Abstract

The mechanical deformations of variable elastomeric diffractive optical elements are calculated by finite element methods. Starting from optimized blazed gratings, the derived profile variations serve as an input for rigorous-coupled-wave analysis to calculate the diffraction efficiency of a spectral band from 200 to 1200 nm. Applied planar strain of up to 80% has little effect on the maximum diffraction efficiency for large grating-period-to-wavelength ratios, g/λ, with only a shift toward shorter wavelengths. With a decreasing g/λ, the maximum efficiency also decreases when stretching the grating structure. Further influences of profile design like the angle of the antiblaze facet and the use of higher-order blaze profiles were investigated. Finally, we simulate the change in the diffraction efficiency at a single wavelength of a flexible blazed grating in direct contact with a rigid glass plate. In this case, the soft matter grating is compressed and deformed to reduce the diffracting properties.

© 2014 Optical Society of America

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References

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  1. S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH, 2003).
  2. H. P. Herzig, Micro-Optics: Elements, Systems and Applications (Taylor & Francis, 1997).
  3. J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley-VCH, 1998).
  4. R. Brunner, “Transferring diffractive optics from research to commercial applications: part I—progress in the patent landscape,” Adv. Opt. Technol. 2, 351–359 (2013).
    [CrossRef]
  5. B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators A 86, 81–85 (2000).
    [CrossRef]
  6. K. Hosakawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices,” J. Micromech. Microeng. 12, 1–6 (2002).
    [CrossRef]
  7. D. M. Burns and V. M. Bright, “Development of microelectromechanical variable blaze gratings,” Sens. Actuators A 64, 7–15 (1998).
    [CrossRef]
  8. O. Solgaard, F. S. A. Sandejas, and D. M. Bloom, “Deformable grating optical modulator,” Opt. Lett. 17, 688–690 (1992).
    [CrossRef]
  9. L. Xiang, L. Tie, M. Anjie, and W. Yuelin, “A compressed wide period-tunable grating working at low voltage,” J. Semicond. 31, 104010 (2010).
    [CrossRef]
  10. C. W. Wong, Y. Jeon, G. Barbastathis, and S.-G. Kim, “Analog piezoelectric-driven tunable gratings with nanometer resolution,” J. Microelectromech. Syst. 13, 998–1005 (2004).
    [CrossRef]
  11. Y. Wang, Y. Kanamori, and K. Hane, “Pitch-variable blazed grating consisting of freestanding silicon beams,” Opt. Express 17, 4419–4426 (2009).
    [CrossRef]
  12. Y. Wang, Y. Kanamori, T. Sasaki, and K. Hane, “Design and fabrication of freestanding pitch-variable blazed gratings on a silicon-on-insulator wafer,” J. Micromech. Microeng. 19, 025019 (2009).
    [CrossRef]
  13. J. A. Rogers, R. J. Jackman, O. J. A. Schueller, and G. M. Whitesides, “Elastomeric diffraction gratings as photothermal detectors,” Appl. Opt. 35, 6641–6647 (1996).
    [CrossRef]
  14. B. Grzybowski, D. Qin, and G. M. Whitesides, “Beam redirection and frequency filtering with transparent elastomeric diffractive elements,” Appl. Opt. 38, 2997–3002 (1999).
    [CrossRef]
  15. A. N. Simonov, O. Akhzar-Mehr, and G. Vdovin, “Light scanner based on a viscoelastic stretchable grating,” Opt. Lett. 30, 949–951 (2005).
    [CrossRef]
  16. ANSYS Inc., http://www.ansys.com/ .
  17. P. A. L. S. Martins, R. M. N. Jorge, and A. J. M. Ferreira, “A comparative study of several material models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues,” Strain 42, 135–147 (2006).
    [CrossRef]
  18. T. K. Kim, J. K. Kim, and O. C. Jeong, “Measurement of nonlinear mechanical properties of PDMS elastomer,” Microelectron. Eng. 88, 1982–1985 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. Dow Corning Corporation, “Product data sheet Sylgard 184,” http://www1.dowcorning.com/DataFiles/090007c8803bb6a1.pdf .
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    [CrossRef]
  26. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997), p. 150.
  27. H. Wolf, Spannungsoptik (Springer, 1976), p. 65.
  28. K. Hoshino and I. Shimoyama, “Analysis of elastic micro optical components under large deformation,” J. Micromech. Microeng. 13, 149–154 (2003).
    [CrossRef]
  29. W. Kuhn and F. Grün, “Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe,” Kolloid-Zeitschrift 101, 248–271 (1942).
  30. R. Brunner, “Diffractive optical elements,” in Springer Handbook of Lasers and Optics, F. Träger, ed., 2nd ed. (Springer, 2012), pp. 454–461.

