Abstract

We analyze the far-field intensity distribution of binary phase gratings whose strips present certain randomness in their height. A statistical analysis based on the mutual coherence function is done in the plane just after the grating. Then, the mutual coherence function is propagated to the far field and the intensity distribution is obtained. Generally, the intensity of the diffraction orders decreases in comparison to that of the ideal perfect grating. Several important limit cases, such as low- and high-randomness perturbed gratings, are analyzed. In the high-randomness limit, the phase grating is equivalent to an amplitude grating plus a “halo.” Although these structures are not purely periodic, they behave approximately as a diffraction grating.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
    [CrossRef]
  4. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  5. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584-586 (1999).
    [CrossRef]
  6. C. G. Someda, “Far field of polarization gratings,” Opt. Lett. 24, 1657-1659 (1999).
    [CrossRef]
  7. G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
    [CrossRef]
  8. 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]
  9. L. M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, “Far field of gratings with rough strips,” J. Opt. Soc. Am. A 25, 828-833 (2008).
    [CrossRef]
  10. Y. Lu, C. Zhou, and H. Luo, “Near field diffraction of irregular phase gratings with multiple phase-shifts,” Opt. Express 13, 6111-6116 (2005).
    [CrossRef]
  11. Y. Sheng and S. Li, “Talbot effect of a grating with different kind of flaws,” J. Opt. Soc. Am. A 22, 2662-2667 (2005).
    [CrossRef]
  12. J. Song and S. He, “Effects of rounded corners on the performance of an echelle diffraction grating demultiplexer,” J. Opt. A Pure Appl. Opt. 6, 769-773 (2004).
    [CrossRef]
  13. H. Wen, D. Pang, and Z. Qiang, “The impact of phase and amplitude errors on an etched diffraction grating demultiplexer,” Opt. Commun. 236, 1-6 (2004).
    [CrossRef]
  14. T. R. Michel, “Resonant light scattering from weakly rough random surfaces and imperfect gratings,” J. Opt. Soc. Am. A 11, 1874-1885 (1994).
    [CrossRef]
  15. M. V. Glazov and S. N. Rashkeev, “Light scattering from rough surfaces with superimposed periodic structures,” Appl. Phys. B 66, 217-223 (1998).
    [CrossRef]
  16. V. A. Doroshenko, “Singular integral equations in the problem of wave diffraction by a grating of imperfect flat irregular strips,” Telecommunications and radio engineering Part 1, Telecommunications 57, 65-72 (2002).
  17. P. P. Naulleau and G. M. Gallatin, “Line-edge roughness transfer function and its application to determining mask effects in EUV resist characterization,” Appl. Opt. 42, 3390-3397 (2003).
    [CrossRef] [PubMed]
  18. 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]
  19. F. J. Torcal-Milla, L. M. Sanchez-Brea, and E. Bernabeu, “Diffraction of gratings with rough edges,” Opt. Express 16, 19757-19769 (2008).
    [CrossRef] [PubMed]
  20. J.Turunen and F.Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, 1997).
  21. B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
    [CrossRef]
  22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  23. J. W. Goodman, Statistical Optics (Wiley, 1985).
  24. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

2008 (3)

2007 (1)

2005 (2)

2004 (2)

J. Song and S. He, “Effects of rounded corners on the performance of an echelle diffraction grating demultiplexer,” J. Opt. A Pure Appl. Opt. 6, 769-773 (2004).
[CrossRef]

H. Wen, D. Pang, and Z. Qiang, “The impact of phase and amplitude errors on an etched diffraction grating demultiplexer,” Opt. Commun. 236, 1-6 (2004).
[CrossRef]

2003 (1)

2002 (1)

V. A. Doroshenko, “Singular integral equations in the problem of wave diffraction by a grating of imperfect flat irregular strips,” Telecommunications and radio engineering Part 1, Telecommunications 57, 65-72 (2002).

2001 (1)

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

1999 (2)

1998 (1)

M. V. Glazov and S. N. Rashkeev, “Light scattering from rough surfaces with superimposed periodic structures,” Appl. Phys. B 66, 217-223 (1998).
[CrossRef]

1994 (1)

Beckmann, P.

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

Bernabeu, E.

Borghi, R.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

Born, M.

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

Doroshenko, V. A.

