Abstract

This paper presents convenient expressions for the monochromatic primary aberrations of a diffractive lens on a spherical substrate having a nonunity refractive index. Sets of nomographs that provide ready estimates for these aberrations are also given. This analysis facilitates structural design of diffractive lenses in various applications—where they are used as stand-alone devices or as one or more components in a multicomponent system.

© 2010 Optical Society of America

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References

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  1. D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test, Vol. TT62 of SPIE Tutorial Text in Optical Engineering (SPIE Press, 2004).
  2. Vide URL: www.canon.com
  3. V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements, Wiley Series in Lasers and Applications (Wiley-Interscience , 2001).
  4. L. N. Hazra and S. Chatterjee, “A prophylactic strategy for global synthesis in lens design,” Opt. Rev. 12, 247–254 (2005).
    [CrossRef]
  5. L. N. Hazra, Y. Han, and C. A. Delisle, “Kinoform lenses: Sweatt model and phase function,” Opt. Commun. 117, 31–36(1995).
    [CrossRef]
  6. W. C. Sweatt, “Describing holographic optical elements as lenses,” J. Opt. Soc. Am. 67, 803–808 (1977).
    [CrossRef]
  7. W. A. Kleinhans, “Aberrations of curved zone plates and Fresnel lenses,” Appl. Opt. 16, 1701–1704 (1977).
    [CrossRef] [PubMed]
  8. W. C. Sweatt, S. A. Kemme, and M. E. Warren , “Diffractive optical elements,” in Optical Engineer’s Desk Reference, W.L.Wolfe, ed. (OSA–SPIE, 2003), p. 347.
  9. T. Stone and N. George, “Hybrid diffractive–refractive lenses and achromats,” Appl. Opt. 27, 2960–2971 (1988).
    [CrossRef] [PubMed]
  10. S. J. Dobson and H. H. Hopkins, “A new rod-lens relay system offering improved image quality,” J. Phys. E 22, 450–455(1989).
    [CrossRef]
  11. H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, 1950).
  12. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  13. H. Coddington, A Treatise on the Reflexion and Refraction of Light (Marshall, 1829).
  14. L. N. Hazra and C. A. Delisle, “Primary aberrations of a thin lens with different object and image space media,” J. Opt. Soc. Am. A 15, 945–953 (1998).
    [CrossRef]
  15. L. N. Hazra and C. A. Delisle, “Higher order kinoform lenses: diffraction efficiency and aberrational properties,” Opt. Eng. 36, 1500–1507 (1997).
    [CrossRef]
  16. D. A. Buralli and G. M. Morris, “Design of diffractive singlets for monochromatic imaging,” Appl. Opt. 30, 2151–2158(1991).
    [CrossRef] [PubMed]
  17. H. P. Herzig, “Design of refractive and diffractive micro-optics,” in Micro-optics, H.P.Herzig, ed. (Taylor & Francis, 1997).
  18. R. V. Shack, “The use of normalization in the application of simple optical systems,” Proc. SPIE 54, 155–162 (1974).

2005 (1)

L. N. Hazra and S. Chatterjee, “A prophylactic strategy for global synthesis in lens design,” Opt. Rev. 12, 247–254 (2005).
[CrossRef]

1998 (1)

1997 (1)

L. N. Hazra and C. A. Delisle, “Higher order kinoform lenses: diffraction efficiency and aberrational properties,” Opt. Eng. 36, 1500–1507 (1997).
[CrossRef]

1995 (1)

L. N. Hazra, Y. Han, and C. A. Delisle, “Kinoform lenses: Sweatt model and phase function,” Opt. Commun. 117, 31–36(1995).
[CrossRef]

1991 (1)

1989 (1)

S. J. Dobson and H. H. Hopkins, “A new rod-lens relay system offering improved image quality,” J. Phys. E 22, 450–455(1989).
[CrossRef]

1988 (1)

1977 (2)

1974 (1)

R. V. Shack, “The use of normalization in the application of simple optical systems,” Proc. SPIE 54, 155–162 (1974).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Buralli, D. A.

Chatterjee, S.

L. N. Hazra and S. Chatterjee, “A prophylactic strategy for global synthesis in lens design,” Opt. Rev. 12, 247–254 (2005).
[CrossRef]

Coddington, H.

H. Coddington, A Treatise on the Reflexion and Refraction of Light (Marshall, 1829).

Delisle, C. A.