2013

R. Brunner, “Transferring diffractive optics from research to commercial applications: part I—progress in the patent landscape,” Adv. Opt. Technol. 2, 351–359 (2013).
[CrossRef]

2011

T. K. Kim, J. K. Kim, and O. C. Jeong, “Measurement of nonlinear mechanical properties of PDMS elastomer,” Microelectron. Eng. 88, 1982–1985 (2011).
[CrossRef]

2010

2009

Y. Wang, Y. Kanamori, and K. Hane, “Pitch-variable blazed grating consisting of freestanding silicon beams,” Opt. Express 17, 4419–4426 (2009).
[CrossRef]

Y. Wang, Y. Kanamori, T. Sasaki, and K. Hane, “Design and fabrication of freestanding pitch-variable blazed gratings on a silicon-on-insulator wafer,” J. Micromech. Microeng. 19, 025019 (2009).
[CrossRef]

F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Process and material properties of polydimethylsiloxane (PDMS) for optical MEMS,” Sens. Actuators A 151, 95–99 (2009).
[CrossRef]

2006

P. A. L. S. Martins, R. M. N. Jorge, and A. J. M. Ferreira, “A comparative study of several material models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues,” Strain 42, 135–147 (2006).
[CrossRef]

O. Sandfuchs, R. Brunner, D. Pätz, S. Sinzinger, and J. Ruoff, “Rigorous analysis of shadowing effects in blazed transmission gratings,” Opt. Lett. 31, 3638–3640 (2006).
[CrossRef]

2005

2004

C. W. Wong, Y. Jeon, G. Barbastathis, and S.-G. Kim, “Analog piezoelectric-driven tunable gratings with nanometer resolution,” J. Microelectromech. Syst. 13, 998–1005 (2004).
[CrossRef]

2003

K. Hoshino and I. Shimoyama, “Analysis of elastic micro optical components under large deformation,” J. Micromech. Microeng. 13, 149–154 (2003).
[CrossRef]

2002

K. Hosakawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices,” J. Micromech. Microeng. 12, 1–6 (2002).
[CrossRef]

2000

B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators A 86, 81–85 (2000).
[CrossRef]

1999

1998

D. M. Burns and V. M. Bright, “Development of microelectromechanical variable blaze gratings,” Sens. Actuators A 64, 7–15 (1998).
[CrossRef]

1996

1992

1942

W. Kuhn and F. Grün, “Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe,” Kolloid-Zeitschrift 101, 248–271 (1942).

Akhzar-Mehr, O.

Anjie, M.

L. Xiang, L. Tie, M. Anjie, and W. Yuelin, “A compressed wide period-tunable grating working at low voltage,” J. Semicond. 31, 104010 (2010).
[CrossRef]

Barbastathis, G.

C. W. Wong, Y. Jeon, G. Barbastathis, and S.-G. Kim, “Analog piezoelectric-driven tunable gratings with nanometer resolution,” J. Microelectromech. Syst. 13, 998–1005 (2004).
[CrossRef]

Bischoff, J.

