Abstract

We analyze the near-field behavior of binary amplitude/phase diffraction gratings, which modulate at the same time the amplitude and phase of the incident light beam. As it is expected, the distance between two consecutive self-images of the grating depends only on the period of the grating and the wavelength of the illumination. However, the location of the self-images depends on the specific properties of the grating. In this work, we analyze the location of the self-images in terms of the Fourier coefficients of the grating, obtaining analytical expressions. This analysis can be useful in applications in which the position of the self-images must be at certain fixed distances from the grating. Finally, an experimental and numerical verification of the proposed theory is performed.

© 2009 Optical Society of America

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References

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  1. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).
  2. C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, 2000).
  3. J. M. Cowley and A. F. Moodie, “Fourier images IV: the phase grating,” Proc. Phys. Soc. Ser. B 76, 378-384 (1960).
    [CrossRef]
  4. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  6. W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401-407 (1836).
  7. K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1-108 (1989).
  8. E. Keren and O. Kafri, “Diffraction effects in moiré deflectometry,” J. Opt. Soc. Am. A 2, 111-120 (1985).
    [CrossRef]
  9. A. P. Smirnov, “Talbot effect for amplitude-phase periodic transparencies,” Opt. Spectrosc. 69, 700-701 (1990).
  10. U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
    [CrossRef]
  11. W. G. Rees, “The validity of the Fresnel approximation,” Eur. J. Phys. 8, 44-48 (1987).
    [CrossRef]
  12. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811-818 (1981).
    [CrossRef]
  13. D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827-1834 (1994).
    [CrossRef]
  14. N. Chateau and J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321-1331 (1994).
    [CrossRef]

2004 (1)

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

1994 (2)

1990 (1)

A. P. Smirnov, “Talbot effect for amplitude-phase periodic transparencies,” Opt. Spectrosc. 69, 700-701 (1990).

1989 (1)

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1-108 (1989).

1987 (1)

W. G. Rees, “The validity of the Fresnel approximation,” Eur. J. Phys. 8, 44-48 (1987).
[CrossRef]

1985 (1)

1981 (1)

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811-818 (1981).
[CrossRef]

1960 (1)

J. M. Cowley and A. F. Moodie, “Fourier images IV: the phase grating,” Proc. Phys. Soc. Ser. B 76, 378-384 (1960).
[CrossRef]

1836 (1)

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401-407 (1836).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Chateau, N.

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “Fourier images IV: the phase grating,” Proc. Phys. Soc. Ser. B 76, 378-384 (1960).
[CrossRef]

Gaylord, T. K.

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811-818 (1981).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Grann, E. B.

Hugonin, J. P.

Kafri, O.

Keren, E.

Levy, U.

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

Marom, E.

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

Mendlovic, D.

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

Moharam, M. G.

D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827-1834 (1994).
[CrossRef]

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811-818 (1981).
[CrossRef]

Moodie, A. F.

J. M. Cowley and A. F. Moodie, “Fourier images IV: the phase grating,” Proc. Phys. Soc. Ser. B 76, 378-384 (1960).
[CrossRef]

Palmer, C.

C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, 2000).

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1-108 (1989).

Pommet, D. A.

Popov, E.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

Rees, W. G.

W. G. Rees, “The validity of the Fresnel approximation,” Eur. J. Phys. 8, 44-48 (1987).
[CrossRef]

Smirnov, A. P.

A. P. Smirnov, “Talbot effect for amplitude-phase periodic transparencies,” Opt. Spectrosc. 69, 700-701 (1990).

Talbot, W. H. F.

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401-407 (1836).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Eur. J. Phys. (1)

W. G. Rees, “The validity of the Fresnel approximation,” Eur. J. Phys. 8, 44-48 (1987).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

Opt. Spectrosc. (1)

A. P. Smirnov, “Talbot effect for amplitude-phase periodic transparencies,” Opt. Spectrosc. 69, 700-701 (1990).

Philos. Mag. (1)

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401-407 (1836).

Proc. Phys. Soc. Ser. B (1)

J. M. Cowley and A. F. Moodie, “Fourier images IV: the phase grating,” Proc. Phys. Soc. Ser. B 76, 378-384 (1960).
[CrossRef]

Prog. Opt. (1)

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1-108 (1989).

Other (4)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, 2000).

