical and physical optics: the k-function and its ramifications (Wiley-VCH, Weinheim, 2006).
  23. J. E. A. Landgrave and J. R. MoyaCessa, “Generalized Coddington equations in ophthalmic lens design,” J. Opt. Soc. Am. A 13(8), 1637–1644 (1996), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-13-8-1637 .
    [CrossRef]
  24. D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical-system,” Appl. Opt. 20(5), 897–909 (1981), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-20-5-897 .
    [CrossRef] [PubMed]
  25. M. P. Keating, Geometric, physical, and visual optics (Butterworth-Heinemann, Boston, 2002).
  26. W. F. Harris, “Wavefronts and their propagation in astigmatic optical systems,” Optom. Vis. Sci. 73(9), 605–612 (1996).
    [CrossRef]
  27. E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vis. Sci. 82(10), 923–932 (2005).
    [CrossRef] [PubMed]
  28. C. E. Campbell, “Generalized Coddington equations found via an operator method,” J. Opt. Soc. Am. A 23(7), 1691–1698 (2006), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-23-7-1691 .
    [CrossRef]

2006 (2)

2005 (1)

E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vis. Sci. 82(10), 923–932 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

2000 (1)

1996 (2)

1992 (1)

1981 (1)

1978 (1)

L. W. Alvarez, “Development of variable- focus lenses and a new refractor,” J. Am. Optom. Assoc. 49(1), 24–29 (1978).
[PubMed]

1976 (1)

W. E. Humphrey, “Remote subjective refractor employing continuously variable sphere-cylinder corrections,” Opt. Eng. 15, 286–291 (1976).

1970 (1)

1964 (1)

J. A. Kneisly, “Local Curvature Of Wavefronts In Optical System,” J. Opt. Soc. Am. A 54(2), 229–235 (1964), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-54-2-229 .
[CrossRef]

1950 (1)

J. B. Keller and H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Opt. Soc. Am. A 40(1), 48–52 (1950), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-40-1-48 .
[CrossRef]

1906 (1)

A. Gullstrand, “Die Reelle Optische Abbildung,” Sven. Vetensk. Handl 41, 1–119 (1906).

1873 (1)

J. C. Maxwell, “On the Focal Lines of a Refracted Pencil,” Proc. London Math. Soc. s1–4, 337–343 (1873).

Acosta, E.

E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vis. Sci. 82(10), 923–932 (2005).
[CrossRef] [PubMed]

Alvarez, L. W.

L. W. Alvarez, “Development of variable- focus lenses and a new refractor,” J. Am. Optom. Assoc. 49(1), 24–29 (1978).
[PubMed]

Avicola, K.

Barton, I. M.

Blendowske, R.

E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vis. Sci. 82(10), 923–932 (2005).
[CrossRef] [PubMed]

Burkhard, D. G.

Campbell, C. E.

Descour, M. R.

Dixit, S. N.

Forbes, G. W.

Gullstrand, A.

A. Gullstrand, “Die Reelle Optische Abbildung,” Sven. Vetensk. Handl 41, 1–119 (1906).

Harris, W. F.

W. F. Harris, “Wavefronts and their propagation in astigmatic optical systems,” Optom. Vis. Sci. 73(9), 605–612 (1996).
[CrossRef]

Humphrey, W. E.

W. E. Humphrey, “Remote subjective refractor employing continuously variable sphere-cylinder corrections,” Opt. Eng. 15, 286–291 (1976).

Keller, H. B.

J. B. Keller and H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Opt. Soc. Am. A 40(1), 48–52 (1950), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-40-1-48 .
[CrossRef]

Keller, J. B.

J. B. Keller and H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Opt. Soc. Am. A 40(1), 48–52 (1950), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-40-1-48 .
[CrossRef]

Kneisly, J. A.

J. A. Kneisly, “Local Curvature Of Wavefronts In Optical System,” J. Opt. Soc. Am. A 54(2), 229–235 (1964), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-54-2-229 .
[CrossRef]

Landgrave, J. E. A.

Lohmann, A. W.

Maxwell, J. C.

J. C. Maxwell, “On the Focal Lines of a Refracted Pencil,” Proc. London Math. Soc. s1–4, 337–343 (1873).

MoyaCessa, J. R.

Rege, S. S.

Rombach, M. C.

Rubinstein, J.

Shealy, D. L.

Simonov, A. N.

Stone, B. D.

Summers, L. J.

Tkaczyk, T. S.

Vdovin, G.

Wilhelmsen, J.

Wolansky, G.

Appl. Opt. (2)

J. Am. Optom. Assoc. (1)

L. W. Alvarez, “Development of variable- focus lenses and a new refractor,” J. Am. Optom. Assoc. 49(1), 24–29 (1978).
[PubMed]

J. Opt. Soc. Am. A (6)

Opt. Eng. (1)

W. E. Humphrey, “Remote subjective refractor employing continuously variable sphere-cylinder corrections,” Opt. Eng. 15, 286–291 (1976).