Bloom, D. M.

Bright, V. M.

D. M. Burns and V. M. Bright, “Development of microelectromechanical variable blaze gratings,” Sens. Actuators A 64, 7–15 (1998).
[CrossRef]

Brunner, R.

R. Brunner, “Transferring diffractive optics from research to commercial applications: part I—progress in the patent landscape,” Adv. Opt. Technol. 2, 351–359 (2013).
[CrossRef]

D. Lehr, M. Helgert, M. Sundermann, C. Morhard, C. Pacholski, J. P. Spatz, and R. Brunner, “Simulating different manufactured antireflective sub-wavelength structures considering the influence of local topographic variations,” Opt. Express 18, 23878–23890 (2010).
[CrossRef]

O. Sandfuchs, R. Brunner, D. Pätz, S. Sinzinger, and J. Ruoff, “Rigorous analysis of shadowing effects in blazed transmission gratings,” Opt. Lett. 31, 3638–3640 (2006).
[CrossRef]

R. Brunner, “Diffractive optical elements,” in Springer Handbook of Lasers and Optics, F. Träger, ed., 2nd ed. (Springer, 2012), pp. 454–461.

Burns, D. M.

D. M. Burns and V. M. Bright, “Development of microelectromechanical variable blaze gratings,” Sens. Actuators A 64, 7–15 (1998).
[CrossRef]

Draheim, J.

F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Process and material properties of polydimethylsiloxane (PDMS) for optical MEMS,” Sens. Actuators A 151, 95–99 (2009).
[CrossRef]

Ferreira, A. J. M.

P. A. L. S. Martins, R. M. N. Jorge, and A. J. M. Ferreira, “A comparative study of several material models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues,” Strain 42, 135–147 (2006).
[CrossRef]

Golub, M.

Grün, F.

W. Kuhn and F. Grün, “Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe,” Kolloid-Zeitschrift 101, 248–271 (1942).

Grzybowski, B.

B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators A 86, 81–85 (2000).
[CrossRef]

B. Grzybowski, D. Qin, and G. M. Whitesides, “Beam redirection and frequency filtering with transparent elastomeric diffractive elements,” Appl. Opt. 38, 2997–3002 (1999).
[CrossRef]

Haag, R.

B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators A 86, 81–85 (2000).
[CrossRef]

Hanada, K.

K. Hosakawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices,” J. Micromech. Microeng. 12, 1–6 (2002).
[CrossRef]

Hane, K.

Y. Wang, Y. Kanamori, and K. Hane, “Pitch-variable blazed grating consisting of freestanding silicon beams,” Opt. Express 17, 4419–4426 (2009).
[CrossRef]

Y. Wang, Y. Kanamori, T. Sasaki, and K. Hane, “Design and fabrication of freestanding pitch-variable blazed gratings on a silicon-on-insulator wafer,” J. Micromech. Microeng. 19, 025019 (2009).
[CrossRef]

Helgert, M.

Herzig, H. P.

H. P. Herzig, Micro-Optics: Elements, Systems and Applications (Taylor & Francis, 1997).

Hosakawa, K.

K. Hosakawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices,” J. Micromech. Microeng. 12, 1–6 (2002).
[CrossRef]

Hoshino, K.

K. Hoshino and I. Shimoyama, “Analysis of elastic micro optical components under large deformation,” J. Micromech. Microeng. 13, 149–154 (2003).
[CrossRef]

Jackman, R. J.

Jahns, J.

S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH, 2003).

Jeon, Y.

C. W. Wong, Y. Jeon, G. Barbastathis, and S.-G. Kim, “Analog piezoelectric-driven tunable gratings with nanometer resolution,” J. Microelectromech. Syst. 13, 998–1005 (2004).
[CrossRef]

Jeong, O. C.

