Abstract

Conventional diffraction gratings composed of a series of equally spaced slits suffer from wavelength overlapping caused by high-order diffraction. Here modulated groove position gratings (MGPGs) are proposed to significantly suppress the high diffraction orders. Both numerical solution and experimental results demonstrate the effectiveness of MGPGs. The suppression ratio is determined by the number of grating grooves used. By using an MGPG with 10,000 grooves, a suppression ratio as high as 18,000 can be obtained. In addition, the minimum linewidth is kept to 1/4 of the grating period, which enables grating realization with high line density employing today’s nanofabrication technology. Our results should be of great interest in both diffraction grating theory and applications, particularly due to MGPGs’ applicability in a wide wavelength range and realizability with high line density.

© 2011 Optical Society of America

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References

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2009

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H. Yan, J. Maser, A. T. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, and H. C. Kang, Phys. Rev. B 76, 115438(2007).
[CrossRef]

E. Gamet, A. V. Tishchenko, and O. Parriaux, Appl. Opt. 46, 6719 (2007).
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L. Cao, E. Förster, A. Fuhrmann, C. Wang, L. Kuang, S. Liu, and Y. Ding, Appl. Phys. Lett. 90, 053501 (2007).
[CrossRef]

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V. Daneu, A. Sanchez, T. Y. Fan, H. K. Choi, G. W. Turner, and C. C. Cook, Opt. Lett. 25, 405 (2000).
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Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
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E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, J. Vac. Sci. Technol. B 18, 2970 (2000).
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[CrossRef]

K. P. Beuermann, R. Lenzen, and H. Bräuninger, Appl. Opt. 16, 1425 (1977).
[CrossRef] [PubMed]

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Anderson, E. H.

W. Chao, J. Kim, S. Rekawa, P. Fischer, and E. H. Anderson, Opt. Express 17, 17669 (2009).
[CrossRef] [PubMed]

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, J. Vac. Sci. Technol. B 18, 2970 (2000).
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E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, J. Vac. Sci. Technol. B 18, 2970 (2000).
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M. Born and E. Wolf, Principles of Optics, 7th ed.(expanded) (Cambridge U. Press, 1997), pp. 446–458.

Bräuninger, H.

Brown, B. R.

Cambril, E.

Cao, L.

L. Kuang, C. Wang, Z. Wang, L. Cao, X. Zhu, C. Xie, S. Liu, and Y. Ding, Rev. Sci. Instrum. 81, 073508 (2010).
[CrossRef] [PubMed]

L. Cao, E. Förster, A. Fuhrmann, C. Wang, L. Kuang, S. Liu, and Y. Ding, Appl. Phys. Lett. 90, 053501 (2007).
[CrossRef]

Ceglio, N. M.

Chao, C. H.

Chao, W.

W. Chao, J. Kim, S. Rekawa, P. Fischer, and E. H. Anderson, Opt. Express 17, 17669 (2009).
[CrossRef] [PubMed]

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, J. Vac. Sci. Technol. B 18, 2970 (2000).
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Collin, S.

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I. Shoshan, N. N. Danon, and U. P. Oppenheim, J. Appl. Phys. 48, 4495 (1977).
[CrossRef]

Denbeaux, G.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, J. Vac. Sci. Technol. B 18, 2970 (2000).
[CrossRef]

Ding, Y.

L. Kuang, C. Wang, Z. Wang, L. Cao, X. Zhu, C. Xie, S. Liu, and Y. Ding, Rev. Sci. Instrum. 81, 073508 (2010).
[CrossRef] [PubMed]

L. Cao, E. Förster, A. Fuhrmann, C. Wang, L. Kuang, S. Liu, and Y. Ding, Appl. Phys. Lett. 90, 053501 (2007).
[CrossRef]

Fan, T. Y.

Fischer, P.

Förster, E.

L. Cao, E. Förster, A. Fuhrmann, C. Wang, L. Kuang, S. Liu, and Y. Ding, Appl. Phys. Lett. 90, 053501 (2007).
[CrossRef]

Fuhrmann, A.

L. Cao, E. Förster, A. Fuhrmann, C. Wang, L. Kuang, S. Liu, and Y. Ding, Appl. Phys. Lett. 90, 053501 (2007).
[CrossRef]

Fujisawa, M.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

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Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 73–83.

