Abstract

A normalized modal analysis of binary gratings under normal TE incidence involving the most condensed set of optogeometrical parameters gives a complete solution to the problem of canceling the 0th transmitted order in phase masks of a low-to-high refractive index ratio down to 0.5 with a large tolerance on the corrugation duty cycle or a large spectral bandwidth. The solution is presented in the form of single normalized 3D charts which shed light on the fulfillment of the 0th-order cancellation condition: balanced excitation and π-phase difference between two grating modes. Examples of tolerant gratings are given.

© 2010 Optical Society of America

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References

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  1. K. Hill, B. Malo, F. Bilodeau, D. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62, 1035–1037 (1993).
    [CrossRef]
  2. E. Gamet, A. V. Tishchenko, and O. Parriaux, “Cancellation of the zeroth order in a phase mask by mode interplay in a high index contrast binary grating,” Appl. Opt. 46, 6719–6726 (2007).
    [CrossRef] [PubMed]
  3. MC-Grating, MC Grating Software Development Company (www.mcgrating.com).
  4. J. Zheng, C. ZhouB. Wang, and J. Feng, “Beam splitting of low-contrast binary gratings under second Bragg angle incidence,” J. Opt. Soc. Am. A 25, 1075–1083 (2008).
    [CrossRef]
  5. J. Feng, C. Zhou, J. Zheng, and B. Wang, “Modal analysis of deep-etched low-contrast two-port beam splitter grating,” Opt. Commun. 281, 5298–5301 (2008).
    [CrossRef]
  6. J. Feng, C. Zhou, J. Zheng, W. Jia, H. Cao, and P. Lv, “Three-port beam splitter of a binary fused-silica grating,” Appl. Opt. 47, 6638–6643 (2008).
    [CrossRef] [PubMed]
  7. J. Feng, C. Zhou, J. Zheng, H. Cao, and P. Lv, “Design and fabrication of a polarization-independent two-port beam splitter,” Appl. Opt. 48, 5636–5641 (2009).
    [CrossRef] [PubMed]
  8. E. Gamet, F. Pigeon, and O. Parriaux, “Duty cycle tolerant binary gratings for fabricable short period phase masks,” J. Eur. Opt. Soc. Rapid Publ. 4, 09047 (2009).
    [CrossRef]
  9. S. Jeon, V. Malyarchuk, J. A. Rogers, and G. P. Wiederrecht, “Fabricating three-dimensional nanostructures using two photon lithography in a single exposure step,” Opt. Express 14, 2300–2308 (2006).
    [CrossRef] [PubMed]
  10. L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28, 413–428 (1981).
    [CrossRef]
  11. A. Tishchenko, “A phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. 37, 309–330 (2005).
    [CrossRef]
  12. P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
    [CrossRef]
  13. M. Foresti, L. Menez, and A. Tishchenko, “Modal method in deep metal-dielectric gratings: the decisive role of hidden modes,” J. Opt. Soc. Am. A 23, 2501–2509 (2006).
    [CrossRef]
  14. M. Moharam and T. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  15. T. Clausnitzer, T. Kämpfe, E. B. Kley, A. Tünnermann, U. Peschel, A. Tishchenko, and O. Parriaux, “An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings,” Opt. Express 13, 10448–10456 (2005).
    [CrossRef] [PubMed]

2009 (2)

J. Feng, C. Zhou, J. Zheng, H. Cao, and P. Lv, “Design and fabrication of a polarization-independent two-port beam splitter,” Appl. Opt. 48, 5636–5641 (2009).
[CrossRef] [PubMed]

E. Gamet, F. Pigeon, and O. Parriaux, “Duty cycle tolerant binary gratings for fabricable short period phase masks,” J. Eur. Opt. Soc. Rapid Publ. 4, 09047 (2009).
[CrossRef]

2008 (3)

2007 (1)

2006 (2)

2005 (2)

1993 (1)

K. Hill, B. Malo, F. Bilodeau, D. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62, 1035–1037 (1993).
[CrossRef]

1982 (1)

P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

1981 (2)

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

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28, 413–428 (1981).
[CrossRef]

Adams, J.

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28, 413–428 (1981).
[CrossRef]

Albert, J.

K. Hill, B. Malo, F. Bilodeau, D. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62, 1035–1037 (1993).
[CrossRef]

Andrewartha, J.

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28, 413–428 (1981).
[CrossRef]

Bilodeau, F.

K. Hill, B. Malo, F. Bilodeau, D. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62, 1035–1037 (1993).
[CrossRef]

Botten, L.

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28, 413–428 (1981).
[CrossRef]

Cao, H.

Clausnitzer, T.

Craig, M.

