Abstract

The use of digital holographic intrerferometry in the testing of simple thin lenses is explored. Focal length, radius of curvature, and refractive index are the lens parameters that can be determined using this method. The digital holograms using the lens under test are recorded at various positions of the test lens using off-axis geometry. This is combined with a digitally computed plane wavefront to determine the curvature of the light beam emerging from the test lens. Focal length is the position of the test lens where a single fringe results. The radius of curvature of the test lens is also determined similarly using a long focal length lens to concentrate a collimated beam onto the test lens. The nonuniformities on the lens surface could also be found by using this method. The implementation of the method is shown by using computer simulations in the case of biconvex lenses. The method can be utilized to measure the parameters of plano–convex and concave lenses also.

© 2007 Optical Society of America

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References

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  1. G. Smith, "Liquid immersion method for the measurement of the refractive index of a lens," Appl. Opt. 21, 755-758 (1982).
    [CrossRef] [PubMed]
  2. R. S. Kasana and K. J. Rosenbrunch, "The use of plane parallel plate for determining the lens' parameters," Opt. Commun. 46, 69-73 (1983).
    [CrossRef]
  3. R. S. Kasana, S. Boseck, and K. J. Oesenbrunch, "Use of a grating in a coherent optical processing configuration for evaluating the refractive index of a lens," Appl. Opt. 23, 757-761 (1984).
    [CrossRef] [PubMed]
  4. V. K. Chhaniwal, A. Anand, and C. S. Narayanamurthy, "Determination of refractive indices of biconvex lenses by use of Michelson interferometer," Appl. Opt. 45, 3985-3990 (2006).
    [CrossRef] [PubMed]
  5. U. Schnars and W. P. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
    [CrossRef]
  6. U. Schnars and W. P. Jueptner, Digital Holography: Digital Recording, Numerical Reconstruction and Related Techniques (Springer-Verlag, 2005).
  7. L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).
    [CrossRef]
  8. Yu. I. Ostrovsky, M. M. Butusov, and G. V. Ostrovskaya, Interferometry by Holography (Springer-Verlag, 1980).
  9. M. Griot, "The practical application of light," in Melles Griot Catalogue for Optical Components (2000), http://www.mellesgriot.com.

2006 (1)

2003 (1)

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).
[CrossRef]

2002 (1)

U. Schnars and W. P. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

2000 (1)

M. Griot, "The practical application of light," in Melles Griot Catalogue for Optical Components (2000), http://www.mellesgriot.com.

1984 (1)

1983 (1)

R. S. Kasana and K. J. Rosenbrunch, "The use of plane parallel plate for determining the lens' parameters," Opt. Commun. 46, 69-73 (1983).
[CrossRef]

1982 (1)

Anand, A.

Boseck, S.

Butusov, M. M.

Yu. I. Ostrovsky, M. M. Butusov, and G. V. Ostrovskaya, Interferometry by Holography (Springer-Verlag, 1980).

Cai, L. Z.

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).
[CrossRef]

Chhaniwal, V. K.

Griot, M.

M. Griot, "The practical application of light," in Melles Griot Catalogue for Optical Components (2000), http://www.mellesgriot.com.

Jueptner, W. P.

U. Schnars and W. P. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

U. Schnars and W. P. Jueptner, Digital Holography: Digital Recording, Numerical Reconstruction and Related Techniques (Springer-Verlag, 2005).

Kasana, R. S.

Liu, Q.

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).
[CrossRef]

Narayanamurthy, C. S.

Oesenbrunch, K. J.

Ostrovskaya, G. V.

Yu. I. Ostrovsky, M. M. Butusov, and G. V. Ostrovskaya, Interferometry by Holography (Springer-Verlag, 1980).

Ostrovsky, Yu. I.

Yu. I. Ostrovsky, M. M. Butusov, and G. V. Ostrovskaya, Interferometry by Holography (Springer-Verlag, 1980).

Rosenbrunch, K. J.

R. S. Kasana and K. J. Rosenbrunch, "The use of plane parallel plate for determining the lens' parameters," Opt. Commun. 46, 69-73 (1983).
[CrossRef]

Schnars, U.