L. N. Hazra and C. A. Delisle, “Primary aberrations of a thin lens with different object and image space media,” J. Opt. Soc. Am. A 15, 945–953 (1998).
[CrossRef]

L. N. Hazra and C. A. Delisle, “Higher order kinoform lenses: diffraction efficiency and aberrational properties,” Opt. Eng. 36, 1500–1507 (1997).
[CrossRef]

L. N. Hazra, Y. Han, and C. A. Delisle, “Kinoform lenses: Sweatt model and phase function,” Opt. Commun. 117, 31–36(1995).
[CrossRef]

Dobson, S. J.

S. J. Dobson and H. H. Hopkins, “A new rod-lens relay system offering improved image quality,” J. Phys. E 22, 450–455(1989).
[CrossRef]

George, N.

Han, Y.

L. N. Hazra, Y. Han, and C. A. Delisle, “Kinoform lenses: Sweatt model and phase function,” Opt. Commun. 117, 31–36(1995).
[CrossRef]

Hazra, L. N.

L. N. Hazra and S. Chatterjee, “A prophylactic strategy for global synthesis in lens design,” Opt. Rev. 12, 247–254 (2005).
[CrossRef]

L. N. Hazra and C. A. Delisle, “Primary aberrations of a thin lens with different object and image space media,” J. Opt. Soc. Am. A 15, 945–953 (1998).
[CrossRef]

L. N. Hazra and C. A. Delisle, “Higher order kinoform lenses: diffraction efficiency and aberrational properties,” Opt. Eng. 36, 1500–1507 (1997).
[CrossRef]

L. N. Hazra, Y. Han, and C. A. Delisle, “Kinoform lenses: Sweatt model and phase function,” Opt. Commun. 117, 31–36(1995).
[CrossRef]

Herzig, H. P.

H. P. Herzig, “Design of refractive and diffractive micro-optics,” in Micro-optics, H.P.Herzig, ed. (Taylor & Francis, 1997).

Hopkins, H. H.

S. J. Dobson and H. H. Hopkins, “A new rod-lens relay system offering improved image quality,” J. Phys. E 22, 450–455(1989).
[CrossRef]

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, 1950).

Kathman, A. D.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test, Vol. TT62 of SPIE Tutorial Text in Optical Engineering (SPIE Press, 2004).

Kemme, S. A.

W. C. Sweatt, S. A. Kemme, and M. E. Warren , “Diffractive optical elements,” in Optical Engineer’s Desk Reference, W.L.Wolfe, ed. (OSA–SPIE, 2003), p. 347.

Kleinhans, W. A.

Morris, G. M.

O’Shea, D. C.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test, Vol. TT62 of SPIE Tutorial Text in Optical Engineering (SPIE Press, 2004).

Prather, D. W.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test, Vol. TT62 of SPIE Tutorial Text in Optical Engineering (SPIE Press, 2004).

Shack, R. V.

R. V. Shack, “The use of normalization in the application of simple optical systems,” Proc. SPIE 54, 155–162 (1974).

Soifer, V. A.

V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements, Wiley Series in Lasers and Applications (Wiley-Interscience , 2001).

Stone, T.

Suleski, T. J.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test, Vol. TT62 of SPIE Tutorial Text in Optical Engineering (SPIE Press, 2004).

Sweatt, W. C.

W. C. Sweatt, “Describing holographic optical elements as lenses,” J. Opt. Soc. Am. 67, 803–808 (1977).
[CrossRef]

W. C. Sweatt, S. A. Kemme, and M. E. Warren , “Diffractive optical elements,” in Optical Engineer’s Desk Reference, W.L.Wolfe, ed. (OSA–SPIE, 2003), p. 347.

Warren, M. E.

W. C. Sweatt, S. A. Kemme, and M. E. Warren , “Diffractive optical elements,” in Optical Engineer’s Desk Reference, W.L.Wolfe, ed. (OSA–SPIE, 2003), p. 347.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. E (1)

S. J. Dobson and H. H. Hopkins, “A new rod-lens relay system offering improved image quality,” J. Phys. E 22, 450–455(1989).
[CrossRef]

Opt. Commun. (1)

L. N. Hazra, Y. Han, and C. A. Delisle, “Kinoform lenses: Sweatt model and phase function,” Opt. Commun. 117, 31–36(1995).
[CrossRef]

Opt. Eng. (1)

L. N. Hazra and C. A. Delisle, “Higher order kinoform lenses: diffraction efficiency and aberrational properties,” Opt. Eng. 36, 1500–1507 (1997).
[CrossRef]

Opt. Rev. (1)

L. N. Hazra and S. Chatterjee, “A prophylactic strategy for global synthesis in lens design,” Opt. Rev. 12, 247–254 (2005).
[CrossRef]

Proc. SPIE (1)

R. V. Shack, “The use of normalization in the application of simple optical systems,” Proc. SPIE 54, 155–162 (1974).

Other (8)

H. P. Herzig, “Design of refractive and diffractive micro-optics,” in Micro-optics, H.P.Herzig, ed. (Taylor & Francis, 1997).

