Abstract

To accurately and easily design the diffraction characteristics of a rectangular transmission grating under the illumination of Littrow mounting, the validity and limitation of the simplified modal method is evaluated by a comparison of diffraction efficiencies predicted by the modal approach to exact results calculated with rigorous coupled-wave analysis. The influence of the grating normalized period, the normalized groove depth, and the fill factor on the accuracy of the modal method is quantitatively determined. More importantly, the reflection effect of two propagating grating modes with the optical thin-film model and the nonsymmetrical Fabry–Perot model is proposed and applied in the modal method to improve the accuracy of the calculated diffraction efficiencies. Generally, it is found that the thin-film model of reflection loss is valid at the smaller normalized period, but the Fabry–Perot model can exactly calculate the reflection loss of grating modes at the larger normalized period. Based on the fact that the validity of the modal approach is determined independently of the incident wavelength, the exact design and analysis of grating diffraction elements can be implemented at different wavelengths by simply scaling the grating parameters. Moreover, the polarization effect of diffraction properties on the limitation of the modal method without and with the reflection loss of grating modes is clearly demonstrated.

© 2014 Optical Society of America

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  1. T. Clausnitzer, J. Limpert, K. Zöllner, H. Zellmer, H.-J. Fuchs, E.-B. Kley, A. Tünnermann, M. Jupé, and D. Ristau, “Highly-efficient transmission gratings in fused silica for chirped pulse amplification systems,” Appl. Opt. 42, 6934–6938 (2003).
    [CrossRef]
  2. F. Zhu, D. Zhang, P. Fan, L. Li, and Y. Guo, “Non-uniform strain measurement along a fiber Bragg grating using optical frequency domain reflectometry,” Chin. Opt. Lett. 11, 100603 (2013).
    [CrossRef]
  3. T. Osuch and Z. Jaroszewicz, “Numerical analysis of apodized fiber Bragg gratings formation using phase mask with variable diffraction efficiency,” Opt. Commun. 284, 567–572 (2011).
    [CrossRef]
  4. P. E. Dyer, R. J. Farley, and R. Giedl, “Analysis of grating formation with excimer laser irradiated phase masks,” Opt. Commun. 115, 327–334 (1995).
    [CrossRef]
  5. T. Osuch and Z. Jaroszewicz, “Analysis of the Talbot effect in apodized diffractive optical elements,” Photonics Lett. Pol. 1, 190–192 (2009).
  6. J. Zheng, C. Zhou, B. 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]
  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]
  8. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  9. J. Ma, S. Liu, Y. Jin, C. Xu, J. Shao, and Z. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
    [CrossRef]
  10. T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, U. Peschel, A. V. Tishchenko, and O. Parriaux, “An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings,” Opt. Express 13, 10448–10456 (2005).
    [CrossRef]
  11. I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  12. A. V. Tishchenko, “Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. 37, 309–330 (2005).
    [CrossRef]
  13. J. Wu, C. Zhou, H. Cao, A. Hu, W. Sun, and W. Jia, “Simplified mode analysis of guided mode resonance gratings with asymmetric coatings,” Chin. Opt. Lett. 11, 060501 (2013).
    [CrossRef]
  14. X. Jing and Y. Jin, “Transmittance analysis of diffraction phase grating,” Appl. Opt. 50, C11–C18 (2011).
    [CrossRef]
  15. 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]
  16. B. Wang, C. Zhou, J. Feng, H. Ru, and J. Zheng, “Wideband two port beam splitter of a binary fused silica phase grating,” Appl. Opt. 47, 4004–4008 (2008).
    [CrossRef]
  17. T. Clausnitzer, T. Kampfe, E. B. Kley, A. Tunnermann, A. V. Tishchenko, and O. Parriaux, “Highly-dispersive dielectric transmission gratings with 100% diffraction efficiency,” Opt. Express 16, 5577–5584 (2008).
    [CrossRef]
  18. I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  19. T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, A. Tishchenko, and O. Parriaux, “Investigation of the polarization-dependent diffraction of deep dielectric rectangular gratings illuminated in Littrow mounting,” Appl. Opt. 46, 819–826 (2007).
    [CrossRef]
  20. J. Feng, C. Zhou, B. Wang, 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]
  21. K. R. Catchpole, “A conceptual model of the diffuse transmittance of lamellar diffraction gratings on solar cells,” J. Appl. Phys. 102, 013102 (2007).
  22. H. T. Nguyen, B. W. Shore, S. J. Bryan, J. A. Britten, R. D. Boyd, and M. D. Perry, “High efficiency fused silica transmission gratings,” Opt. Lett. 22, 142–144 (1997).
    [CrossRef]
  23. H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, 2001), p. 41.