U. Schnars and W. P. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

U. Schnars and W. P. Jueptner, Digital Holography: Digital Recording, Numerical Reconstruction and Related Techniques (Springer-Verlag, 2005).

Smith, G.

Wang, Y. R.

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).
[CrossRef]

Yang, X. L.

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).
[CrossRef]

Appl. Opt. (3)

Meas. Sci. Technol. (1)

U. Schnars and W. P. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Opt. Commun. (2)

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).
[CrossRef]

R. S. Kasana and K. J. Rosenbrunch, "The use of plane parallel plate for determining the lens' parameters," Opt. Commun. 46, 69-73 (1983).
[CrossRef]

Other (3)

Yu. I. Ostrovsky, M. M. Butusov, and G. V. Ostrovskaya, Interferometry by Holography (Springer-Verlag, 1980).

M. Griot, "The practical application of light," in Melles Griot Catalogue for Optical Components (2000), http://www.mellesgriot.com.

U. Schnars and W. P. Jueptner, Digital Holography: Digital Recording, Numerical Reconstruction and Related Techniques (Springer-Verlag, 2005).

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Figures (10)

Fig. 1
Fig. 1

Experimental setup for the measurement of focal length.

Fig. 2
Fig. 2

Coordinate system of the recording as well as reconstruction geometry. The wavefront due to a lens placed just inside the focus is also shown. The y and η axes are into the plane of the paper.

Fig. 3
Fig. 3

Simulated hologram for a test lens of 50   mm focal length defocused by 0 .1   mm .

Fig. 4
Fig. 4

Change of the radius of curvature of the wavefront with defocusing of the test lens.

Fig. 5
Fig. 5

Phase maps for different defocusing for a test lens with a focal length of 50 mm. (a) 0 .5   mm inside focus, (b) 0.1 mm inside focus, (c) 0 .02   mm inside focus, (d) at focus, (e) 0 .02   mm outside focus, (f) 0 .1   mm outside focus, and (g) 0.5 mm outside focus.

Fig. 6
Fig. 6

Change in sensitivity in the measurement of focal length.

Fig. 7
Fig. 7

Phase maps obtained with a tilted digitally introduced plane wavefront used for comparison to determine the sign of the curvature of the wavefront (tilt was 0.2° with y axis).

Fig. 8
Fig. 8

Experimental setup for the determination of the radius of curvature of the lenses.

Fig. 9
Fig. 9

Phase maps obtained for various positions of the test lens (focal length of 50   mm ) from the autocollimating lens. (a) z = 48   mm , (b) z = z 1 = 48.5   mm , (c) z = 50   mm , (d) z = 99   mm , (e) z = z 2 = 100   mm , and (f) z = 100.5   mm .

Fig. 10
Fig. 10

Error in the determination of the refractive index. The differences of focal lengths between the test lens and the autocollimating lens were (A) 10   mm and (B) 50   mm .

Equations (9)

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Γ ( ξ , η ) = ( i λ d exp ( i 2 π λ d ) U l ( x , y ) × exp [ i π λ d ( ( ξ x ) 2 + ( η y ) 2 ) ] d x d y ) + A exp ( i 2 π λ ξ sin θ ) ,
ϕ S ( x , y ) = tan 1 ( Im ( U l ( x , y ) ) Re ( U l ( x , y ) ) ) .
R w = f 2 / Δ f ,
ϕ S ( x , y ) = 2 π λ x 2 + y 2 + R w 2 .
Δ ϕ ( x , y )
= { ϕ R ( x , y ) ϕ S ( x , y ) if   ϕ R ( x , y ) ϕ S ( x , y ) ϕ R ( x , y ) ϕ S ( x , y ) + 2 π if   ϕ R ( x , y ) < ϕ S ( x , y ) .
R l = z 1 z 2 .
n = 1 + R l 1 R l 2 f ( R l 2 R l 1 ) ,
d n = ± [ 1 f { d R l 1 ( R l 2 R l 2 R l 1 ) + d R l 2 ( R l 1 R l 2 R l 1 ) } ( n 1 ) d f f ] ,

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