Opt. Express (2)

Opt. Lett. (1)

Optom. Vis. Sci. (2)

W. F. Harris, “Wavefronts and their propagation in astigmatic optical systems,” Optom. Vis. Sci. 73(9), 605–612 (1996).
[CrossRef]

E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vis. Sci. 82(10), 923–932 (2005).
[CrossRef] [PubMed]

Proc. London Math. Soc. (1)

J. C. Maxwell, “On the Focal Lines of a Refracted Pencil,” Proc. London Math. Soc. s1–4, 337–343 (1873).

Sven. Vetensk. Handl (1)

A. Gullstrand, “Die Reelle Optische Abbildung,” Sven. Vetensk. Handl 41, 1–119 (1906).

Other (11)

G. L. Van Der Heijde, “Artificial intraocular lens e.g. Alvarez-type lens for implantation in eye, comprises two lens elements with optical thickness such that power of the lens changes by transversal displacement of one lens element relative to the other element,” Int Patent 2006/025726 (2006).

O. N. Stavroudis, The mathematics of geometrical and physical optics: the k-function and its ramifications (Wiley-VCH, Weinheim, 2006).

M. P. Keating, Geometric, physical, and visual optics (Butterworth-Heinemann, Boston, 2002).

A. Walther, The ray and wave theory of lenses (Cambridge University Press, 1995).

L. W. Alvarez, and W. E. Humphrey, “Variable power lens and system,” US Patent 3,507,565 (1970).

W. E. Humphrey, “Apparatus for opthalmological prescription readout,” US Patent 3,927,933 (1974).

H. Mukaiyama, K. Kato, and A. Komatsu, “Variable focus visual power correction device - has optical lenses superposed so that main meridians are coincident with one another,” Int Patent 93/15432 (1994).

G. L. van Der Heijde, “Universal spectacles for children in developing countries,” in Mopane: Conference on Visual Optics (Kruger National Park, South Africa, 2006).

B. Spivey, “Lens for spectacles, has thickness designed so that by adjusting relative positions of two lenses in direction perpendicular to viewing direction, combined focus of two lenses is changed,” US Patent 151184 (2008).

C. F. Cheung, and W. B. Lee, Surface generation in ultra-precision diamond turning: modelling and practices (Professional Engineering Publ., London 2003).

J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, San Francisco, 1968).

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Figures (10)

Fig. 1
Fig. 1

(a) Alvarez cubic surface profile. (b) Lohmann cubic surface profile. The colormap mapping of the surfaces is proportional to the local curvature.

Fig. 2
Fig. 2

Astigmatic localized wavefronts before and after refraction through an optical surface. The arrows represent the base rays associated to each localized wavefront. The colormap is proportional to local curvature.

Fig. 3
Fig. 3

Outer cubic surfaces configuration at: (a) neutral position, (b) negative power addition, (c) positive power addition. Inner cubic surfaces configuration at: (d) neutral position, (e) negative power addition and (f) positive power addition. Note that for the outer cubic surfaces configuration there must be a space between both lenses to avoid collision when the shift is done to achieve positive power addition (f).

Fig. 7
Fig. 7

Alvarez-type surfaces: (a-b-c) Alvarez lens example nº1, (d-e-f) new Alvarez lens optimized with ray tracing, (g-h-i) new Alvarez lens obtained with Eq. (19). (a-d-g) 3-D surfaces plot. (b-e-h) X-Z lateral view, (c-f-i) Y-Z lateral view. The lateral views show the 2-D projection of the intersection between the Alvarez surface and a plane z = 0 (blue area) and a plane z = R (violet area).

Fig. 8
Fig. 8

Lohmann-type surfaces: (a-b-c) Lohmann lens, (d-e-f) new Lohmann lens optimized with ray tracing, (g-h-i) new Lohmann lens obtained with Eq. (19). (a-d-g) 3-D surfaces plot. (b-e-h) X-Z lateral view, (c-f-i) Y-Z lateral view. The lateral views show the 2-D projection of the intersection between the Lohmann surface and a plane z = 0 (blue area) and a plane z = R (violet area).

Fig. 4
Fig. 4

Sphere power versus lateral shift (δ) computed using the thin lens and normal incident approximation (Eq. (2) red line) and using the exact Eqs. (9)(11) (blue line) in: (a) Alvarez example lens nº 1 (c) Alvarez example lens nº 2. Sphere power versus lens thickness for zero lateral shift (δ) using exact equations in: (b) Alvarez example lens nº 1 (d) Alvarez example lens nº 2.

Fig. 5
Fig. 5

Flow chart of the ray tracing intersection point iterative routine.