T. K. Kim, J. K. Kim, and O. C. Jeong, “Measurement of nonlinear mechanical properties of PDMS elastomer,” Microelectron. Eng. 88, 1982–1985 (2011).
[CrossRef]

Jorge, R. M. N.

P. A. L. S. Martins, R. M. N. Jorge, and A. J. M. Ferreira, “A comparative study of several material models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues,” Strain 42, 135–147 (2006).
[CrossRef]

Kamberger, R.

F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Process and material properties of polydimethylsiloxane (PDMS) for optical MEMS,” Sens. Actuators A 151, 95–99 (2009).
[CrossRef]

Kanamori, Y.

Y. Wang, Y. Kanamori, and K. Hane, “Pitch-variable blazed grating consisting of freestanding silicon beams,” Opt. Express 17, 4419–4426 (2009).
[CrossRef]

Y. Wang, Y. Kanamori, T. Sasaki, and K. Hane, “Design and fabrication of freestanding pitch-variable blazed gratings on a silicon-on-insulator wafer,” J. Micromech. Microeng. 19, 025019 (2009).
[CrossRef]

Kim, J. K.

T. K. Kim, J. K. Kim, and O. C. Jeong, “Measurement of nonlinear mechanical properties of PDMS elastomer,” Microelectron. Eng. 88, 1982–1985 (2011).
[CrossRef]

Kim, S.-G.

C. W. Wong, Y. Jeon, G. Barbastathis, and S.-G. Kim, “Analog piezoelectric-driven tunable gratings with nanometer resolution,” J. Microelectromech. Syst. 13, 998–1005 (2004).
[CrossRef]

Kim, T. K.

T. K. Kim, J. K. Kim, and O. C. Jeong, “Measurement of nonlinear mechanical properties of PDMS elastomer,” Microelectron. Eng. 88, 1982–1985 (2011).
[CrossRef]

Kuhn, W.

W. Kuhn and F. Grün, “Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe,” Kolloid-Zeitschrift 101, 248–271 (1942).

Lehr, D.

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997), p. 150.

Maeda, R.

K. Hosakawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices,” J. Micromech. Microeng. 12, 1–6 (2002).
[CrossRef]

Martins, P. A. L. S.

P. A. L. S. Martins, R. M. N. Jorge, and A. J. M. Ferreira, “A comparative study of several material models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues,” Strain 42, 135–147 (2006).
[CrossRef]

Morhard, C.

Pacholski, C.

Pätz, D.

Popov, E.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997), p. 150.

Qin, D.

B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators A 86, 81–85 (2000).
[CrossRef]

B. Grzybowski, D. Qin, and G. M. Whitesides, “Beam redirection and frequency filtering with transparent elastomeric diffractive elements,” Appl. Opt. 38, 2997–3002 (1999).
[CrossRef]

Rogers, J. A.

Ruoff, J.

Sandejas, F. S. A.

Sandfuchs, O.

Sasaki, T.

Y. Wang, Y. Kanamori, T. Sasaki, and K. Hane, “Design and fabrication of freestanding pitch-variable blazed gratings on a silicon-on-insulator wafer,” J. Micromech. Microeng. 19, 025019 (2009).
[CrossRef]

Schneider, F.

F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Process and material properties of polydimethylsiloxane (PDMS) for optical MEMS,” Sens. Actuators A 151, 95–99 (2009).
[CrossRef]

Schueller, O. J. A.

Shimoyama, I.

K. Hoshino and I. Shimoyama, “Analysis of elastic micro optical components under large deformation,” J. Micromech. Microeng. 13, 149–154 (2003).
[CrossRef]

Simonov, A. N.

Sinzinger, S.

Solgaard, O.

Spatz, J. P.

Sundermann, M.

Tie, L.

L. Xiang, L. Tie, M. Anjie, and W. Yuelin, “A compressed wide period-tunable grating working at low voltage,” J. Semicond. 31, 104010 (2010).
[CrossRef]

Turunen, J.