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28, 413–428 (1981).
[CrossRef]

Feng, J.

Foresti, M.

Gamet, E.

E. Gamet, F. Pigeon, and O. Parriaux, “Duty cycle tolerant binary gratings for fabricable short period phase masks,” J. Eur. Opt. Soc. Rapid Publ. 4, 09047 (2009).
[CrossRef]

E. Gamet, A. V. Tishchenko, and O. Parriaux, “Cancellation of the zeroth order in a phase mask by mode interplay in a high index contrast binary grating,” Appl. Opt. 46, 6719–6726 (2007).
[CrossRef] [PubMed]

Gaylord, T.

Hill, K.

K. Hill, B. Malo, F. Bilodeau, D. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62, 1035–1037 (1993).
[CrossRef]

Jeon, S.

Jia, W.

Johnson, D.

K. Hill, B. Malo, F. Bilodeau, D. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62, 1035–1037 (1993).
[CrossRef]

Kämpfe, T.

Kley, E. B.

Lv, P.

Malo, B.

K. Hill, B. Malo, F. Bilodeau, D. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62, 1035–1037 (1993).
[CrossRef]

Malyarchuk, V.

McPhedran, R.

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28, 413–428 (1981).
[CrossRef]

Menez, L.

Moharam, M.

Parriaux, O.

Peschel, U.

Pigeon, F.

E. Gamet, F. Pigeon, and O. Parriaux, “Duty cycle tolerant binary gratings for fabricable short period phase masks,” J. Eur. Opt. Soc. Rapid Publ. 4, 09047 (2009).
[CrossRef]

Rogers, J. A.

Sanda, P.

P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Sheng, P.

P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Stepleman, R.

P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Tishchenko, A.

Tishchenko, A. V.

Tünnermann, A.

Wang, B.

J. Feng, C. Zhou, J. Zheng, and B. Wang, “Modal analysis of deep-etched low-contrast two-port beam splitter grating,” Opt. Commun. 281, 5298–5301 (2008).
[CrossRef]

J. Zheng, C. ZhouB. Wang, and J. Feng, “Beam splitting of low-contrast binary gratings under second Bragg angle incidence,” J. Opt. Soc. Am. A 25, 1075–1083 (2008).
[CrossRef]

Wiederrecht, G. P.

Zheng, J.

Zhou, C.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

K. Hill, B. Malo, F. Bilodeau, D. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62, 1035–1037 (1993).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ. (1)

E. Gamet, F. Pigeon, and O. Parriaux, “Duty cycle tolerant binary gratings for fabricable short period phase masks,” J. Eur. Opt. Soc. Rapid Publ. 4, 09047 (2009).
[CrossRef]

J. Mod. Opt. (1)

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28, 413–428 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

J. Feng, C. Zhou, J. Zheng, and B. Wang, “Modal analysis of deep-etched low-contrast two-port beam splitter grating,” Opt. Commun. 281, 5298–5301 (2008).
[CrossRef]

Opt. Express (2)

Opt. Quantum Electron. (1)

A. Tishchenko, “A phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. 37, 309–330 (2005).
[CrossRef]

Phys. Rev. B (1)

P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Other (1)

MC-Grating, MC Grating Software Development Company (www.mcgrating.com).

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Figures (14)

Fig. 1
Fig. 1

Binary corrugation grating with ridge and groove refractive indices n r and n g , respectively; period Λ; line width r; groove width g = Λ r ; and height h under normal TE-illumination.

Fig. 2
Fig. 2

Graph of the dispersion Eq. (11) versus the square of the normalized modal effective index ν in a binary grating under normal incidence.

Fig. 3
Fig. 3

Two-beam interferometer analogy of light transmission through a binary corrugation. t ci and t is are the transmission coefficients at the cover–grating and grating–substrate interfaces, respectively, for mode i.

Fig. 4
Fig. 4

Effective indices ν 0 and ν 2 (upper plot) and transmission t 0 and t 2 (lower plot) of the first two even grating modes as a function of the duty cycle d for p = 2 and c = 0.5 . Shaded region: duty cycle tolerant domain of the phase difference and balance, respectively.

Fig. 5
Fig. 5

Three-dimensional visualization of ( d , c , p ) -triplets fulfilling separately the condition of large phase difference and large balance tolerance with regard to duty cycle variations (upper plot) and fulfilling both conditions at the same time for a weakened zero criterion, with the shading (color online) scale indicating the weighted sum of both conditions (lower plot). On the side faces of the cube shadow projections of the ( d , c , p ) -triplets are depicted.

Fig. 6
Fig. 6

Three-dimensional visualization of ( d , c , p ) -triplets fulfilling the balanced transmission ratio condition.