Fig. 6
Fig. 6

Ray tracing through Alvarez lens example nº 2. (a) Lateral view. Blue lines are ray trajectories and red stars are intersections of rays with Alvarez surfaces (b) Spot diagram in image plane (500 microns scale).

Fig. 9
Fig. 9

Root mean square error (µm) versus sphere power (D). Pupil radius: 3 mm. (a) Alvarez lens example nº1 (black line), new Alvarez lens obtained with Eq. (19) (red line) and new Alvarez lens optimized with ray tracing (blue stars). (b) Lohmann lens (black line), new Lohmann lens obtained Eq. (19) (red line) and new Lohmann lens optimized with ray tracing (blue stars).

Fig. 10
Fig. 10

Root mean square error (µm) versus sphere power (D). Pupil radius: 3 mm. (a) New Alvarez lens obtained with Eq. (19) using outer cubic configuration (red line) versus inner cubic configuration (blue line). (b) New Lohmann lens obtained with Eq. (19) using outer cubic configuration (red line) versus inner cubic configuration (blue line).

Tables (1)

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Table 1 Coefficients of the different Alvarez and Lohmann lensesa

Equations (22)

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z=Axx3+Ax'yx2+Ayy3+Ay'xy2+Bx2+Cxy+Dy2+Ex+Fy+G
f=14Aδ(n1)
z=A((xδ)33+(xδ)y2)+C(xδ)y+E(xδ)+Fy+G
z=A((x+δ)33+(x+δ)y2)+C(x+δ)y+E(x+δ)+Fy+G
zx=A((xδ)2+y2)+Cy+E  (1º surf) or  A((x+δ)2+y2)+Cy+E (2º surf)zy= (2Ay+C)(xδ)+F (1º surf) or (2Ay+C)(x+δ)+F  (2º surf) 2zx2=2zy2=2A(xδ) (1º surf) or 2A(x+δ) (2º surf)zyx=zxy=C (1º & 2º surf) 
z=A(xδ)33+A(yδ)33+C(xδ)(yδ)+E(xδ)+F(yδ)+G
z=A(x+δ)33+A(y+δ)33+C(x+δ)(y+δ)+E(x+δ)+F(y+δ)+G
zx=A(xδ)2+C(yδ)+E (1º surf) or A(x+δ)2+C(y+δ)+E (2º surf)zy=A(yδ)2+C(xδ)+F  (1º surf) or A(y+δ)2+C(x+δ)+F (2º surf) 2zx2=2zy2=2A(xδ) (1º surf) or 2A(x+δ) (2º surf)zyx=zxy=C (1º & 2º surf) 
L˜1(x0,y0)=(ncos(φref)cos(φin))R˜(θs)K˜0(x0,y0)R˜(θs)1+                   R˜(θin,φin)L˜0(x0,y0)R˜(θin,φin)1
L˜2(x1,y1)=L˜1(x0,y0)Δtdet(L˜1(x0,y0))I˜Δ
L˜3(x1,y1)=(cos(φref)ncos(φin))R˜(θs)K˜1(x1,y1)R˜(θs)1+                  R˜(θin,φin)L˜2(x1,y1)R˜(θin,φin)1
L˜1(x0,y0)=(n1)K˜0(x0,y0)+L˜0(x0,y0)=                  (n1)(2z(0,0)x2z(0,0)xyz(0,0)xy2z(0,0)y2)+L˜0(x0,y0)=                  (n1)(2AδCC2Aδ)+L˜0(x0,y0)
L˜3(x0,y0)=(1n)K˜1(x0,y0)+L˜1(x0,y0)=                   (1n)(2z(0,0)x2z(0,0)xyz(0,0)xy2z(0,0)y2)+L˜1(x0,y0)=                   (1n)(2AδCC2Aδ)+L˜1(x0,y0)
L˜3(x0,y0)=(1n)(4AδCC4Aδ)+L˜0(x0,y0)
z=0.06((xδ)33+(xδ)y2)0.106(xδ)+0.01
x=x0+tα            y=y0+tβ                   z=z0+tφ
minΩ{A(xy2+x33)+Ex}dxdy          (Alvarez lens) minΩ{A(y33+x33)+E(x+y)}dxdy  (Lohmann lens) 
  min0Rdx0Rdy{A(xy2+x33)+Ex}=minAR54+ER32       (Alvarez lens)  min0Rdx0Rdy{A(y33+x33)+E(x+y)}=minAR56+ER3 (Lohmann lens) 
   E=AR22     (Alvarez lens)    E=AR26   (Lohmann lens) 
   min0Rdx0R2x2dy{A(xy2+x33)+Ex}=minAR59+ER33           (Alvarez lens)  min0Rdx0R2x2dy{A(y33+x33)+E(x+y)}=min4AR545+2ER33 (Lohmann lens) 
   E=AR23     (Alvarez lens)    E=2AR215   (Lohmann lens) 
MF=iRMSi

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