Abstract

A novel technique for focal-length measurements with a circular Dammann grating is presented. In the back focal plane of the lens under test, a one-order circular Dammann grating with limited aperture will produce double-humped radial rings. The separation between the two lobes varies with the displacement of the observed plane from the focal plane of the lens. By searching for the position at which the separation is minimal, the focal point of the lens can be located and hence the back focal length can be determined. Experimental results demonstrated that this method is efficient and can be used effectively for a quick check of focal length.

© 2007 Optical Society of America

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  1. R. Kingslake, ed., Applied Optics and Optical Engineering (Academic, 1965).
  2. I. Geatt and O. Kafri, "Determination of the focal length of nonparaxial lenses by moiré deflectometry," Appl. Opt. 26, 2507-2508 (1987).
    [CrossRef]
  3. E. Keren, M. K. Kreske, and O. Kafri, "Universal method for determining the focal length of optical systems by moiré deflectometry," Appl. Opt. 27, 1383-1385 (1988).
    [CrossRef] [PubMed]
  4. Y. Nakano and K. Murata, "Talbot interferometry for measuring the focal length of a lens," Appl. Opt. 24, 3162-3166 (1985).
    [CrossRef] [PubMed]
  5. C. W. Chang and D. C. Su, "An improved technique of measuring the focal length of a lens," Opt. Commun. 73, 257-262 (1989).
    [CrossRef]
  6. K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, "Direct determination of focal length by using Talbot interferometry," Appl. Opt. 31, 5984-5987 (1992).
    [CrossRef] [PubMed]
  7. P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, "Measurement of focal length with phase-shifting Talbot interferometry," Appl. Opt. 44, 1572-1576 (2005).
    [CrossRef] [PubMed]
  8. L. Angel, M. Tebaldi, and R. Henao, "Phase stepping in Lau interferometry," Opt. Commun. 164, 247-255 (1999).
    [CrossRef]
  9. M. Thakur and C. Shakher, "Evaluation of the focal distance of lenses by white-light Lau phase interferometry," Appl. Opt. 41, 1841-1845 (2002).
    [CrossRef] [PubMed]
  10. M. de Angelis, S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "A new approach to high accuracy measurement of the focal lengths of lenses using a digital Fourier transform," Opt. Commun. 136, 370-374 (1997).
    [CrossRef]
  11. S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect," Opt. Commun. 132, 432-436 (1996).
    [CrossRef]
  12. I. K. Ilev, "Simple fiber-optic autocollimation method for determining the focal lengths of optical elements," Opt. Lett. 20, 527-529 (1995).
    [CrossRef] [PubMed]
  13. B. DeBoo and J. Sasian, "Precise focal-length measurement technique with a reflective Fresnel-zone hologram," Appl. Opt. 42, 3903-3909 (2003).
    [CrossRef] [PubMed]
  14. S. Zhao and P. S. Chung, "Design of a circular Dammann grating," Opt. Lett. 31, 2387-2389 (2006).
    [CrossRef] [PubMed]

2006 (1)

2005 (1)

P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, "Measurement of focal length with phase-shifting Talbot interferometry," Appl. Opt. 44, 1572-1576 (2005).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

1999 (1)

L. Angel, M. Tebaldi, and R. Henao, "Phase stepping in Lau interferometry," Opt. Commun. 164, 247-255 (1999).
[CrossRef]

1997 (1)

M. de Angelis, S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "A new approach to high accuracy measurement of the focal lengths of lenses using a digital Fourier transform," Opt. Commun. 136, 370-374 (1997).
[CrossRef]

1996 (1)

S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect," Opt. Commun. 132, 432-436 (1996).
[CrossRef]

1995 (1)

1992 (1)

1989 (1)

C. W. Chang and D. C. Su, "An improved technique of measuring the focal length of a lens," Opt. Commun. 73, 257-262 (1989).
[CrossRef]

1988 (1)

1987 (1)

1985 (1)

1965 (1)

R. Kingslake, ed., Applied Optics and Optical Engineering (Academic, 1965).

Angel, L.

L. Angel, M. Tebaldi, and R. Henao, "Phase stepping in Lau interferometry," Opt. Commun. 164, 247-255 (1999).
[CrossRef]

Chang, C. W.

