Abstract

Pattern generation in Talbot planes has generally been interpreted in terms of image formation, the repetitive slits are said to make repetitive images of themselves. In this context, Fourier optics developments have correctly predicted the positions of some but not all of the Talbot planes. Now, wave-optics methods are used to obtain general expressions for the positions of all known Talbot planes and the lateral positions of the diffraction fringes within them. These equations predict the key features of the Talbot effect, and they better relate multiple-slit diffraction in the Fresnel and Fraunhofer domains.

© 1992 Optical Society of America

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References

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  1. F. Talbot, “Facts relating to optical science No. IV,” Philos. Mag. 9, 401–407 (1836).
  2. Rayleigh, “On copying diffraction-gratings, and on some phenomenon connected therewith,” Philos. Mag. 11, 196–205 (1881).
    [CrossRef]
  3. J. M. Cowley, A. F. Moodie, “Fourier images: I-the point source,” Proc. R. Soc. London Ser. B 70, 486–496 (1957).
    [CrossRef]
  4. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–381 (1965).
    [CrossRef]
  5. E. A. Hiedemann, M. A. Breazeale, “Secondary interference in the Fresnel zone of gratings,” J. Opt. Soc. Am. 49, 372–375 (1959).
    [CrossRef]
  6. G. L. Rogers, “Calculations of intermediate Fourier images of a finite line grating on a digital computer, with an application to an unusual case,” Brit. J. Appl. Phys. 14, 657–661 (1963).
    [CrossRef]
  7. D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11, 2613–2624 (1972).
    [CrossRef] [PubMed]
  8. Y. Nukano, K. Murata, “Talbot interferometer for measuring the focal length of a lens,” Appl. Opt. 24, 3162–3166 (1985).
    [CrossRef]
  9. E. Keren, I. Glatt, O. Kafri, “Propagator for the modulating transfer function of a wide-angle scatterer,” Opt. Lett. 11, 554–556 (1986).
    [CrossRef] [PubMed]
  10. K. Patorski, “Talbot interferometry with increased shear: part 3,” Appl. Opt. 27, 3875–3878 (1988).
    [CrossRef] [PubMed]
  11. E. Tepichin, J. Ojeda-Castaneda, “Talbot interferometer with simultaneous dark and bright fields,” Appl. Opt. 28, 1517–1520 (1989).
    [CrossRef] [PubMed]
  12. O. Kafri, E. Keren, K. Kreske, Y. Zag., “Moiré defiectometry with a focused beam: radius of curvature, microscopy, and thickness analysis,” Appl. Opt. 29, 133–136 (1990).
    [CrossRef] [PubMed]
  13. F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, New York, 1957).
  14. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1959), p. 382.

1990 (1)

1989 (1)

1988 (1)

1986 (1)

1985 (1)

1972 (1)

1965 (1)

1963 (1)

G. L. Rogers, “Calculations of intermediate Fourier images of a finite line grating on a digital computer, with an application to an unusual case,” Brit. J. Appl. Phys. 14, 657–661 (1963).
[CrossRef]

1959 (1)

1957 (1)

J. M. Cowley, A. F. Moodie, “Fourier images: I-the point source,” Proc. R. Soc. London Ser. B 70, 486–496 (1957).
[CrossRef]

1881 (1)

Rayleigh, “On copying diffraction-gratings, and on some phenomenon connected therewith,” Philos. Mag. 11, 196–205 (1881).
[CrossRef]

1836 (1)

F. Talbot, “Facts relating to optical science No. IV,” Philos. Mag. 9, 401–407 (1836).

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1959), p. 382.

Breazeale, M. A.

Cowley, J. M.

J. M. Cowley, A. F. Moodie, “Fourier images: I-the point source,” Proc. R. Soc. London Ser. B 70, 486–496 (1957).
[CrossRef]

Glatt, I.

Hiedemann, E. A.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, New York, 1957).

Kafri, O.

Keren, E.

Kreske, K.

Moodie, A. F.

J. M. Cowley, A. F. Moodie, “Fourier images: I-the point source,” Proc. R. Soc. London Ser. B 70, 486–496 (1957).
[CrossRef]

Murata, K.

Nukano, Y.

Ojeda-Castaneda, J.

Patorski, K.

Rayleigh,

Rayleigh, “On copying diffraction-gratings, and on some phenomenon connected therewith,” Philos. Mag. 11, 196–205 (1881).
[CrossRef]

Rogers, G. L.

G. L. Rogers, “Calculations of intermediate Fourier images of a finite line grating on a digital computer, with an application to an unusual case,” Brit. J. Appl. Phys. 14, 657–661 (1963).
[CrossRef]

Silva, D. E.

Talbot, F.

F. Talbot, “Facts relating to optical science No. IV,” Philos. Mag. 9, 401–407 (1836).

Tepichin, E.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, New York, 1957).

Winthrop, J. T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1959), p. 382.

Worthington, C. R.

Zag, Y.