V. A. Doroshenko, “Singular integral equations in the problem of wave diffraction by a grating of imperfect flat irregular strips,” Telecommunications and radio engineering Part 1, Telecommunications 57, 65-72 (2002).

Gallatin, G. M.

Glazov, M. V.

M. V. Glazov and S. N. Rashkeev, “Light scattering from rough surfaces with superimposed periodic structures,” Appl. Phys. B 66, 217-223 (1998).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

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

Gori, F.

He, S.

J. Song and S. He, “Effects of rounded corners on the performance of an echelle diffraction grating demultiplexer,” J. Opt. A Pure Appl. Opt. 6, 769-773 (2004).
[CrossRef]

Li, S.

Loewen, E. G.

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

Lu, Y.

Luo, H.

Michel, T. R.

Mondello, A.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

Naulleau, P. P.

Palmer, C.

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

Pang, D.

H. Wen, D. Pang, and Z. Qiang, “The impact of phase and amplitude errors on an etched diffraction grating demultiplexer,” Opt. Commun. 236, 1-6 (2004).
[CrossRef]

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

Piquero, G.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

Popov, E.

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

Qiang, Z.

H. Wen, D. Pang, and Z. Qiang, “The impact of phase and amplitude errors on an etched diffraction grating demultiplexer,” Opt. Commun. 236, 1-6 (2004).
[CrossRef]

Rashkeev, S. N.

M. V. Glazov and S. N. Rashkeev, “Light scattering from rough surfaces with superimposed periodic structures,” Appl. Phys. B 66, 217-223 (1998).
[CrossRef]

Saleh, B. E.

B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Sanchez-Brea, L. M.

Santarsiero, M.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

Sheng, Y.

Someda, C. G.

Song, J.

J. Song and S. He, “Effects of rounded corners on the performance of an echelle diffraction grating demultiplexer,” J. Opt. A Pure Appl. Opt. 6, 769-773 (2004).
[CrossRef]

Spizzichino, A.

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

Teich, M. C.

B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Torcal-Milla, F. J.

Wen, H.

H. Wen, D. Pang, and Z. Qiang, “The impact of phase and amplitude errors on an etched diffraction grating demultiplexer,” Opt. Commun. 236, 1-6 (2004).
[CrossRef]

Wolf, E.

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

Zhou, C.

Appl. Opt. (2)

Appl. Phys. B (1)

M. V. Glazov and S. N. Rashkeev, “Light scattering from rough surfaces with superimposed periodic structures,” Appl. Phys. B 66, 217-223 (1998).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

J. Song and S. He, “Effects of rounded corners on the performance of an echelle diffraction grating demultiplexer,” J. Opt. A Pure Appl. Opt. 6, 769-773 (2004).
[CrossRef]

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

Opt. Commun. (2)

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

H. Wen, D. Pang, and Z. Qiang, “The impact of phase and amplitude errors on an etched diffraction grating demultiplexer,” Opt. Commun. 236, 1-6 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Telecommunications and radio engineering Part 1, Telecommunications (1)

V. A. Doroshenko, “Singular integral equations in the problem of wave diffraction by a grating of imperfect flat irregular strips,” Telecommunications and radio engineering Part 1, Telecommunications 57, 65-72 (2002).

Other (9)

J.Turunen and F.Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, 1997).

B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

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

J. W. Goodman, Statistical Optics (Wiley, 1985).

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

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

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

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

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

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 (8)

Fig. 1
Fig. 1

Scheme of the phase grating with random heights. The dashed lines represent the grating without randomness and the solid lines are realizations of the grating where odd strips present certain randomness. The period of the unperturbed grating is p and the refraction index is n.

Fig. 2
Fig. 2

Γ matrix corresponding to high-randomness limit σ λ . For this example, the unperturbed grating has N = 31 steps, the refraction index is n = 1.5 , the wavelength is λ = 680 nm , and σ = 5 λ . The x m and x n axes are normalized to p / 2 . For all the figures and simulations the following parameters have been used: the wavelength of the incident beam is λ = 0.68 μm , the period of the grating is p = 25 μm , the refraction index is n = 1.5 , and the number of periods is N = 31 .

Fig. 3
Fig. 3

(a) Average intensity for high randomness σ = 5 λ when Δ = π / 2 . The period of the grating studied in every figure is p = 25 μm . (b) Intensity for the same grating but without randomness σ = 0 .