Guérineau, N.

Haidar, R.

Harada, H.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Harteneck, B.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, J. Vac. Sci. Technol. B 18, 2970 (2000).
[CrossRef]

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Hong, C. Y.

Horng, H. E.

Hu, S. T.

Huang, J. C.

Huang, Y. W.

Johnson, L.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, J. Vac. Sci. Technol. B 18, 2970 (2000).
[CrossRef]

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H. Yan, J. Maser, A. T. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, and H. C. Kang, Phys. Rev. B 76, 115438(2007).
[CrossRef]

Kim, J.

Kimura, H.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Kotsugi, M.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Kuang, L.

L. Kuang, C. Wang, Z. Wang, L. Cao, X. Zhu, C. Xie, S. Liu, and Y. Ding, Rev. Sci. Instrum. 81, 073508 (2010).
[CrossRef] [PubMed]

L. Cao, E. Förster, A. Fuhrmann, C. Wang, L. Kuang, S. Liu, and Y. Ding, Appl. Phys. Lett. 90, 053501 (2007).
[CrossRef]

Külhne, M.

Lenzen, R.

Lin, C. F.

Liu, S.

L. Kuang, C. Wang, Z. Wang, L. Cao, X. Zhu, C. Xie, S. Liu, and Y. Ding, Rev. Sci. Instrum. 81, 073508 (2010).
[CrossRef] [PubMed]

L. Cao, E. Förster, A. Fuhrmann, C. Wang, L. Kuang, S. Liu, and Y. Ding, Appl. Phys. Lett. 90, 053501 (2007).
[CrossRef]

Lohmann, A. W.

Lucero, A.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, J. Vac. Sci. Technol. B 18, 2970 (2000).
[CrossRef]

Macrander, A. T.

H. Yan, J. Maser, A. T. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, and H. C. Kang, Phys. Rev. B 76, 115438(2007).
[CrossRef]

Maser, J.

H. Yan, J. Maser, A. T. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, and H. C. Kang, Phys. Rev. B 76, 115438(2007).
[CrossRef]

Matsushita, T.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Miyahara, T.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Morita, R.

Muro, T.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Müuller, P.

Nakatani, T.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Oka, K.

Olynick, D. L.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, J. Vac. Sci. Technol. B 18, 2970 (2000).
[CrossRef]

Oppenheim, U. P.

I. Shoshan, N. N. Danon, and U. P. Oppenheim, J. Appl. Phys. 48, 4495 (1977).
[CrossRef]

Parriaux, O.

Pelouard, J. L.

Primot, J.

Rekawa, S.

Saitoh, Y.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Sanchez, A.

Sanchez-Brea, L. M.

Sekiyama, A.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Shen, Q.

H. Yan, J. Maser, A. T. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, and H. C. Kang, Phys. Rev. B 76, 115438(2007).
[CrossRef]

Shoshan, I.

I. Shoshan, N. N. Danon, and U. P. Oppenheim, J. Appl. Phys. 48, 4495 (1977).
[CrossRef]

Smith, R. E.

Soda, K.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Stearns, D. G.

Stephenson, G. B.

H. Yan, J. Maser, A. T. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, and H. C. Kang, Phys. Rev. B 76, 115438(2007).
[CrossRef]

Suga, S.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Suguro, A.

Suzuki, Y.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Tishchenko, A. V.

Torcal-Milla, F. J.

Turner, G. W.

Ueda, S.

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Vawter, G. A.

Veklerov, E.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, J. Vac. Sci. Technol. B 18, 2970 (2000).
[CrossRef]

Vincent, G.

Vogt, S.

H. Yan, J. Maser, A. T. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, and H. C. Kang, Phys. Rev. B 76, 115438(2007).
[CrossRef]

Wang, C.

L. Kuang, C. Wang, Z. Wang, L. Cao, X. Zhu, C. Xie, S. Liu, and Y. Ding, Rev. Sci. Instrum. 81, 073508 (2010).
[CrossRef] [PubMed]

L. Cao, E. Förster, A. Fuhrmann, C. Wang, L. Kuang, S. Liu, and Y. Ding, Appl. Phys. Lett. 90, 053501 (2007).
[CrossRef]

Wang, Z.

