Abstract

In this work we investigate, analytically and numerically, the effect on the diffracted field produced by typical fabrication errors in sawtooth gratings. The analysis is carried out for the near and far field, showing the effects on the intensity and on the diffraction orders efficiency. When the grating profile is not perfect but presents a curved profile or overdevelopment error, some different diffraction orders appear, changing the intensity and the efficiency of each order. In addition, when roughness is present, a decreasing of efficiency is produced, but without generating different diffraction orders than the first one. We show the analytical dependence of these modifications in terms of the profile of the grating, cor roborating the results with numerical methods.

© 2010 Optical Society of America

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References

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    [CrossRef] [PubMed]
  8. D. L. Brundrett, T. K. Gaylord, and E. N. Glytsis, “Polarizing mirror/absorber for visible wavelengths based on a silicon subwavelength grating: design and fabrication,” Appl. Opt. 37, 2534-2541 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  18. J. M. Rico-Garcia and L. M. Sanchez-Brea, “Binary gratings with random heights,” Appl. Opt. 48, 3062-3069 (2009).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  25. F. J. Torcal-Milla, L. M. Sanchez-Brea, and E. Bernabeu, “Self-imaging of gratings with rough strips,” J. Opt. Soc. Am. A 25, 2390-2394 (2008).
    [CrossRef]
  26. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, 1987).

2009

2008

2007

F. J. Torcal-Milla, L. M. Sanchez-Brea, and E. Bernabeu, “Talbot effect with rough reflection gratings,” Appl. Opt. 46, 3668-3673 (2007).
[CrossRef] [PubMed]

L. M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, “Talbot effect in metallic gratings under Gaussian illumination,” Opt. Commun. 278, 23-27 (2007).
[CrossRef]

2006

2004

2001

S. Tibuleac and R. Magnusson, “Narrow-linewidth bandpass filters with diffractive thin-film layers,” Opt. Lett. 26, 584-586(2001).
[CrossRef]

Y. Li, D. Chen, and C. Yang, “Sub-microns period grating couplers fabricated by silicon mold,” Opt. Laser Technol. 33, 623-626 (2001).
[CrossRef]

2000

1998

1995

1989

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

1986

S. Ura, T. Suhara, H. Nishihara, and J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913-918 (1986).
[CrossRef]

T. Suhara and H. Nishihara, “Integrated optics components and devices using periodic structures,” IEEE J. Quantum Electron. 22, 845-867 (1986).
[CrossRef]

1982

1972

1836

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

Auslender, M.

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, 1987).

Bernabeu, E.

Brundrett, D. L.

Chen, D.

Y. Li, D. Chen, and C. Yang, “Sub-microns period grating couplers fabricated by silicon mold,” Opt. Laser Technol. 33, 623-626 (2001).
[CrossRef]

Connell, J. R.

Dübendorfer, J.

Fujita, T.

Gaylord, T. K.

Glytsis, E. N.

Harrison, G. R.

Hava, S.

Hegedus, Z.

Hirai, Y.

Kazukonis, H.

Kikuta, H.

Koyama, J.

S. Ura, T. Suhara, H. Nishihara, and J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913-918 (1986).
[CrossRef]

T. Fujita, H. Nishihara, and J. Koyama, “Blazed gratings and Fresnel lenses fabricated by electron-beam lithography,” Opt. Lett. 7, 578-580 (1982).
[CrossRef] [PubMed]

Kunz, R. E.

Li, Y.

Y. Li, D. Chen, and C. Yang, “Sub-microns period grating couplers fabricated by silicon mold,” Opt. Laser Technol. 33, 623-626 (2001).
[CrossRef]

Loewen, E. G.

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

Magnusson, R.

McMullin, J. N.

Netterfield, R.

Nishihara, H.

S. Ura, T. Suhara, H. Nishihara, and J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913-918 (1986).
[CrossRef]

T. Suhara and H. Nishihara, “Integrated optics components and devices using periodic structures,” IEEE J. Quantum Electron. 22, 845-867 (1986).
[CrossRef]

T. Fujita, H. Nishihara, and J. Koyama, “Blazed gratings and Fresnel lenses fabricated by electron-beam lithography,” Opt. Lett. 7, 578-580 (1982).
[CrossRef] [PubMed]

Okano, M.

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).
[CrossRef]

Popov, E.

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

Rajkumar, N.

Rico-Garcia, J. M.

Sanchez-Brea, L. M.

Shen, F.

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, 1987).

Suhara, T.

T. Suhara and H. Nishihara, “Integrated optics components and devices using periodic structures,” IEEE J. Quantum Electron. 22, 845-867 (1986).
[CrossRef]

S. Ura, T. Suhara, H. Nishihara, and J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913-918 (1986).
[CrossRef]

Talbot, W. H. F.

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

Thompson, S. W.

Tibuleac, S.

Torcal-Milla, F. J.