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley-VCH, 1998).

Vdovin, G.

Wallrabe, U.

F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Process and material properties of polydimethylsiloxane (PDMS) for optical MEMS,” Sens. Actuators A 151, 95–99 (2009).
[CrossRef]

Wang, Y.

Y. Wang, Y. Kanamori, and K. Hane, “Pitch-variable blazed grating consisting of freestanding silicon beams,” Opt. Express 17, 4419–4426 (2009).
[CrossRef]

Y. Wang, Y. Kanamori, T. Sasaki, and K. Hane, “Design and fabrication of freestanding pitch-variable blazed gratings on a silicon-on-insulator wafer,” J. Micromech. Microeng. 19, 025019 (2009).
[CrossRef]

Whitesides, G. M.

Wolf, H.

H. Wolf, Spannungsoptik (Springer, 1976), p. 65.

Wong, C. W.

C. W. Wong, Y. Jeon, G. Barbastathis, and S.-G. Kim, “Analog piezoelectric-driven tunable gratings with nanometer resolution,” J. Microelectromech. Syst. 13, 998–1005 (2004).
[CrossRef]

Wyrowski, F.

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley-VCH, 1998).

Xiang, L.

L. Xiang, L. Tie, M. Anjie, and W. Yuelin, “A compressed wide period-tunable grating working at low voltage,” J. Semicond. 31, 104010 (2010).
[CrossRef]

Yuelin, W.

L. Xiang, L. Tie, M. Anjie, and W. Yuelin, “A compressed wide period-tunable grating working at low voltage,” J. Semicond. 31, 104010 (2010).
[CrossRef]

Adv. Opt. Technol.

R. Brunner, “Transferring diffractive optics from research to commercial applications: part I—progress in the patent landscape,” Adv. Opt. Technol. 2, 351–359 (2013).
[CrossRef]

Appl. Opt.

J. Microelectromech. Syst.

C. W. Wong, Y. Jeon, G. Barbastathis, and S.-G. Kim, “Analog piezoelectric-driven tunable gratings with nanometer resolution,” J. Microelectromech. Syst. 13, 998–1005 (2004).
[CrossRef]

J. Micromech. Microeng.

Y. Wang, Y. Kanamori, T. Sasaki, and K. Hane, “Design and fabrication of freestanding pitch-variable blazed gratings on a silicon-on-insulator wafer,” J. Micromech. Microeng. 19, 025019 (2009).
[CrossRef]

K. Hosakawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices,” J. Micromech. Microeng. 12, 1–6 (2002).
[CrossRef]

K. Hoshino and I. Shimoyama, “Analysis of elastic micro optical components under large deformation,” J. Micromech. Microeng. 13, 149–154 (2003).
[CrossRef]

J. Opt. Soc. Am. A

J. Semicond.

L. Xiang, L. Tie, M. Anjie, and W. Yuelin, “A compressed wide period-tunable grating working at low voltage,” J. Semicond. 31, 104010 (2010).
[CrossRef]

Kolloid-Zeitschrift

W. Kuhn and F. Grün, “Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe,” Kolloid-Zeitschrift 101, 248–271 (1942).

Microelectron. Eng.

T. K. Kim, J. K. Kim, and O. C. Jeong, “Measurement of nonlinear mechanical properties of PDMS elastomer,” Microelectron. Eng. 88, 1982–1985 (2011).
[CrossRef]

Opt. Express

Opt. Lett.

Sens. Actuators A

F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Process and material properties of polydimethylsiloxane (PDMS) for optical MEMS,” Sens. Actuators A 151, 95–99 (2009).
[CrossRef]

D. M. Burns and V. M. Bright, “Development of microelectromechanical variable blaze gratings,” Sens. Actuators A 64, 7–15 (1998).
[CrossRef]

B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators A 86, 81–85 (2000).
[CrossRef]

Strain

P. A. L. S. Martins, R. M. N. Jorge, and A. J. M. Ferreira, “A comparative study of several material models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues,” Strain 42, 135–147 (2006).
[CrossRef]

Other

ANSYS Inc., http://www.ansys.com/ .