C. W. Chang and D. C. Su, "An improved technique of measuring the focal length of a lens," Opt. Commun. 73, 257-262 (1989).
[CrossRef]

Chung, P. S.

de Angelis, M.

M. de Angelis, S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "A new approach to high accuracy measurement of the focal lengths of lenses using a digital Fourier transform," Opt. Commun. 136, 370-374 (1997).
[CrossRef]

De Nicola, S.

M. de Angelis, S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "A new approach to high accuracy measurement of the focal lengths of lenses using a digital Fourier transform," Opt. Commun. 136, 370-374 (1997).
[CrossRef]

S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect," Opt. Commun. 132, 432-436 (1996).
[CrossRef]

DeBoo, B.

Faridi, M. S.

P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, "Measurement of focal length with phase-shifting Talbot interferometry," Appl. Opt. 44, 1572-1576 (2005).
[CrossRef] [PubMed]

Ferrao, P.

M. de Angelis, S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "A new approach to high accuracy measurement of the focal lengths of lenses using a digital Fourier transform," Opt. Commun. 136, 370-374 (1997).
[CrossRef]

S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect," Opt. Commun. 132, 432-436 (1996).
[CrossRef]

Finizio, A.

M. de Angelis, S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "A new approach to high accuracy measurement of the focal lengths of lenses using a digital Fourier transform," Opt. Commun. 136, 370-374 (1997).
[CrossRef]

S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect," Opt. Commun. 132, 432-436 (1996).
[CrossRef]

Geatt, I.

Henao, R.

L. Angel, M. Tebaldi, and R. Henao, "Phase stepping in Lau interferometry," Opt. Commun. 164, 247-255 (1999).
[CrossRef]

Ilev, I. K.

Kafri, O.

Keren, E.

Kingslake, R.

R. Kingslake, ed., Applied Optics and Optical Engineering (Academic, 1965).

Kothiyal, M. P.

Kreske, M. K.

Murata, K.

Nakano, Y.

Pierattini, G.

M. de Angelis, S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "A new approach to high accuracy measurement of the focal lengths of lenses using a digital Fourier transform," Opt. Commun. 136, 370-374 (1997).
[CrossRef]

S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect," Opt. Commun. 132, 432-436 (1996).
[CrossRef]

Sasian, J.

Shakher, C.

P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, "Measurement of focal length with phase-shifting Talbot interferometry," Appl. Opt. 44, 1572-1576 (2005).
[CrossRef] [PubMed]

M. Thakur and C. Shakher, "Evaluation of the focal distance of lenses by white-light Lau phase interferometry," Appl. Opt. 41, 1841-1845 (2002).
[CrossRef] [PubMed]

Singh, P.

P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, "Measurement of focal length with phase-shifting Talbot interferometry," Appl. Opt. 44, 1572-1576 (2005).
[CrossRef] [PubMed]

Sirohi, R. S.

P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, "Measurement of focal length with phase-shifting Talbot interferometry," Appl. Opt. 44, 1572-1576 (2005).
[CrossRef] [PubMed]

K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, "Direct determination of focal length by using Talbot interferometry," Appl. Opt. 31, 5984-5987 (1992).
[CrossRef] [PubMed]

Sriram, K. V.

Su, D. C.

C. W. Chang and D. C. Su, "An improved technique of measuring the focal length of a lens," Opt. Commun. 73, 257-262 (1989).
[CrossRef]

Tebaldi, M.

L. Angel, M. Tebaldi, and R. Henao, "Phase stepping in Lau interferometry," Opt. Commun. 164, 247-255 (1999).
[CrossRef]

Thakur, M.

Zhao, S.

Appl. Opt. (1)

P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, "Measurement of focal length with phase-shifting Talbot interferometry," Appl. Opt. 44, 1572-1576 (2005).
[CrossRef] [PubMed]

Appl. Opt. (6)

Opt. Commun. (2)

C. W. Chang and D. C. Su, "An improved technique of measuring the focal length of a lens," Opt. Commun. 73, 257-262 (1989).
[CrossRef]

L. Angel, M. Tebaldi, and R. Henao, "Phase stepping in Lau interferometry," Opt. Commun. 164, 247-255 (1999).
[CrossRef]

Opt. Commun. (2)

M. de Angelis, S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "A new approach to high accuracy measurement of the focal lengths of lenses using a digital Fourier transform," Opt. Commun. 136, 370-374 (1997).
[CrossRef]

S. De Nicola, P. Ferrao, A. Finizio, and G. Pierattini, "Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect," Opt. Commun. 132, 432-436 (1996).
[CrossRef]

Opt. Lett. (2)

Other (1)

R. Kingslake, ed., Applied Optics and Optical Engineering (Academic, 1965).

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Figures (11)

Fig. 1
Fig. 1

Cross section of a one-order binary phase (0, π) CDG.