Ura, S.

S. Ura, T. Suhara, H. Nishihara, and J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913-918 (1986).
[CrossRef]

Wang, A.

Wiki, M.

Yamamoto, K.

Yang, C.

Y. Li, D. Chen, and C. Yang, “Sub-microns period grating couplers fabricated by silicon mold,” Opt. Laser Technol. 33, 623-626 (2001).
[CrossRef]

Yotsuya, T.

Appl. Opt.

J. Dübendorfer and R. E. Kunz, “Compact integrated optical immunosensor using replicated chirped grating coupler sensor chips,” Appl. Opt. 37, 1890-1894 (1998).
[CrossRef]

S. Hava and M. Auslender, “Silicon grating-based mirror for 1.3 μm polarized beams: MATLAB-aided design,” Appl. Opt. 34, 1053-1058 (1995).
[CrossRef] [PubMed]

D. L. Brundrett, T. K. Gaylord, and E. N. Glytsis, “Polarizing mirror/absorber for visible wavelengths based on a silicon subwavelength grating: design and fabrication,” Appl. Opt. 37, 2534-2541 (1998).
[CrossRef]

N. Rajkumar and J. N. McMullin, “V-groove gratings on silicon for infrared beam splitting,” Appl. Opt. 34, 2556-2559 (1995).
[CrossRef] [PubMed]

Z. Hegedus and R. Netterfield, “Low sideband guided-mode resonant filter,” Appl. Opt. 39, 1469-1473 (2000).
[CrossRef]

M. Okano, H. Kikuta, Y. Hirai, K. Yamamoto, and T. Yotsuya, “Optimization of diffraction grating profiles in fabrication by electron-beam lithography,” Appl. Opt. 43, 5137-5142 (2004).
[CrossRef] [PubMed]

F. J. Torcal-Milla, L. M. Sanchez-Brea, and E. Bernabeu, “Talbot effect with rough reflection gratings,” Appl. Opt. 46, 3668-3673 (2007).
[CrossRef] [PubMed]

J. M. Rico-Garcia and L. M. Sanchez-Brea, “Binary gratings with random heights,” Appl. Opt. 48, 3062-3069 (2009).
[CrossRef] [PubMed]

F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 451102-1110 (2006).
[CrossRef] [PubMed]

IEEE J. Quantum Electron.

T. Suhara and H. Nishihara, “Integrated optics components and devices using periodic structures,” IEEE J. Quantum Electron. 22, 845-867 (1986).
[CrossRef]

J. Lightwave Technol.

S. Ura, T. Suhara, H. Nishihara, and J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913-918 (1986).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

L. M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, “Talbot effect in metallic gratings under Gaussian illumination,” Opt. Commun. 278, 23-27 (2007).
[CrossRef]

Opt. Express

Opt. Laser Technol.

Y. Li, D. Chen, and C. Yang, “Sub-microns period grating couplers fabricated by silicon mold,” Opt. Laser Technol. 33, 623-626 (2001).
[CrossRef]

Opt. Lett.

Philos. Mag.

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

Prog. Opt.

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

Other

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, 1987).

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

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

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Figures (7)

Fig. 1
Fig. 1

Sawtooth grating profile: perfect grating (solid line) and overdeveloped grating (dashed line).

Fig. 2
Fig. 2

Near-field diffraction pattern for an overdeveloped grating with β = 1 / 2 , h = λ / ( n ˜ 1 ) , and p = 100 μm : (a) analytical n , n = 20 , , 20 and (b) numerical.

Fig. 3
Fig. 3

Efficiency of the orders depicted in the figure in terms of β (overdevelopment error), p = 100 μm : (a) analytical and (b) differences between analytical and numerical results.

Fig. 4
Fig. 4

(a) Example of profiles defined using Eq. (12) and (b) approximations by means of straight segments, D = 10 .

Fig. 5
Fig. 5

Near-field diffraction pattern for two gratings with different curvatures along their profiles, n , n = 20 , , 20 , h = λ / ( n ˜ 1 ) , λ = 633 nm , n ˜ = 1.5 , and p = 100 μm : (a)  α = 1 / 2 , D = 100 (analytical), (b)  α = 2 , D = 100 (analytical), (c)  α = 1 / 2 (numerical), and (d)  α = 2 (numerical).

Fig. 6
Fig. 6

Efficiency of the orders depicted in the figure in terms of α (curved profile error), h = λ / ( n ˜ 1 ) , λ = 633 nm , n ˜ = 1.5 , and p = 100 μm : (a)  D = 100 (analytical) and (b) differences between analytical and numerical results.

Fig. 7
Fig. 7

(a) Analytical first-order efficiency for a sawtooth grating with roughness in terms of the standard deviation, σ, L = 10 mm , p = 100 μm , T 0 = 1 μm , h = λ / ( n ˜ 1 ) , λ = 633 nm , and n ˜ = 1.5 and (b) differences between analytical and numerical results.