2013

2011

X. Jing and Y. Jin, “Transmittance analysis of diffraction phase grating,” Appl. Opt. 50, C11–C18 (2011).
[CrossRef]

T. Osuch and Z. Jaroszewicz, “Numerical analysis of apodized fiber Bragg gratings formation using phase mask with variable diffraction efficiency,” Opt. Commun. 284, 567–572 (2011).
[CrossRef]

2009

T. Osuch and Z. Jaroszewicz, “Analysis of the Talbot effect in apodized diffractive optical elements,” Photonics Lett. Pol. 1, 190–192 (2009).

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]

2008

2007

2005

2003

1997

1995

P. E. Dyer, R. J. Farley, and R. Giedl, “Analysis of grating formation with excimer laser irradiated phase masks,” Opt. Commun. 115, 327–334 (1995).
[CrossRef]

1982

1981

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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

Adams, J. L.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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

Andrewartha, J. R.

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

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Botten, I. C.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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

Boyd, R. D.

Britten, J. A.

Bryan, S. J.

Cao, H.

Catchpole, K. R.

K. R. Catchpole, “A conceptual model of the diffuse transmittance of lamellar diffraction gratings on solar cells,” J. Appl. Phys. 102, 013102 (2007).

Clausnitzer, T.

Craig, M. S.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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

Dyer, P. E.

P. E. Dyer, R. J. Farley, and R. Giedl, “Analysis of grating formation with excimer laser irradiated phase masks,” Opt. Commun. 115, 327–334 (1995).
[CrossRef]

Fan, P.

Fan, Z.

J. Ma, S. Liu, Y. Jin, C. Xu, J. Shao, and Z. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Farley, R. J.

P. E. Dyer, R. J. Farley, and R. Giedl, “Analysis of grating formation with excimer laser irradiated phase masks,” Opt. Commun. 115, 327–334 (1995).
[CrossRef]

Feng, J.

Fuchs, H.-J.

Gaylord, T. K.

Giedl, R.

P. E. Dyer, R. J. Farley, and R. Giedl, “Analysis of grating formation with excimer laser irradiated phase masks,” Opt. Commun. 115, 327–334 (1995).
[CrossRef]

Guo, Y.

Hu, A.

Jaroszewicz, Z.

T. Osuch and Z. Jaroszewicz, “Numerical analysis of apodized fiber Bragg gratings formation using phase mask with variable diffraction efficiency,” Opt. Commun. 284, 567–572 (2011).
[CrossRef]

T. Osuch and Z. Jaroszewicz, “Analysis of the Talbot effect in apodized diffractive optical elements,” Photonics Lett. Pol. 1, 190–192 (2009).

Jia, W.

Jin, Y.

X. Jing and Y. Jin, “Transmittance analysis of diffraction phase grating,” Appl. Opt. 50, C11–C18 (2011).
[CrossRef]

J. Ma, S. Liu, Y. Jin, C. Xu, J. Shao, and Z. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Jing, X.

Jupé, M.

Kampfe, T.

Kämpfe, T.

Kley, E. B.

Kley, E.-B.

Li, L.

Limpert, J.

Liu, S.

J. Ma, S. Liu, Y. Jin, C. Xu, J. Shao, and Z. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Lv, P.

Ma, J.

J. Ma, S. Liu, Y. Jin, C. Xu, J. Shao, and Z. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, 2001), p. 41.

McPhedran, R. C.

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

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Moharam, M. G.

Nguyen, H. T.

Osuch, T.

T. Osuch and Z. Jaroszewicz, “Numerical analysis of apodized fiber Bragg gratings formation using phase mask with variable diffraction efficiency,” Opt. Commun. 284, 567–572 (2011).
[CrossRef]

T. Osuch and Z. Jaroszewicz, “Analysis of the Talbot effect in apodized diffractive optical elements,” Photonics Lett. Pol. 1, 190–192 (2009).

Parriaux, O.

Perry, M. D.

Peschel, U.

Ristau, D.

Ru, H.

Shao, J.

J. Ma, S. Liu, Y. Jin, C. Xu, J. Shao, and Z. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Shore, B. W.

Sun, W.

Tishchenko, A.

Tishchenko, A. V.

Tunnermann, A.

Tünnermann, A.

Wang, B.

Wu, J.

Xu, C.

J. Ma, S. Liu, Y. Jin, C. Xu, J. Shao, and Z. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Zellmer, H.

Zhang, D.

Zheng, J.

Zhou, C.

Zhu, F.

Zöllner, K.

Appl. Opt.

Chin. Opt. Lett.

J. Appl. Phys.

K. R. Catchpole, “A conceptual model of the diffuse transmittance of lamellar diffraction gratings on solar cells,” J. Appl. Phys. 102, 013102 (2007).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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

Opt. Commun.