W. C. Sweatt, S. A. Kemme, and M. E. Warren , “Diffractive optical elements,” in Optical Engineer’s Desk Reference, W.L.Wolfe, ed. (OSA–SPIE, 2003), p. 347.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test, Vol. TT62 of SPIE Tutorial Text in Optical Engineering (SPIE Press, 2004).

Vide URL: www.canon.com

V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements, Wiley Series in Lasers and Applications (Wiley-Interscience , 2001).

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, 1950).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

H. Coddington, A Treatise on the Reflexion and Refraction of Light (Marshall, 1829).

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Figures (8)

Fig. 1
Fig. 1

(a) Hybrid lens modeled as the cascade of a thin refractive lens (RL) of refractive index μ and thickness d 0 and a diffractive lens (DL). The two lenses are separated by an air medium of zero thickness. Object and image spaces for each of the lens are air. (b) Diffractive lens on a curved substrate is modeled as the combination of a diffractive lens (DL) and a refractive interface having a nonunity refractive index in the object space. The two are separated by an air medium of zero thickness. Object and image spaces for the DL are air. For the refracting interface, the image space is air. (c) Diffractive lens on a curved substrate. Object space of DL is of a nonunity refractive index, image space of DL is air.

Fig. 2
Fig. 2

PMR and PPR through a single lens of refractive index μ, with n and n being the object and image space refractive indices, respectively. Equivalent thin lens for the singlet assumes d 0 . Stop is on the lens.

Fig. 3
Fig. 3

Variation of I with shape factor B, substrate refractive index n, and image space refractive index, n = 1 for values of Y: (a) 3 , (b) 2 , (c) 1 , (d) 0, (e) + 1 , (f) + 2 , and (g) + 3 . Diamonds, n = 1.4 ; triangles, n = 1.6 ; exes, n = 1.8 ; pluses, n = 2.0 .

Fig. 4
Fig. 4

Variation of I with conjugate variable Y and substrate refractive index n for values of shape factor B = 0 , 1, and image space refractive index n = 1 .

Fig. 5
Fig. 5

Variation of I I with shape factor B, substrate refractive index n, and image space refractive index n = 1 for values of Y: (a) 3 , (b) 2 , (c) 1 , (d) 0 (e) + 1 , (f) + 2 , and (g) + 3 . Diamonds, n = 1.4 ; triangles, n = 1.6 ; exes, n = 1.8 ; boxes, n = 2.0 .

Fig. 6
Fig. 6

Variation of I I I with conjugate variable Y for values of substrate refractive indices n and image space refractive index n = 1 . Diamonds, n = 1.4 ; triangles, n = 1.6 ; exes, n = 1.8 ; pluses, n = 2.0 .

Fig. 7
Fig. 7

Variation of I V with shape factor B, substrate refractive index n, and image space refractive index, n = 1 . Diamonds, n = 1.4 ; triangles, n = 1.6 ; exes, n = 1.8 ; pluses, n = 2.0 .

Fig. 8
Fig. 8

Variation of V with substrate refractive index n and image space refractive index n = 1 .

Equations (49)

