Abstract

Refractive and profilometric analyses of lenses with large radii of curvature and/or large focal distance were performed through photorefractive holography using a Bi12TiO20 crystal as the recording medium and two red diode lasers as light sources. Both lasers were properly aligned and tuned in order to provide submillimetric synthetic wavelengths providing real-time interferograms in a two-color holography experiment. The resulting contour interferogram describes the form of the wavefront after the beam traveled back and forth through the lens. The fringe quantitative evaluation was carried out through the four-stepping technique, and the resulting phase map and the branch-cut method were employed for phase unwrapping. Exact ray tracing calculation was performed in order to establish a relation between the output wavefront geometry and the lens parameters such as radii of curvature, thickness, and refractive index. By quantitatively comparing the theoretically calculated wavefront geometry with the experimental results, errors below 1% for both refractive index and focal length were obtained.

© 2009 Optical Society of America

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References

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  1. M. de Angelis, S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, and T. Hessler, “Interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5-9 (1999).
    [CrossRef]
  2. C. J. Tay, M. Thakur, L. Chen, and C. Shakher, “Measurement of focal length of lens using phase shifting Lau phase interferometry,” Opt. Commun. 248, 339-345 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  6. C. Quan, S. H. Wang, C. J. Tay, I. Reading, and Z. P. Fang, “Integrated optical inspection on surface geometry and refractive index distribution of a microlens array,” Opt. Commun. 225, 223-231 (2003).
    [CrossRef]
  7. E. A. Barbosa and S. C. dos Santos, “Refractive and geometric lens characterization through multiwavelength digital speckle pattern interferometry, “Opt. Commun. 281, 1022-1029(2008).
    [CrossRef]
  8. E. A. Barbosa, “Holographic imaging with multimode, large free spectral range lasers in photorefractive sillenite crystals,” Appl. Phys. B 80, 345-350 (2005).
    [CrossRef]
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    [CrossRef]
  10. E. A. Barbosa and J. F. Carvalho, “Surface analysis by two-diode laser photorefractive holography,” Appl. Phys. B 87, 417-423 (2007).
    [CrossRef]
  11. E. A. Barbosa, R. Verzini, and J. F. Carvalho, “Multi-wavelength holography in Bi12TiO20 crystals: applications in refractometry,” Opt. Commun. 263, 189-196 (2006).
    [CrossRef]
  12. K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349-393 (1988).
    [CrossRef]
  13. B. Gutmann and H. Weber, “Phase unwrapping with the branch-cut method: clustering of discontinuity sources and reverse simulated annealing,” Appl. Opt. 38, 5577-5593 (1999).
    [CrossRef]
  14. A. A. Kamshilin and M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53, 23-26 (1985).
    [CrossRef]
  15. E. Hecht, Optics (Addison-Wesley, 1998).

2008

E. A. Barbosa and S. C. dos Santos, “Refractive and geometric lens characterization through multiwavelength digital speckle pattern interferometry, “Opt. Commun. 281, 1022-1029(2008).
[CrossRef]

2007

2006

E. A. Barbosa, R. Verzini, and J. F. Carvalho, “Multi-wavelength holography in Bi12TiO20 crystals: applications in refractometry,” Opt. Commun. 263, 189-196 (2006).
[CrossRef]

2005

E. A. Barbosa, “Holographic imaging with multimode, large free spectral range lasers in photorefractive sillenite crystals,” Appl. Phys. B 80, 345-350 (2005).
[CrossRef]

E. A. Barbosa, A. A. V. Filho, M. R. R. Gesualdi, B. G. Curcio, M. Muramatsu, and D. Soga, “Single-exposure, photorefractive holographic surface contouring with multiwavelength diode lasers,” J. Opt. Soc. Am. A 22, 2872-2879 (2005).
[CrossRef]

C. J. Tay, M. Thakur, L. Chen, and C. Shakher, “Measurement of focal length of lens using phase shifting Lau phase interferometry,” Opt. Commun. 248, 339-345 (2005).
[CrossRef]

2003

C. Quan, S. H. Wang, C. J. Tay, I. Reading, and Z. P. Fang, “Integrated optical inspection on surface geometry and refractive index distribution of a microlens array,” Opt. Commun. 225, 223-231 (2003).
[CrossRef]