Fig. 4
Fig. 4

Γ corresponding to σ λ . The x m and x n axes are normalized to p / 2 .

Fig. 5
Fig. 5

Far-field pattern of (a) the perturbed grating at low randomness, σ = 0.25 λ , Δ = π versus (b) the unperturbed structure, σ = 0 , Δ = π . Although the pattern has been normalized, its maximum is higher than the maximum at high randomness.

Fig. 6
Fig. 6

Intensity of the diffraction orders 0, ± 1 , and ± 3 when the randomness of the strips σ is varied. (a)  Δ = 0 and (b)  Δ = π .

Fig. 7
Fig. 7

(a) Diffraction grating with random heights and (b) intensity distribution for this diffraction grating (realization) numerically. The phase shift of the unperturbed grating is Δ = π / 2 and the randomness parameter is σ = 5 λ .

Fig. 8
Fig. 8

Average intensity distribution for several gratings with randomness. (a)  Δ = 0 , σ = 0.25 λ , (b)  Δ = 0 , σ = 5 λ , (c)  Δ = π , σ = 0.25 λ , and (d)  Δ = π , σ = 5 λ . The number of experiments of the averaging process is N = 100 . There is a coincidence of the peaks corresponding to the diffraction orders between the theoretical results and the simulation.

Equations (20)

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

h ( x ) = h u ( x ) + Δ h ( x ) ,
Δ T step = e i ϕ ,
T 0 ( x ) = j a j e i q j x ,
Δ T ( x ) = m = 0 N s m Π ( x x m p / 2 ) ,
s m = { 1 if     m   is odd e i ϕ if     m   is even .
J ( x , x ) = | A 0 | 2 T ( x ) T * ( x ) ,
J ( x , x ) = | A 0 | 2 T 0 ( x ) T * 0 ( x ) m = 0 N n = 0 N Γ n m Π ( x x m p / 2 ) Π ( x x n p / 2 ) ,
Δ T step = e i ϕ = w ( z ) e i ϕ ( z ) d z .
Γ m n ( α ) = { 1 m = n 1 m n , ( m , n )   odd α 2 m n , ( m , n )   even α m   even , n   odd   or   m   odd , n   even ,
J ( x 2 , x 2 ) = e i k 2 z ( x 2 2 x 2 2 ) J ( x 1 , x 1 ) e i k z ( x 2 x 1 x 2 x 1 ) d x 1 d x 1 .
J ( θ 2 , θ 2 ) ¯ = m = 0 N n = 0 N Γ m n { T * 0 ( x 1 ) Π m ( x 1 ) e i k z x 2 x 1 d x 1 T 0 ( x 1 ) Π n ( x 1 ) e i k z x 2 x 1 d x 1 } ,
J ( θ 2 , θ 2 ) ¯ = j , w a j a * w M j w ( x 2 , x 2 ) sinc [ π 2 ( p θ 2 / λ j ) ] sinc [ π 2 ( p θ 2 / λ w ) ] ,
M j w ( θ 2 , θ 2 ) = n , m Γ n m e i [ ϕ m j ( θ 2 ) ϕ n w ( θ 2 ) ] .
I ( θ 2 ) ¯ = J ( θ 2 , θ 2 ) ¯ = j , w a j a * w M j w ( θ 2 ) sinc [ π 2 ( p θ 2 λ j ) ] sinc [ π 2 ( p θ 2 λ w ) ] .
I ( θ 2 ) ¯ j | a j | 2 M j j ( θ 2 ) sinc 2 [ π 2 ( p θ 2 λ j ) ] .
M j j ( θ 2 ) = n , m e i ( m n ) ( q j k θ 2 ) p / 2 = sin 2 [ π N 2 ( p θ 2 λ j ) ] sin 2 [ π 2 ( p θ 2 λ j ) ] ,
I ( θ 2 ) ¯ = j | a j | 2 sinc 2 [ π N 2 ( p θ λ j ) ] .
Γ ( α 1 ) Λ + C O O ( 1 ) .
Λ m n = { 1 n = m ( n , m ) even 0 otherwise , C O O m n ( 1 ) = { 1 ( n , m ) odd 0 otherwise .
Γ ( g 1 ) C ( 1 ) + g P ,

Metrics