T. Osuch and Z. Jaroszewicz, “Numerical analysis of apodized fiber Bragg gratings formation using phase mask with variable diffraction efficiency,” Opt. Commun. 284, 567–572 (2011).
[CrossRef]

P. E. Dyer, R. J. Farley, and R. Giedl, “Analysis of grating formation with excimer laser irradiated phase masks,” Opt. Commun. 115, 327–334 (1995).
[CrossRef]

J. Ma, S. Liu, Y. Jin, C. Xu, J. Shao, and Z. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

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

Opt. Lett.

Opt. Quantum Electron.

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

Photonics Lett. Pol.

T. Osuch and Z. Jaroszewicz, “Analysis of the Talbot effect in apodized diffractive optical elements,” Photonics Lett. Pol. 1, 190–192 (2009).

Other

H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, 2001), p. 41.

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Figures (12)

Fig. 1.
Fig. 1.

Schematic of a rectangular grating under the Littrow configuration illumination.

Fig. 2.
Fig. 2.

Comparison of diffraction efficiencies between the modal method and RCWA versus the normalized grating period with a fill factor of 0.5. (a) TE polarization and (b) TM polarization.

Fig. 3.
Fig. 3.

Comparison of transmittance characteristics between the modal method and RCWA as a function of the normalized groove depth at a normalized period of 0.7. (a) TE polarization and (b) TM polarization.

Fig. 4.
Fig. 4.

Reflection of the two propagating grating modes at the air–grating and the grating–substrate interfaces with a grating period of 800 nm. (a) TE polarization and (b) TM polarization.

Fig. 5.
Fig. 5.

Simulation of the reflection losses based on Eq. (8) as a function of the groove depth and the normalized period for TE polarization with a fill factor of 0.5. (a) The zeroth mode and (b) the first mode.

Fig. 6.
Fig. 6.

Reflection of grating modes calculated by Eq. (8) with a grating period of 800 nm for (a), (b) TE polarization and (c), (d) TM polarization.

Fig. 7.
Fig. 7.

Reflection of grating modes calculated by Eq. (9) with a grating period of 800 nm for (a), (b) TE polarization and (c), (d) TM polarization.

Fig. 8.
Fig. 8.

Comparison of transmittance characteristics between the modal method with the consideration of the reflection loss of grating modes and RCWA. (a) TE polarization and (b) TM polarization.

Fig. 9.
Fig. 9.

Comparison of diffraction efficiencies between RCWA and the modal method at a normalized period of 0.7. The key for the lines is the same as in Fig. 8. (a) TE polarization and (b) TM polarization.

Fig. 10.
Fig. 10.

Error of diffraction efficiency comparison between the simplified modal method with the consideration of reflection loss by the thin-film model and the Fabry–Perot model and RCWA versus the normalized period. (a) TE polarization and (b) TM polarization.

Fig. 11.
Fig. 11.

Error of diffraction efficiency comparison between the simplified modal method with the consideration of reflection loss of the thin-film model and Fabry–Perot model and RCWA versus the normalized groove depth. (a) TE polarization and (b) TM polarization.

Fig. 12.
Fig. 12.

Simulation of the diffraction efficiencies of a rectangular transmission grating with a 600 nm period as a function of the groove depth and the fill factor for TE and TM polarization. (a), (b) The numerical calculation by RCWA; (c), (d) the results according to the optical thin-film model of Eq. (10); (e), (f) the results according to the nonsymmetrical Fabry–Perot resonator model of Eq. (10).

Equations (12)

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

η0T=cos2(Δφ/2),
η1T=sin2(Δφ/2)
Δφ=k0(neff0neff1)h=2π(neff0neff1)q,
F(neff2)=cos(2πpsinθi),
F(neff2)=cos(2πβpf)cos[2πγp(1f)]β2+γ22βγsin(2πβpf)sin[2πγp(1f)],
F(neff2)=cos(2πβpf)cos[2πγp(1f)]ng4β2+n04γ22ng2n02βγsin(2πβpf)sin[2πγp(1f)].
Rair/subm=(neffmneffair/subneffmneffair/sub)2,
[BC]=[cosδisinδ/ηiηsinδcosδ][1ηs],
RTHm=12·(η0neffairYneffsubη0neffair+Yneffsub)(η0neffairYneffsubη0neffair+Yneffsub)*.
RFPm=12·(RairmRsubm)2+4RairmRsubmsin2(2πλneffmh)(1RairmRsubm)2+4RairmRsubmsin2(2πλneffmh).
η0TH/FP=η0T·(1RTH/FP0)·(1RTH/FP1),
η1TH/FP=η1T·(1RTH/FP0)·(1RTH/FP1),

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