1999

M. de Angelis, S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, and T. Hessler, “Interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5-9 (1999).
[CrossRef]

B. Gutmann and H. Weber, “Phase unwrapping with the branch-cut method: clustering of discontinuity sources and reverse simulated annealing,” Appl. Opt. 38, 5577-5593 (1999).
[CrossRef]

1998

R. A. Arizaga, J. A. Pomarico, and R. D. Torroba, “Digital technique for high accuracy focal length measurements,” Opt. Commun. 152, 6-10 (1998).
[CrossRef]

1988

1985

A. A. Kamshilin and M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53, 23-26 (1985).
[CrossRef]

Anand, A.

Arizaga, R. A.

R. A. Arizaga, J. A. Pomarico, and R. D. Torroba, “Digital technique for high accuracy focal length measurements,” Opt. Commun. 152, 6-10 (1998).
[CrossRef]

Barbosa, E. A.

E. A. Barbosa and S. C. dos Santos, “Refractive and geometric lens characterization through multiwavelength digital speckle pattern interferometry, “Opt. Commun. 281, 1022-1029(2008).
[CrossRef]

E. A. Barbosa and J. F. Carvalho, “Surface analysis by two-diode laser photorefractive holography,” Appl. Phys. B 87, 417-423 (2007).
[CrossRef]

E. A. Barbosa, R. Verzini, and J. F. Carvalho, “Multi-wavelength holography in Bi12TiO20 crystals: applications in refractometry,” Opt. Commun. 263, 189-196 (2006).
[CrossRef]

E. A. Barbosa, “Holographic imaging with multimode, large free spectral range lasers in photorefractive sillenite crystals,” Appl. Phys. B 80, 345-350 (2005).
[CrossRef]

E. A. Barbosa, A. A. V. Filho, M. R. R. Gesualdi, B. G. Curcio, M. Muramatsu, and D. Soga, “Single-exposure, photorefractive holographic surface contouring with multiwavelength diode lasers,” J. Opt. Soc. Am. A 22, 2872-2879 (2005).
[CrossRef]

Carvalho, J. F.

E. A. Barbosa and J. F. Carvalho, “Surface analysis by two-diode laser photorefractive holography,” Appl. Phys. B 87, 417-423 (2007).
[CrossRef]

E. A. Barbosa, R. Verzini, and J. F. Carvalho, “Multi-wavelength holography in Bi12TiO20 crystals: applications in refractometry,” Opt. Commun. 263, 189-196 (2006).
[CrossRef]

Chen, L.

C. J. Tay, M. Thakur, L. Chen, and C. Shakher, “Measurement of focal length of lens using phase shifting Lau phase interferometry,” Opt. Commun. 248, 339-345 (2005).
[CrossRef]

Chhaniwal, V.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349-393 (1988).
[CrossRef]

Curcio, B. G.

de Angelis, M.

M. de Angelis, S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, and T. Hessler, “Interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5-9 (1999).
[CrossRef]

de Nicola, S.

M. de Angelis, S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, and T. Hessler, “Interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5-9 (1999).
[CrossRef]

dos Santos, S. C.

E. A. Barbosa and S. C. dos Santos, “Refractive and geometric lens characterization through multiwavelength digital speckle pattern interferometry, “Opt. Commun. 281, 1022-1029(2008).
[CrossRef]

Fang, Z. P.

C. Quan, S. H. Wang, C. J. Tay, I. Reading, and Z. P. Fang, “Integrated optical inspection on surface geometry and refractive index distribution of a microlens array,” Opt. Commun. 225, 223-231 (2003).
[CrossRef]

Ferraro, P.

M. de Angelis, S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, and T. Hessler, “Interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5-9 (1999).
[CrossRef]

Filho, A. A. V.

Finizio, A.

M. de Angelis, S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, and T. Hessler, “Interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5-9 (1999).
[CrossRef]

Gesualdi, M. R. R.

Gutmann, B.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 1998).

Hessler, T.

M. de Angelis, S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, and T. Hessler, “Interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5-9 (1999).
[CrossRef]

Kafri, O.

Kamshilin, A. A.

A. A. Kamshilin and M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53, 23-26 (1985).
[CrossRef]

Keren, E.

Kreske, K. M.