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Figures (8)

Fig. 1
Fig. 1

Scheme of the amplitude/phase grating with the parameters used to calculate the Fourier coefficients of the grating.

Fig. 2
Fig. 2

Contrast along z of the self-images produced by a diffraction grating with period p = 5 μm , fill factor α = 1 / 2 , and phase retardation δ = π / 2 , where we consider a = 1 : (a)  b = 0 , (b)  b = 1 , (c)  b = 0.35 , and (d)  b = 0.65 .

Fig. 3
Fig. 3

Contrast along z of the self-images obtained using (RCWA). The period of the grating is p = 5 μm , the fill factor is α = 1 / 2 , and the phase retardation is δ = π / 2 , where we consider a = 1 . We have simulated the change of b by means of the thickness of the chromium layer (h): solid curve ( h = 100 nm ), dashed-dotted curve ( h = 10 nm ), and dashed curve ( h = 1 nm ).

Fig. 4
Fig. 4

Contrast of the fringes produced by a diffraction grating with period p = 5 μm , fill factor α = 1 / 2 , and a = 1 in terms of z and b for (a)  δ = 0 , (b)  δ = π / 2 , (c)  δ = 3 π / 2 , and (d)  δ = π . The solid curves in (b) and (c) are plotted using Eq. (12).

Fig. 5
Fig. 5

(a) Maximum contrast position along z in terms of b and δ. (b) Maximum contrast value in terms of b and δ.

Fig. 6
Fig. 6

Experimental Talbot effect for several ratios of amplitude–phase in the diffraction grating: (a) pure-amplitude grating, (b) amplitude/phase grating, and (c) pure-phase grating ( p = 20 μm , α = 0.5 , δ = π / 2 ).

Fig. 7
Fig. 7

Experimental contrasts corresponding to situations depicted in Fig. 6: solid curve [Fig. 6a], dashed-dotted curve [Fig. 6b], and dashed curve [Fig. 6c].

Fig. 8
Fig. 8

Dependence of the transmittance of chrome in terms of the thickness of the layer.

Equations (12)

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t ( ξ ) = n = c n exp ( i q n ξ ) ,
c 0 = α ( a e i δ b ) + b , c n = α ( a e i δ b ) sinc ( π n α ) .
U ( x , z ) = exp ( i k z ) i λ z U 0 ( ξ ) t ( ξ ) exp [ i k 2 z ( x ξ ) 2 ] d ξ .
U ( x , z ) = A 0 n c n exp ( i q x n ) exp ( i π n 2 z / z T ) ,
I ^ ( z , x ) = n n c n c n * exp [ i q x ( n n ) ] exp [ i π ( n 2 n 2 ) z / z T ] .
I ^ ( z , x ) = n n | c n | | c n | exp [ i q x ( n n ) ] exp { i [ π ( n 2 n 2 ) z z T + ( β n β n ) ] } .
I ^ ( z , x ) | c 0 | 2 + 2 | c 1 | 2 + 4 | c 0 | | c 1 | cos ( q x ) cos ( π z z T + β 0 β 1 ) + 2 | c 1 | 2 cos ( 2 q x ) .
c 0 = 1 2 1 + b 2 + 2 b cos δ exp [ i atan ( sin δ b + cos δ ) ] , c 1 = 1 π 1 + b 2 2 b cos δ exp [ i atan ( sin δ b + cos δ ) ] .
I ^ ( z , x ) 1 4 ( 1 + b 2 + 2 b cos δ ) + 2 π 1 + b 4 2 b 2 cos ( 2 δ ) cos ( q x ) cos [ π z T ( z + z S ) ] + 2 π 2 ( 1 + b 2 2 b cos δ ) [ 1 + cos ( 2 q x ) ] ,
z S = z T π ( β n β n ) = z T π { acot [ ( b + cos δ ) csc δ ] acot ( cot δ b csc δ ) }
z S = { 2 b sin δ 2 3 b 3 sin ( 3 δ ) f o r     b 0 = 0 , χ ( b 1 ) csc δ + ( b 1 ) 2 2 csc δ f o r     b 0 = 1 ,
C ( z ) = 16 1 + b 4 2 b 2 cos ( 2 δ ) ( 1 + b 2 ) ( 8 + π 2 ) + 2 b ( 8 + π 2 ) cos ( δ ) cos [ π z T ( z + z S ) ] .

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