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K = ( μ n ) c 1 ( μ n ) c 2 .
X = ( μ n ) c 1 + ( μ n ) c 2 K .
Y = n u + n u n u n u = n u + n u h K .
H = n u ¯ h = n u ¯ h .
K = ( μ n ) c 1 ( μ n ) c 2 = μ ( c 1 c 2 ) + ( n c 2 n c 1 ) .
K = K d + ( n n ) c s .
K = K d + ( 1 n ) c s .
K ˜ = K ˜ d + ( 1 n ˜ ) c ˜ s .
c ˜ s = ( 1 n ) ( 1 n ˜ ) c s .
K ˜ d = K d + ( n ˜ n ) c s .
B = X μ = c 1 + c 2 K ( n c 1 + n c 2 ) K μ .
ϕ ( r ) = 2 π λ i = 1 I a i r 2 i .
S I = 1 8 h 4 K 3 ( a ^ 1 B 3 + a ^ 2 Y 3 + a ^ 3 B 2 Y + a ^ 4 B Y 2 + a ^ 5 B 2 + a ^ 6 Y 2 + a ^ 7 B Y + a ^ 8 B + a ^ 9 Y + a ^ 10 ) ,
a ^ 1 = [ n ( n μ ) ( 1 n μ ) 2 n ( n μ ) ( 1 n μ ) 2 ] ,
a ^ 2 = [ 1 ( n μ ) 2 n 2 1 ( n μ ) 2 ( n ) 2 ] ,
a ^ 3 = [ 1 + 3 ( n μ ) 1 ( n μ ) 1 + 3 ( n μ ) 1 ( n μ ) ] ,
a ^ 4 = [ 2 + 3 ( n μ ) n 2 + 3 ( n μ ) n ] ,
a ^ 5 = [ 1 + 2 ( n μ ) ( 1 n μ ) 2 + 1 + 2 ( n μ ) ( 1 n μ ) 2 ] ,
a ^ 6 = [ 3 + 2 ( n μ ) n 2 + 3 + 2 ( n μ ) ( n ) 2 ] ,
a ^ 7 = 4 [ 1 + ( n μ ) n ( 1 n μ ) + 1 + ( n μ ) n ( 1 n μ ) ] ,
a ^ 8 = [ 2 + ( n μ ) n ( 1 n μ ) 2 2 + ( n μ ) n ( 1 n μ ) 2 ] ,
a ^ 9 = [ 3 + ( n μ ) n 2 ( 1 n μ ) 3 + ( n μ ) n 2 ( 1 n μ ) ] ,
a ^ 10 = [ 1 n 2 ( 1 n μ ) 2 + 1 n 2 ( 1 n μ ) 2 ] ,
( S I ) P K = 1 4 h 4 K 3 [ 1 2 ( 1 n 2 1 ) Y 3 + ( 1 n 1 ) B Y 2 + B 2 + 3 2 ( 1 n 2 + 1 ) Y 2 2 ( 1 n + 1 ) B Y + ( 1 n 1 ) B 3 2 ( 1 n 2 1 ) Y + 1 2 ( 1 n 2 + 1 ) ] .
( S I ) N P K = ( S I ) P K 8 a 2 h 4 .
( S I ) 0 = 1 4 h 4 K 3 ( 1 + B 2 4 B Y + 3 Y 2 ) 8 a 2 h 4 .
I = ( S I ) P K h 4 K 3 .
I = 1 4 ( B 2 ) 2 .
S I I = 1 4 h 2 K 2 H ( p ^ 1 B 2 + p ^ 2 Y 2 + p ^ 3 B Y + p ^ 4 B + p ^ 5 Y + p ^ 6 ) ,
p ^ 1 = [ n μ 1 ( n μ ) n μ 1 ( n μ ) ] ,
p ^ 2 = [ 1 ( n μ ) 2 n 2 1 ( n μ ) 2 ( n ) 2 ] ,
p ^ 3 = [ 1 + 2 ( n μ ) n 1 + 2 ( n μ ) n ] ,
p ^ 4 = [ 1 + ( n μ ) n ( 1 n μ ) + 1 + ( n μ ) n ( 1 n μ ) ] ,
p ^ 5 = [ 2 + ( n μ ) n 2 + 2 + ( n μ ) ( n ) 2 ] ,
p ^ 6 = [ 1 n 2 ( 1 n μ ) 1 ( n ) 2 ( 1 n μ ) ] .
S I I = 1 4 h 2 K 2 H [ ( 1 n 2 1 ) Y 2 ( 1 n 1 ) B Y + ( 1 n + 1 ) B 2 ( 1 n 2 + 1 ) Y + ( 1 n 2 1 ) ] .
( S I I ) 0 = 1 2 h 2 K 2 H ( B 2 Y ) .
I I = S I I h 2 K 2 H .
I I = 1 4 ( 1 n n ) { Y ( 1 + n ) ( 1 n ) } [ ( 1 + n n ) Y ( 1 n n ) B ] .
I I = 1 4 ( 1 + n ) n [ B + ( 1 n ) n ] .
I I = 1 2 ( B 2 ) .
S I I I = 1 2 H 2 K [ ( 1 n 2 + 1 n 2 ) ( 1 n 2 1 n 2 ) Y ] .
S I I I = 1 2 H 2 K [ ( 1 n 2 + 1 ) ( 1 n 2 1 ) Y ] .
I I I = S I I I H 2 K .
S I V = H 2 K 2 μ [ 1 n + 1 n ] + H 2 K 2 [ 1 n 1 n ] B .
I V = S I V H 2 K = 1 2 ( 1 n 1 ) B .
S V = H 3 h 2 [ 1 n 2 1 n 2 ] .
S V = H 3 h 2 ( 1 n 2 1 ) .
V = S V h 2 H 3 = [ 1 n 2 1 ] .

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