Muramatsu, M.

Petrov, M. P.

A. A. Kamshilin and M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53, 23-26 (1985).
[CrossRef]

Pierattini, G.

M. de Angelis, S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, and T. Hessler, “Interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5-9 (1999).
[CrossRef]

Pomarico, J. A.

R. A. Arizaga, J. A. Pomarico, and R. D. Torroba, “Digital technique for high accuracy focal length measurements,” Opt. Commun. 152, 6-10 (1998).
[CrossRef]

Quan, C.

C. Quan, S. H. Wang, C. J. Tay, I. Reading, and Z. P. Fang, “Integrated optical inspection on surface geometry and refractive index distribution of a microlens array,” Opt. Commun. 225, 223-231 (2003).
[CrossRef]

Reading, I.

C. Quan, S. H. Wang, C. J. Tay, I. Reading, and Z. P. Fang, “Integrated optical inspection on surface geometry and refractive index distribution of a microlens array,” Opt. Commun. 225, 223-231 (2003).
[CrossRef]

Shakher, C.

C. J. Tay, M. Thakur, L. Chen, and C. Shakher, “Measurement of focal length of lens using phase shifting Lau phase interferometry,” Opt. Commun. 248, 339-345 (2005).
[CrossRef]

Soga, D.

Tay, C. J.

C. J. Tay, M. Thakur, L. Chen, and C. Shakher, “Measurement of focal length of lens using phase shifting Lau phase interferometry,” Opt. Commun. 248, 339-345 (2005).
[CrossRef]

C. Quan, S. H. Wang, C. J. Tay, I. Reading, and Z. P. Fang, “Integrated optical inspection on surface geometry and refractive index distribution of a microlens array,” Opt. Commun. 225, 223-231 (2003).
[CrossRef]

Thakur, M.

C. J. Tay, M. Thakur, L. Chen, and C. Shakher, “Measurement of focal length of lens using phase shifting Lau phase interferometry,” Opt. Commun. 248, 339-345 (2005).
[CrossRef]

Torroba, R. D.

R. A. Arizaga, J. A. Pomarico, and R. D. Torroba, “Digital technique for high accuracy focal length measurements,” Opt. Commun. 152, 6-10 (1998).
[CrossRef]

Verzini, R.

E. A. Barbosa, R. Verzini, and J. F. Carvalho, “Multi-wavelength holography in Bi12TiO20 crystals: applications in refractometry,” Opt. Commun. 263, 189-196 (2006).
[CrossRef]

Wang, S. H.

C. Quan, S. H. Wang, C. J. Tay, I. Reading, and Z. P. Fang, “Integrated optical inspection on surface geometry and refractive index distribution of a microlens array,” Opt. Commun. 225, 223-231 (2003).
[CrossRef]

Weber, H.

Appl. Opt.

Appl. Phys. B

E. A. Barbosa and J. F. Carvalho, “Surface analysis by two-diode laser photorefractive holography,” Appl. Phys. B 87, 417-423 (2007).
[CrossRef]

E. A. Barbosa, “Holographic imaging with multimode, large free spectral range lasers in photorefractive sillenite crystals,” Appl. Phys. B 80, 345-350 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

C. Quan, S. H. Wang, C. J. Tay, I. Reading, and Z. P. Fang, “Integrated optical inspection on surface geometry and refractive index distribution of a microlens array,” Opt. Commun. 225, 223-231 (2003).
[CrossRef]

E. A. Barbosa and S. C. dos Santos, “Refractive and geometric lens characterization through multiwavelength digital speckle pattern interferometry, “Opt. Commun. 281, 1022-1029(2008).
[CrossRef]

R. A. Arizaga, J. A. Pomarico, and R. D. Torroba, “Digital technique for high accuracy focal length measurements,” Opt. Commun. 152, 6-10 (1998).
[CrossRef]

M. de Angelis, S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, and T. Hessler, “Interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5-9 (1999).
[CrossRef]

C. J. Tay, M. Thakur, L. Chen, and C. Shakher, “Measurement of focal length of lens using phase shifting Lau phase interferometry,” Opt. Commun. 248, 339-345 (2005).
[CrossRef]

E. A. Barbosa, R. Verzini, and J. F. Carvalho, “Multi-wavelength holography in Bi12TiO20 crystals: applications in refractometry,” Opt. Commun. 263, 189-196 (2006).
[CrossRef]

A. A. Kamshilin and M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53, 23-26 (1985).
[CrossRef]

Prog. Opt.

