Abstract

A simple and precise technique, based on Talbot interferometry, of measuring the focal distance of a simple lens or a compound lens system is described. To establish the foundations of the technique, properties of self-images in free space and of their images, when formed by a lens, are first examined and experimentally demonstrated.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Talbot, “Facts Relating to Optical Science,” Philos. Mag. 9, 401 (1936).
  2. Y. Cohen-Sabban, D. Joyeux, “Aberration-Free Nonparaxial Self-Imaging,” J. Opt. Soc. Am. 73, 707 (1983).
    [CrossRef]
  3. A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971); “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1973).
    [CrossRef]
  4. K. Patorski, “Talbot Interferometry with Increased Shear,” Appl. Opt. 24, 4448 (1985); “Talbot Interferometry with Increased Shear: Further Considerations,” Appl. Opt. 25, 1111 (1986).
    [CrossRef] [PubMed]
  5. D. E. Silva, “Talbot Interferometry for Radial and Lateral Derivatives,” Appl. Opt. 11, 2613 (1972).
    [CrossRef] [PubMed]
  6. Y. Nakano, K. Murata, “Talbot Interferometry for Measuring the Focal Length of a Lens,” Appl. Opt. 24, 3162 (1985).
    [CrossRef] [PubMed]
  7. See, e.g., W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).
  8. W. D. Montgomery, “Self-Imaging Objects of Infinite Aperture,” J. Opt. Soc. Am. 57, 772 (1967).
    [CrossRef]

1985 (2)

1983 (1)

1972 (1)

1971 (1)

A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971); “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1973).
[CrossRef]

1967 (1)

1936 (1)

F. Talbot, “Facts Relating to Optical Science,” Philos. Mag. 9, 401 (1936).

Cathey, W. T.

See, e.g., W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

Cohen-Sabban, Y.

Joyeux, D.

Lohmann, A. W.

A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971); “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1973).
[CrossRef]

Montgomery, W. D.

Murata, K.

Nakano, Y.

Patorski, K.

Silva, D. E.

D. E. Silva, “Talbot Interferometry for Radial and Lateral Derivatives,” Appl. Opt. 11, 2613 (1972).
[CrossRef] [PubMed]

A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971); “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1973).
[CrossRef]

Talbot, F.

F. Talbot, “Facts Relating to Optical Science,” Philos. Mag. 9, 401 (1936).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Relative positioning of different distributions characterized by μm(m = 0,±1,…); virtual distributions have order m < 0. N m + = N m + 1 and O m + = O m + 1 .

Fig. 2
Fig. 2

Irradiance distributions: (a) Po (object); (b) O o +; (c) N o +; (d) P1. Away from the boundaries, irradiances of the same type μ are identical, independent of the order m.

Fig. 3
Fig. 3

Spectra of the irradiances I2, I3, and I4 for positive frequencies. They are symmetric for negative frequencies. (a) Ĩ2 for a specified value of ϕ; (b) Ĩ3; (c) Ĩ4, where s = cosϕ and the frequencies are according to increasing value (for ω2 < ω1): 0 ω1ω2, 2(ω1ω2); 2ω2ω1, ω2, ω1, 2ω1ω2; 2ω2, ω1 + ω2, 2ω1; 2ω2 + ω1, 2ω1 + ω2, and 2(ω1 + ω2). Some of the components of the spectra vanish when ϕ = [(2m ± 1)/2]π.

Fig. 4
Fig. 4

Geometry representing the transformation of distribution U1 in U3 by a lens of focal distance f. Illumination is made by a plane wave from the left.

Fig. 5
Fig. 5

Irradiance distribution observed at the distance d2 = 2f from the lens when (a) d1 = 2f ± Δ/4; (b) d1 = 2f ± Δ/2; (c) d1 = 2f ± 3Δ/4; (d) d1 = 2f ± Δ. For d1 = 2f the object is fully reproduced.

Fig. 6
Fig. 6

Schematic representation of the location of the object distribution Po with real (P1,P2) and virtual (P−1,P−2,P−3) self-images and respective real images I1, I2 and I−1, I−2, I−2, formed by a positive lens of focal distance f; d1 = d2 = 2f and Δ = f/2.

Fig. 7
Fig. 7

Moire patterns at the (x3,y3) plane behind transparency t3 when (a) d1 = 80 mm, d2 = 200 mm ≃ 2f, (b) d1 = 110 mm, d2 = 170 mm.