L. Kuang, C. Wang, Z. Wang, L. Cao, X. Zhu, C. Xie, S. Liu, and Y. Ding, Rev. Sci. Instrum. 81, 073508 (2010).
[CrossRef] [PubMed]

Warren, M. E.

Wendt, J. R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed.(expanded) (Cambridge U. Press, 1997), pp. 446–458.

Xie, C.

L. Kuang, C. Wang, Z. Wang, L. Cao, X. Zhu, C. Xie, S. Liu, and Y. Ding, Rev. Sci. Instrum. 81, 073508 (2010).
[CrossRef] [PubMed]

Yamane, K.

Yamashita, M.

Yan, H.

H. Yan, J. Maser, A. T. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, and H. C. Kang, Phys. Rev. B 76, 115438(2007).
[CrossRef]

Yang, H. C.

Yang, S. Y.

Zhang, Z.

Zhu, X.

L. Kuang, C. Wang, Z. Wang, L. Cao, X. Zhu, C. Xie, S. Liu, and Y. Ding, Rev. Sci. Instrum. 81, 073508 (2010).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

L. Cao, E. Förster, A. Fuhrmann, C. Wang, L. Kuang, S. Liu, and Y. Ding, Appl. Phys. Lett. 90, 053501 (2007).
[CrossRef]

J. Appl. Phys.

I. Shoshan, N. N. Danon, and U. P. Oppenheim, J. Appl. Phys. 48, 4495 (1977).
[CrossRef]

J. Opt. Soc. Am. B

J. Vac. Sci. Technol. B

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, J. Vac. Sci. Technol. B 18, 2970 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

H. Yan, J. Maser, A. T. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, and H. C. Kang, Phys. Rev. B 76, 115438(2007).
[CrossRef]

Rev. Sci. Instrum.

L. Kuang, C. Wang, Z. Wang, L. Cao, X. Zhu, C. Xie, S. Liu, and Y. Ding, Rev. Sci. Instrum. 81, 073508 (2010).
[CrossRef] [PubMed]

Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, K. Soda, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. 71, 3254 (2000).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 7th ed.(expanded) (Cambridge U. Press, 1997), pp. 446–458.

Here ρ(s) is the probability density function of the continuous random variable s, and the normalizing condition is satisfied that ∫ρ(s)ds=1.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 73–83.

D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge University, 1999), p. 55.

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Figures (3)

Fig. 1
Fig. 1

(a) Transmittance function of MGPGs. (b) Schematic of the diffraction from an MGPG. (c) Simulated intensity distribution along the ξ axis.

Fig. 2
Fig. 2

(a) Experimental setup for optical demonstration of MGPGs. (b) Microscopy image of the fabricated MGPG with 80 μm period and 20 μm minimum linewidth. (c)  Microscopy image of a conventional grating with the same period. (d) Recorded diffraction intensity profile of the MGPG. (e) Recorded diffraction intensity profile of the conventional grating. In the experiment, the utilized size of the grating is about 5 cm , and the period number is about 625.

Fig. 3
Fig. 3

(a) Average transmittance function normalized to 1 of an MGPG with 1000 periods (blue solid curve) and the ideal sinusoidal function (red dashed curve). (b) Log scale plot of the L2-norm of t A ( x ) t I ( x ) and the suppression ratio with different N.

Equations (4)

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

ρ ( s ) = π D cos ( 2 π s D ) rect ( 2 s D 1 ) ,
t ( x ) = n rect ( x x n D / 2 ) .
U ( ξ ) t ( x ) exp ( i 2 π ξ λ z x ) d x = D 2 sinc ( D ξ 2 λ z ) n exp ( i 2 π ξ λ z x n ) ,
U ( ξ ) D 2 sinc ( D ξ 2 λ z ) n exp { i 2 π m D [ s n + ( n 1 ) D ] } = D 2 sinc ( m 2 ) n exp ( i 2 π m s n D ) = D 2 sinc ( m 2 ) 1 D 0 D ρ ( s ) exp ( i 2 π m s D ) d s = π 2 D sinc ( m 2 ) D 4 3 D 4 cos ( 2 π s D ) exp ( i 2 π m s D ) d s = 1 2 δ m , 0 1 4 δ m , 1 1 4 δ m , 1 ,

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