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349-393 (1988).
[CrossRef]

Other

E. Hecht, Optics (Addison-Wesley, 1998).

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Figures (7)

Fig. 1
Fig. 1

Beam path into the spherical test lens.

Fig. 2
Fig. 2

Optical setup: M1–M5, mirrors; L1– L3, lenses; P1 and P2, polarizers; BS1, beam splitter; PR, 90 ° prism; BTO, Bi 12 Ti O 20 crystal; CCD, camera.

Fig. 3
Fig. 3

(a) Holographic image of the biconvex lens generated by only one laser ( Δ z = 3.3 mm ). (b) Two-diode laser holographic image of the same surface for Δ z = 0.32 mm . (c) Phase map obtained through the four-stepping technique. (d) Reconstructed wavefront after phase unwrapping. (e) y coordinate (thick black curve) averaged along radial directions and a fitting curve (thin gray curve) providing R F = 33.2 mm .

Fig. 4
Fig. 4

(a) Contour fringe pattern for Δ z = 0.24 mm when the illuminating beam traveled through the lens. (b) Averaged y coordinate of the reconstructed wavefront providing n = 1.50 ± 0.01 .

Fig. 5
Fig. 5

Plano–convex achromatic doublet.

Fig. 6
Fig. 6

Averaged profile of the doublet spherical surface (thick black curve) and fitting curve providing R D = 21.53 mm (thin gray curve).

Fig. 7
Fig. 7

Optical path of the reconstructed wavefront after the light propagates through the uncovered lens providing n eq = 1.64 ± 0.01 .

Equations (19)

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R N = R 0 ( e i k 1 Γ R + e i k 2 Γ R ) m = N 1 2 m = N 1 2 A m e i ( m Δ k Γ R + ϕ m ) , S N = S 0 ( e i k 1 Γ S + e i k 2 Γ S ) m = N 1 2 m = N 1 2 A m e i ( m Δ k Γ S + ϕ m ) ,
I S = η 0 ( 1 + χ 2 + 2 | χ | cos 2 φ ) [ sin ( N γ ) sin ( γ ) ] 2 I R ,
ϕ 4 step ( p , q ) = arctan ( I 3 ( p , q ) I 1 ( p , q ) I 0 ( p , q ) I 2 ( p , q ) ) ,
Γ 1 2 = ( O A + n A B + B C ) ,
θ = arctan [ ( y F x ) x = x A 1 ] = arctan [ x A ( R F 2 x A 2 ) 1 / 2 ] ,
θ = arcsin { n 1 sin [ arctan ( x A ( R F 2 x A 2 ) 1 / 2 ) ] } .
y h ( x ) = x tan γ x A tan γ + y A .
x B = b 2 a [ ( b 2 a ) 2 c a ] 1 / 2 , y B = ( x B x A ) tan ( θ θ ) + y A ,
a 1 / tan 2 ( θ θ ) + 1 , b 2 / tan ( θ θ ) [ x A / tan ( θ θ ) + y A t + R R ] , c [ x A / tan ( θ θ ) + y A t + R R ] 2 R R 2 .
A B = [ ( x B x A ) 2 + ( y B y A ) 2 ] 1 / 2 .
α = θ θ + β ,
β = | arctan [ x B ( R F 2 x B 2 ) 1 / 2 ] | .
x C = x B + ( y B t ) tan ( α β ) , y C = t ,
B C = [ ( x C x B ) 2 + ( y C y B ) 2 ] 1 / 2 .
Γ 1 2 = y A + n [ ( x B x A ) 2 + ( y B y A ) 2 ] 1 / 2 + [ ( x C x B ) 2 + ( y C y B ) 2 ] 1 / 2 .
1 f = ( n 1 ) [ 1 R F 1 R R + t R F R R ( n 1 ) n ] .
1 f = n a n b R D + ( n a 1 R D ) [ 1 ( n a n b n a ) t a R D ] ,
1 f = n eq 1 R D ,
n eq = 2 n a n b ( n a n b ) ( n a 1 ) n a t a R D .

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