Tables (3)

Tables Icon

Table I Values of αμm and ναμmαμo for Different Distributions μm

Tables Icon

Table II Values of F1, F2, F3 and ωL > 0 for d2 > 0

Tables Icon

Table III Experimental Values of f Calculated from Eq. (38) Using a Combination of the Measured Values of F 2 , F 2, and ɛ Shown in Each Row

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

t 1 ( x 1 , y 1 ) = 1 + cos ω 1 x 1 ,
U 1 ( x 1 , y 1 ) = 1 + cos ω 1 x 1 .
U 2 ( x 2 , y 2 ) = exp ( i k L ) i λ L exp ( i k x 2 2 + y 2 2 2 L ) U 1 ( x 1 , y 1 ) × exp ( i k x 1 2 + y 1 2 2 L ) exp ( i k x 1 x 2 + y 1 y 2 L ) d x 1 d y 1 ,
U 2 ( x 2 , y 2 ) = exp ( i γ ) [ 1 + exp ( i ϕ ) cos ω 1 x 2 ] ,
γ = k L ,
ϕ = π λ L f 1 2 .
ϕ = 2 m π ( m = 0 , ± 1 , + 2 , )
Δ = 2 Λ 1 2 λ ,
I 2 ( x 2 , y 2 ) = 1 + 2 cos ϕ cos ω 1 x 2 + cos 2 ω 1 x 2 ,
I P m ( x 2 , y 2 ) = ( 1 + cos ω 1 x 2 ) 2 ,
I 0 m ± ( x 2 , y 2 ) = 1 + cos 2 ω 1 x 2 ,
I N m ± ( x 2 , y 2 ) = ( 1 cos ω 1 x 2 ) 2 .
I 4 ( x 2 , y 2 ) = I 2 I 3 = ( 1 + 2 cos ϕ cos ω 1 x 2 + cos 2 ω 1 x 2 ) ( 1 + cos ω 2 x 2 ) 2 .
U 3 ( x 3 , y 3 ) = i w λ d 1 d 2 exp [ i δ ( x 3 2 + y 3 2 ) ] U 1 ( x 1 , y 1 ) × exp [ i η ( x 1 2 + y 1 2 ) ] × exp [ i 2 π w λ d 1 d 2 ( x 1 x 3 + y 1 y 3 ) ] d x 1 d y 1 ,
δ k 2 d 2 ( l w d 2 ) ,
η k 2 d 1 ( l w d 1 ) ,
w 1 = d 1 1 + d 2 1 f 1 .
U 3 ( x 3 , y 3 ) = C exp [ i γ w ( x 2 + y 2 ) ] [ 1 + exp ( i θ ) cos ω 1 x 3 ] ,
C w w d 1 d 2 d 1 ,
γ w k w d 1 | d 1 w d 1 w 2 d 2 ( d 1 w ) | ,
θ 2 π Δ d 1 2 d 1 w ,
ω 1 ω 1 | f d 2 f | .
1 d 1 P o + 1 d 2 μ ( 1 ± α μ Δ 2 d 1 P o ) 1 f ( 1 ± α μ Δ 2 d 1 P o ) = 0 ,
1 d 1 μ + 1 d 2 μ ( 1 ± α μ α μ 2 d 1 μ Δ ) 1 f ( 1 ± α μ α μ 2 d 1 μ ) = 0 .
1 d 1 μ + 1 d 2 μ 1 f = 0 ,
d 2 μ m d 2 μ o = ± ν Δ M o 2 d 1 μ o ± ν Δ ( 1 M o ) ,
M 0 d 2 μ o d 1 μ o ,
Δ m = ± ν Δ M o 2 d 1 μ o ± ν Δ ( 1 M o ) n = 1 m 1 Δ n .
t 3 = t 1 = 1 + cos ω 1 x 3
I 5 ( x 3 , y 3 ) = ( 1 + 2 cos θ cos ω 1 x 3 + cos 2 ω 1 x 3 ) ( 1 + cos ω 1 x 3 ) 2 ,
ω L 1 = { 2 ( ω 1 ω 1 ) ; if ω 1 < 3 2 ω 1 , ω 1 if 3 2 ω 1 < ω 1 < 2 ω 1 , ω 1 if ω 1 > 2 ω 1 .
ω L 2 = { ω 1 ω 1 if ω 1 < 3 2 ω 1 , 2 ω 1 ω 1 if ω 1 > 3 2 ω 1 .
F = ω 1 ω 1 ω 1
F = { F 1 = d 2 f d 2 , if 0 < d 2 < f , F 2 = 2 f d 2 d 2 f , if d 2 > f > 0 , F 3 = d 2 d 2 + | f | , if f < 0 , d 2 > 0 .
f = d 2 1 + F 2 2 + F 2 .
f = ( 1 + F 2 ) ( 1 + F 2 ) F 2 F 2 ε ,
Δ f f = Δ N 2 | 1 N 2 ( 1 + N 2 ) + 1 N 2 ( 1 + N 2 ) + ( N 2 N 2 + N 2 N 2 ) 1 N 2 N 2 | + Δ ε ε ,

Metrics