Appl. Opt. (5)

Brit. J. Appl. Phys. (1)

G. L. Rogers, “Calculations of intermediate Fourier images of a finite line grating on a digital computer, with an application to an unusual case,” Brit. J. Appl. Phys. 14, 657–661 (1963).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Lett. (1)

Philos. Mag. (2)

F. Talbot, “Facts relating to optical science No. IV,” Philos. Mag. 9, 401–407 (1836).

Rayleigh, “On copying diffraction-gratings, and on some phenomenon connected therewith,” Philos. Mag. 11, 196–205 (1881).
[CrossRef]

Proc. R. Soc. London Ser. B (1)

J. M. Cowley, A. F. Moodie, “Fourier images: I-the point source,” Proc. R. Soc. London Ser. B 70, 486–496 (1957).
[CrossRef]

Other (2)

F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, New York, 1957).

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1959), p. 382.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the apparatus for producing the Talbot effect. A light source P1 illuminates a part of a grating that generates a bright spot at point P2 in a Talbot plane. P1 and P2 are distances R1 and R2 from the grating. Some of the light goes through slit d, which is a distance y1 from a perpendicular dropped from P1 to the grating, a distance y2 from the perpendicular from P2.

Fig. 2
Fig. 2

Some of the predictions by a numerical method [Eqs. (1) and (2)] of the brightest fringes or images (dots) beyond a 20-slit grating that is normally illuminated by a remote source. The fringes occur in Talbot planes; the n values of some of these planes are indicated.

Fig. 3
Fig. 3

Fringe generation by parts of two gratings as illuminated normally by remote sources. In each case, the illumination of P2 in a Talbot plane is only by slits c and υ. On the left, υ = d (j = 1); on the right υ = e (j = 2). Constructive interference lines, slope = Yaυ/R2, are shown as dotted lines. On the left, this line intersects the grating at a point midway between the slits. On the right, the line intersects a slit d, which is midway between the generating slits (j = 2) c and e. The slits containing small circles generate the dotted lines.

Fig. 4
Fig. 4

A pair of slits, normally and remotely illuminated. Emanating from a point midway between them are constructive interference lines of the labeled slope indices m = 0 to ±4. Bright double-slit fringes could be observed in any plane of observation at an arbitrary distance from the grating where these lines intersect this plane.

Fig. 5
Fig. 5

Four slits of a normally and remotely illuminated grating. These slits are interpreted as three overlapping pairs, j = 1. Each pair, c and d, etc. generates a set of constructive interference lines: m = 0, ±1, ±2, etc. When two or more of these lines intersect at a plane indicated by a vertical dashed line, the waves from all the parent slits that are involved constructively interfere. Bright fringes occur at these junctions. When such lines of slope indices m and m′ intersect, R2 = nT, where n = j2/q = 1/q and q = m = m′. There is also an unlabeled Talbot plane at n = j2/q = 4/6 = 2/3. It is omitted here since fringes in this plane are actually generated by wave interactions in the j = 2 mode.

Fig. 6
Fig. 6

Fringe formation by waves interacting in the j = 2 mode. Two different sets of constructive interference lines are generated by slit combinations involving every other slit. Slits d, f, h, etc., generate the solid constructive interference lines that emanate, respectively, from slits e, g, etc. Similarly dotted constructive interference lines from slits d, f, h, etc., are generated by waves from slits c, e, g, i. Some of the Talbot planes, which contain fringes generated by these lines, are marked by vertical solid lines. Some of the line junctions at the level of slit h have also been labeled with a square, diamonds, triangles, and circles. At most junctions, both solid and dotted lines intersect. Then the resultant disturbances depend on the relative phases of the waves from the two series of slits.

Fig. 7
Fig. 7

A grating with j = 1 constructive interference lines in the format of Fig. 6. The labels of the special line junctions of Fig. 6 have been copied. Fringe generation is constrained by requirements that n = j2/q and that the junctions of constructive interference lines of the same series should cross and these lines should be generated by continuously overlapping pairs of slits. In general, wave interactions of the lowest possible mode, j value, are the most important (see text).

Equations (16)

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A = h = 1 H A h exp ( i ϕ h ) ,
Δ = y 2 / ( 2 R ) ,
Δ υ c = ( y υ 2 y c 2 ) / ( 2 R ) = y a υ j a / R ,
ϕ = 2 π m = 2 π Δ / λ ,
y a υ = m λ R 2 / j a = m a R 2 / j T = max / j T ,
d sin θ = m λ .
Δ 1 + Δ 2 = m λ .
( j a y 1 a υ ) / R 1 + ( j a y 2 a υ ) / R 2 = m λ .
d [ sin ( i ) + sin ( θ ) ] = m λ .
j a ( y 1 a υ c d ) / R 1 + j a ( y 2 a υ c d ) / R 2 = m c d λ,
j a ( y 1 a υ d e ) / R 1 + j a ( y 2 a υ d e ) / R 2 = m d e λ .
1 / R 1 + 1 / R 2 = 1 / ( n T ) ,
y 2 a υ = m n a / j = m a j / q .
y 2 a υ = m a / q .
b = a j q ( R 1 + R 2 R 1 ) .
b = a q ( R 1 + R 2 ) R 1 .

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