S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH, 2003).

H. P. Herzig, Micro-Optics: Elements, Systems and Applications (Taylor & Francis, 1997).

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley-VCH, 1998).

R. Brunner, “Diffractive optical elements,” in Springer Handbook of Lasers and Optics, F. Träger, ed., 2nd ed. (Springer, 2012), pp. 454–461.

Dow Corning Corporation, “Product data sheet Sylgard 184,” http://www1.dowcorning.com/DataFiles/090007c8803bb6a1.pdf .

K. Hehl, “Unigit—versatile rigorous grating solver,” http://www.unigit.com/ .

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997), p. 150.

H. Wolf, Spannungsoptik (Springer, 1976), p. 65.

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Figures (9)

Fig. 1.
Fig. 1.

Schematic drawing of the load cases to deform blazed gratings in soft material: (a) stretching in direction of the grating and (b) pressing a hard plate (glass) in normal direction against the blazed grating surface.

Fig. 2.
Fig. 2.

Schematic drawing showing the interaction of incoming light with a blazed grating profile.

Fig. 3.
Fig. 3.

FEM-mesh grating profile (g=10, α=171,6°, β=70°). (a) Original mesh distribution with high node density at the edges and vertices, (b) deformation and normal stress distribution for a 80% stretched structure, and (c) deformation and normal stress distribution for 25% compressed structure.

Fig. 4.
Fig. 4.

Original 10 μm periodic blazed grating stretched in discrete steps (0%–80% length expansion): (a) calculated profile geometries by FEM and (b) RCWA calculated first-order diffraction efficiency as a function of the wavelength for the differently expanded structures.

Fig. 5.
Fig. 5.

Deformation of different periodicities including initial optimal blazed profiles (solid lines) and the same profiles expanded by 80% calculated with FEM (dashed lines). As a reference, comparable optimal blazed profiles with the expanded periodicity are also shown (two-dotted lines). The vertical chain dotted line separates the stretched profile in positive and negative facet. The initial geometries of the optimum blazed gratings follow Eqs. (3) and (4). (a) 50 μm, (b) 10 μm, (c) 5 μm, and (d) 2 μm.

Fig. 6.
Fig. 6.

First-order diffraction efficiency as a function of wavelength for original designed (solid lines) and stretched profile geometries (dashed lines) of 50, 10, 5, and 2 μm periodicities.

Fig. 7.
Fig. 7.

Influence of the variation of the antiblaze angle. (a) For an initial period of 10 μm different angles for the antiblaze facet were chosen (87.5°50°) (solid lines). The dashed lines show the calculated profiles for an expansion of 80%. (b) First-order diffraction efficiency as a function of the wavelength for the initial and stretched blaze gratings and varying antiblaze facet angle.

Fig. 8.
Fig. 8.

Influence on different diffraction orders for initial 50 μm periodic optimum blaze gratings (solid lines) and 80% expanded gratings (dashed lines). (a) Original blazed profile shape optimized for different diffraction orders and calculated shape for the expanded gratings. (b) Diffraction efficiency as a function of the wavelength for the different diffraction orders.

Fig. 9.
Fig. 9.

Pressing a solid plane against a soft 10 μm periodic blaze grating. (a) Shape of the original and the calculated deformed profile. (b) Corresponding efficiency for various diffraction orders for a reference wavelength of 632.8 nm.

Equations (7)

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

tanα=mλg(ncosθmcosθ),
cosβsinθm1n(sinθ+mλg).
n1=n0+C1σ1+C2σ2,
n2=n0+C1σ2+C2σ1,
n1n2=(C1C2)(σ1σ2)=C(σ1σ2).
tanαmλg(n1),
cosβmλgn.

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