Fig. 7
Fig. 7

Three-dimensional visualization of ( d , c , p ) -triplets fulfilling both the tolerance condition with regard to duty cycle changes (Fig. 5) and the balance condition (Fig. 6).

Fig. 8
Fig. 8

Three-dimensional visualization of ( d , c , p ) -triplets fulfilling separately the conditions of broadband tolerance of the phase difference and the balance.

Fig. 9
Fig. 9

Three-dimensional visualization of ( d , c , p ) -triplets showing broadband behavior (Fig. 8) and fulfilling the balance condition (Fig. 6).

Fig. 10
Fig. 10

Transmission to the 0 th and 1 st orders (calculated with the RCWA) of a duty cycle-tolerant (upper plot) and a wavelength-tolerant (lower plot) grating, using parameters from the line-shaped subsets in Fig. 7 and Fig. 9.

Fig. 11
Fig. 11

Comparison of the transmission (calculated with the RCWA) of the duty cycle-tolerant grating from Fig. 10 (solid curve) to gratings with different periods (dotted and dashed curves).

Fig. 12
Fig. 12

Comparison of the transmission (calculated with the RCWA) of the wavelength-tolerant grating from Fig. 10 (solid curve) to gratings with different duty cycle (dotted and dashed curves).

Fig. 13
Fig. 13

Transmission to the 0 th order (calculated with the RCWA) for optimized duty cycle-tolerant (upper plot) and wavelength-tolerant (lower plot) gratings from the line-shaped subset of Fig. 7 and Fig. 9, respectively, for different values of the index ratio.

Fig. 14
Fig. 14

Influence of the material dispersion on the transmission (calculated with the RCWA) of wavelength-tolerant gratings for Si O 2 and Si 3 N 4 corrugated layer materials.

Equations (19)

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E g ( x , z ) = q = 0 E q ( x ) ( a q + e i k z , q z + a q e i k z , q z ) , 0 < z < h .
E q ( x ) = { a q r + e i k x , q , r ( x r 2 ) + a q r e i k x , q , r ( x r 2 ) , m Λ x < m Λ + r a q g + e i k x , q , g ( x r 2 Λ 2 ) + a q g e i k x , q , g ( x r 2 Λ 2 ) , m Λ + r x < m ( Λ + 1 ) } ,
cos ( k x , q , r r ) cos ( k x , q , g g ) 1 2 ( k x , q , r k x , q , g + k x , q , g k x , q , r ) sin ( k x , q , r r ) sin ( k x , q , g g ) = cos ( k x , in Λ ) ,
duty cycle : d = r Λ ,
refractive index ratio : c = n g n r ,
relative period in ridges : p = Λ ( λ n r ) .
ν = n n r .
E q even ( u ) = { a q r cos [ 2 π p 1 ν q 2 ( u 1 2 d ) ] , 0 < u < d a q g cos [ 2 π p c 2 ν q 2 ( u 1 2 d 1 2 ) ] , d < u < 1 } .
0 1 E q ( u ) E p * ( u ) = δ p q ( δ p q : kronecker-delta ) ,
a q g = 2 ( cos 2 ( w 2 , q 2 ) cos 2 ( w 1 , q 2 ) ( d sin ( w 1 , q ) w 1 , q + d ) + ( 1 d ) sin ( w 2 , q ) w 2 , q + 1 d ) , a q r = a q g cos ( w 2 , q 2 ) cos ( w 1 , q 2 ) ,
F ( ν q 2 ) = cos ( w 1 , q ) cos ( w 2 , q ) 1 2 ( 1 ν q 2 c 2 ν q 2 + c 2 ν q 2 1 ν q 2 ) sin ( w 1 , q ) sin ( w 2 , q ) 1 = 0.
ϕ dif = 2 π λ h ( n 0 n 2 ) + arg ( t c 0 ) + arg ( t 0 s ) arg ( t c 2 ) arg ( t 2 s ) .
ϕ dif = 2 π Λ p h ν dif ,
α q , p = 1 Λ 0 Λ E q ( x ) E p * ( x ) d x .
α q = 0 1 E q ( u ) d u .
α q = 2 [ a q r d w 1 , q sin ( w 1 , q 2 ) + a q g ( 1 d ) w 2 , q sin ( w 2 , q 2 ) ] ,
β 1 , 2 = 2 n 2 n 1 + n 2 .
β c , q = 2 c ( ν q + c ) , β q , s = 2 ν q ( ν q + 1 ) .
b = t 0 t 2 = t c 0 t 0 s t c 2 t 2 s = β c , 0 β 0 , s α 0 2 β c , 2 β 2 , s α 2 2 .

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