Fig. 2
Fig. 2

Theoretical normalized intensity of a one-order CDG with (a) an infinite aperture and (b) a limited aperture.

Fig. 3
Fig. 3

Dependence of the radial separation on grating aperture and focal length, where N is the number of the periods.

Fig. 4
Fig. 4

Simulated diffraction patterns with different displacements: (a) Δ z = 0 , (b) Δ z = 1.5 mm , (c) Δ z = 3 mm , (d) Δ z = 7.5 mm , where f = 150 mm , T = 50 μm , and the grating aperture was 15 mm .

Fig. 5
Fig. 5

Radial separation between the two rings at different positions.

Fig. 6
Fig. 6

Graphic representation of | j i n c ( 2 a λ f ρ ) exp ( i k 2 Δ z ρ 2 ) | with different displacements: (a) Δ z = 0 , (b) Δ, z = 0.75 mm , (c) Δ z = 1.5 mm .

Fig. 7
Fig. 7

Schematic of the experimental arrangement for measuring the BFL of a lens.

Fig. 8
Fig. 8

Smallest grating apertures necessary to combine the two rings for different lenses (f varies between 10  and  1000 mm ).

Fig. 9
Fig. 9

Experimental images captured (a) at the focal plane and (b) with Δ z = 1.5 mm .

Fig. 10
Fig. 10

Separation between the two rings corresponding to different focal lengths, where Δ z = 0.01 f .

Fig. 11
Fig. 11

Schematic of the modified experimental arrangement for negative lens measurements.

Equations (11)

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G ( q ) = 1 π n = 1 c n n / T ( n / T + q ) 3 / 2 × δ ( 1 / 2 ) ( q n / T ) ,
c n = 2 T T g ( r ) sin ( 2 π n T r ) d r ,
c n = 2 n π [ 1 ( 1 ) n ] .
G ( q ) = 4 π 3 / 2 1 / T ( 1 / T + q ) 3 / 2 × δ ( 1 / 2 ) ( q 1 / T ) .
G 1 ( q ) = F [ g ( r ) p ( r ) ] = G ( q ) 4 a 2 j i n c ( 2 a q ) = C 1 j i n c ( 1 / 2 ) [ 2 a ( q 1 / T ) ] ,
D a t α f ( t ) = lim h 0 1 h α j = 0 [ ( t a ) / h ] ( 1 ) j ( α j ) f ( t j h ) ,
G 2 ( ρ ) = G 1 ( ρ ) exp ( i k Δ z ) i λ Δ z exp ( i k 2 Δ z ρ 2 ) = G ( ρ ) [ j i n c ( 2 a λ f ρ ) exp ( i k 2 Δ z ρ 2 ) ] = C 2 { j i n c [ 2 a ( ρ λ f 1 T ) ] exp [ i k 2 Δ z ( ρ λ f T ) 2 ] } ( 1 / 2 ) ,
G 3 ( ρ ) = { [ g ( r ) p ( r ) ] [ exp ( i k d ) i λ d exp ( i k 2 d r 2 ) ] × exp ( i k 2 f r 2 ) exp [ i k A sph f ( r f ) 4 ] } [ exp ( i k f ) i λ f exp ( i k 2 f r 2 ) ] [ exp ( i k Δ z ) i λ Δ z exp ( i k 2 Δ z r 2 ) ] ,
G 3 ( ρ ) = F { g ( r ) p ( r ) exp [ i k A sph f ( r f ) 4 ] } [ exp ( i k Δ z ) i λ Δ z exp ( i k 2 Δ z r 2 ) ] = C 3 × G ( ρ ) { j i n c ( 2 a λ f ρ ) exp ( i k 2 Δ z ρ 2 ) F [ exp ( i k A sph f ( r f ) 4 ) ] } ,
Δ Z f = 2 A sph f ( w b f ) 2 ,
Δ = f A 2 x F ,

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