Equations (24)

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t ( ξ ) = n = a n exp ( i q n ξ ) ,
U NF ( x , z ) = 1 + i 2 λ z U 0 exp ( i k z ) t ( ξ ) exp [ i k 2 z ( x ξ ) 2 ] d ξ ,
U NF ( x ^ , z ^ ) = i U 0 exp ( 4 i p 2 π z ^ λ 2 ) n a n exp ( 2 i π n x ^ ) exp ( 2 i π n 2 z ^ ) ,
I NF ( x ^ , z ^ ) = I 0 n , n a n a n * exp [ 2 i π x ^ ( n n ) ] exp [ 2 i π z ^ ( n 2 n 2 ) ] ,
U FF ( x , z ) = 1 i λ z U 0 exp ( i k z ) L / 2 L / 2 t ( ξ ) exp ( i k x z ξ ) d ξ .
U FF ( θ ) = U 0 ( 1 ) 3 / 4 p λ / z π exp [ i π z λ ( 2 + sin 2 θ ) ] n a n sin [ L π ( n λ p sin θ ) / p λ ] ( n λ p sin θ ) ,
I FF ( θ ) = I 0 p 2 λ π 2 z n , n a n a n * sin [ L π ( n λ p sin θ ) / p λ ] ( n λ p sin θ ) sin [ L π ( n λ p sin θ ) / p λ ] ( n λ p sin θ ) .
a n = 1 p 0 p exp [ i k ( n ˜ 1 ) y ( ξ ) ] exp ( 2 i π n ξ p ) d ξ ,
a n = exp { i [ h k ( n ˜ 1 ) 2 n π ] } sinc [ h k ( n ˜ 1 ) 2 n π ] .
y ( ξ ) = { 0 0 < ξ / p < β h p ( ξ β p ) β < ξ / p < 1 .
a 0 = β 2 e i 2 h k ( n ˜ 1 ) ( β 1 ) h k ( n ˜ 1 ) sin [ h k ( n ˜ 1 ) ( β 1 ) 2 ] ; a n = e i n π β sin ( n π β ) n π + 2 e i 2 [ h k ( n ˜ 1 ) ( β 1 ) + 2 n π ( β + 1 ) ] h k ( n ˜ 1 ) + 2 n π sin { ( β 1 ) 2 [ h k ( n ˜ 1 ) 2 n π ] } .
y ( ξ ) = h ( ξ p ) α .
y m ( ξ m ) = h ( m 1 D ) α + h p D α 1 [ m α ( m 1 ) α ] [ ξ m ( m 1 ) p D ] ,
a n = 1 p m = 1 D ( m 1 ) p / D m p / D exp [ i k ( n ˜ 1 ) y m ( ξ m ) ] exp ( 2 i π n ξ m p ) d ξ m ,
a n = m = 1 D 1 D e i { ( 1 2 m ) π n D + h k 2 ( n ˜ 1 ) [ ( m 1 D ) α + ( m D ) α ] } sinc { h k ( n ˜ 1 ) 2 [ ( m 1 D ) α ( m D ) α ] + π n D }
t T ( ξ ) = t ( ξ ) t R ( ξ ) ,
I NF ( x , z ) = 1 λ z I 0 n , n a n a n * M ( ξ , ξ ) e i q ( n ξ n ξ ) e i k 2 z [ ( x ξ ) 2 ( x ξ ) 2 ] d ξ d ξ ,
I FF ( θ ) = 1 λ z I 0 n , n a n a n * L / 2 L / 2 L / 2 L / 2 M ( ξ , ξ ) e i q ( n ξ n ξ ) e i k sin θ ( ξ ξ ) d ξ d ξ .
w ( z 1 , z 2 ) = 1 2 π σ 1 C 2 exp [ z 1 2 2 C z 1 z 2 + z 2 2 2 σ 2 ( 1 C 2 ) ] ,
M ( ξ , ξ ) = exp { g [ 1 C ( τ ) ] } = e g m = 0 g m m ! exp ( m τ 2 T 0 2 ) .
I NF ( x ^ , z ^ ) = I 0 e g n , n a n a n * e 2 i π x ^ ( n n ) e 2 i π z ^ ( n 2 n 2 ) m = 0 g m m ! e 4 m [ ( n n ) z ^ T 0 / p ] 2 ,
I FF ( θ ) = I 0 L λ z e g ( L + T 0 m = 1 g m m ! m { π m Erf ( m T 0 / L ) + T 0 L [ e m ( T 0 / L ) 2 1 ] } ) δ ( θ θ 1 ) ,
I FF ( θ ) L I 0 λ z e g ( L + T 0 π m = 1 g m m ! m ) δ ( θ θ 1 ) .
I FF ( θ ) I 0 e g δ ( θ θ 1 ) .

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