Abstract

The Ronchi grating is well known for its many applications in areas such as spectroscopy, grating interferometry, and Talbot interferometry. On the other hand, the checker grating has attracted very little attention. A checker grating also self-images at equidistant planes; the separation between these planes is a quarter of the Talbot distance of a Ronchi grating of the same period. To understand this and several other features, a transition from Ronchi grating (a one-dimensional grating) to checker grating (a two-dimensional grating) has been both theoretically and experimentally studied and results are presented. Because the checker grating self-images closer to the grating and its transmittance is higher than that of a Ronchi grating, the use of its self-image planes for array generation is also emphasized.

© 1997 Optical Society of America

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References

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  1. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–451 (1964).
    [CrossRef]
  2. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (Elseiver Science, New York, 1989), Vol. 27, pp. 1–108.
  3. E. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass.1963), Chap. 7, p. 113.
  4. V. Arrizon, E. Lopez-Olazagasti, “Binary phase grating for array generation at 1/16 of the Talbot length,” J. Opt. Soc. Am. A 12, 801–804 (1995).
    [CrossRef]
  5. R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
    [CrossRef]
  6. F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, Singapore, 1981), Chap. 15, p. 324.
  7. J. R. Leger, G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett. 15, 288–290 (1990).
    [CrossRef] [PubMed]
  8. N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
    [CrossRef]
  9. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).

1995 (1)

1990 (1)

1989 (1)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

1988 (1)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).

1969 (1)

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
[CrossRef]

1964 (1)

Arrizon, V.

Edgar, R. F.

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, Singapore, 1981), Chap. 15, p. 324.

Leger, J. R.

Lohmann, A. W.

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).

Lopez-Olazagasti, E.

O’Neill, E.

E. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass.1963), Chap. 7, p. 113.

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (Elseiver Science, New York, 1989), Vol. 27, pp. 1–108.

Ronchi, V.

Streibl, N.

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

Swanson, G. J.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, Singapore, 1981), Chap. 15, p. 324.

Appl. Opt. (1)

J. Mod. Opt. (1)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
[CrossRef]

Opt. Lett. (1)

Optik (1)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).

Other (3)

F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, Singapore, 1981), Chap. 15, p. 324.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (Elseiver Science, New York, 1989), Vol. 27, pp. 1–108.

E. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass.1963), Chap. 7, p. 113.

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Figures (9)

Fig. 1
Fig. 1

Checker grating; wx/dx, wy/dy are opening ratios in x and y directions, d0 is length of diagonal.

Fig. 2
Fig. 2

Sketch of intermediate gratings; d0, y0 shift in x and y directions.

Fig. 3
Fig. 3

Self-images of gratings at z = 395 mm: (a) A1 grating, (b) A2 grating, (c) A3 grating.

Fig. 4
Fig. 4

Self-images of gratings at z = 198 mm: (a) A2 grating, (b) A3 grating.

Fig. 5
Fig. 5

Phase variation at the quarter-Talbot-plane of a Ronchi grating: (a) portion of an amplitude Ronchi, (b) phase image at quarter-Talbot-plane.

Fig. 6
Fig. 6

(a) Intermediate grating consisting of two Ronchi gratings, A and B; (b) phase variation at quarter-Talbot-plane of grating A; (c) phase variation at quarter-Talbot-plane of grating B; (d) phase difference at quarter-Talbot-plane resulting from both grating A and grating B.

Fig. 7
Fig. 7

Fourier spectra: (a) grating A1, (b) grating A2, (c) grating A3.

Fig. 8
Fig. 8

(a) Compressed spots obtained at z = 55 mm behind checker grating; (b) simulated intensity distribution over an area of 2 mm × 2 mm at z = 55 mm behind checker grating.

Fig. 9
Fig. 9

(a) Self-image of inner circle checker grating at z = 198 mm; at top is pictorial representation of single aperture of inner circle checker grating; (b) self-image of outer circle checker grating at z = 198 mm; at top is pictorial representation of single aperture of outer circle checker grating.

Tables (2)

Tables Icon

Table 1 Self-Imaging Conditions for Intermediate and Checker Gratings

Tables Icon

Table 2 Percentage Transmittance Values for Various Amplitude Gratings

Equations (13)

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tx, y=rectxwx  1dxcombxdx×rectywy  1dycombydy=mn aman expi2π mxdx+nydy,
am=wxdxsincmwxdx, an=wydysincnwydy,
tx, y=mnaman expi2πdmx+ny+mnaman expi2πdmx-x0+ny-y0=mn aman expi2πdmx+ny×1+exp-i2πdmx0+ny0,
tx, y=mnaman expi2πdmx+ny×1+-1m+n.
aTx, y, z=mnaman expi2πdmx+ny×exp-iπλd2m2+n2z×1+exp-i2πdmx0+ny0.
Ix, y, z=aTx, y, z2.
zT=2d2λ.
a1ab sincμasincνb,
AT=a11+expiδx+expi2δx++expiN-1δx×1+expiδy+expi2δy++expiM-1δy+a1 expiδ1+δ21+expiδx+expi2δx++expiN-1δx×1+expiδy+expi2δy++expiM-1δy,
I4a2b2 sinc2μasinc2νbcos2δ1+δ22×sin2Nδx/2sin2δx/2sin2Mδy/2sin2δy/2.
I4a4 sinc2μasinc2νacos2 πx0μ+aν×sin2Nδx/2sin2δx/2sin2Mδy/2sin2δy/2.
tx, y=circx2+y21/2ρ1dxdycombxdxcombydy.
tx, y=nmρdJ12πρ/dn2+m21/2n2+m21/2×expi2πdmx+ny.

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