Abstract

Ultrafast laser inscription of volume phase gratings with low index contrast and self-images with visibility of 0.96 is demonstrated. It is also demonstrated that phase differences of π/2 for visible light are achievable with only one layer of structures induced in bulk borosilicate glass by direct laser writing. The fabrication method avoids the stitching of several layers of structures and significantly reduces the time of process. The increment of visibility with the induced phase difference is proved and results are compared with the expected for planar phase gratings.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
  3. I. Spaleniak, S. Gross, N. Jovanovic, R. J. Williams, J. S. Lawrence, M. J. Ireland, and M. J. Withford, “Multiband processing of multimode light: Combining 3D photonic lanterns with waveguide Bragg gratings,” Laser Photonics Rev. 8(1), L1–L5 (2014).
    [Crossref]
  4. A. Rodríguez, M. C. Morant-mi, A. Dias-ponte, M. Martínez-Calderón, M. Gómez-aranzadi, and S. M. Olaizola, “Femtosecond laser-induced periodic surface nanostructuring of sputtered platinum thin films,” Appl. Surf. Sci. 351, 135–139 (2015).
    [Crossref]
  5. A. Pan, A. Dias, M. Gomez-Aranzadi, S. M. Olaizola, and A. Rodriguez, “Formation of laser-induced periodic surface structures on niobium by femtosecond laser irradiation,” J. Appl. Phys. 115(17), 173101 (2014).
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  8. M. Mikutis, T. Kudrius, G. Šlekys, D. Paipulas, and S. Juodkazis, “High 90% efficiency Bragg gratings formed in fused silica by femtosecond Gauss-Bessel laser beams,” Opt. Mater. Express 3(11), 1862–1871 (2013).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2015 (1)

A. Rodríguez, M. C. Morant-mi, A. Dias-ponte, M. Martínez-Calderón, M. Gómez-aranzadi, and S. M. Olaizola, “Femtosecond laser-induced periodic surface nanostructuring of sputtered platinum thin films,” Appl. Surf. Sci. 351, 135–139 (2015).
[Crossref]

2014 (3)

A. Pan, A. Dias, M. Gomez-Aranzadi, S. M. Olaizola, and A. Rodriguez, “Formation of laser-induced periodic surface structures on niobium by femtosecond laser irradiation,” J. Appl. Phys. 115(17), 173101 (2014).
[Crossref]

I. Spaleniak, S. Gross, N. Jovanovic, R. J. Williams, J. S. Lawrence, M. J. Ireland, and M. J. Withford, “Multiband processing of multimode light: Combining 3D photonic lanterns with waveguide Bragg gratings,” Laser Photonics Rev. 8(1), L1–L5 (2014).
[Crossref]

A. Arriola, S. Mukherjee, D. Choudhury, L. Labadie, and R. R. Thomson, “Ultrafast laser inscription of mid-IR directional couplers for stellar interferometry,” Opt. Lett. 39(16), 4820–4822 (2014).
[Crossref] [PubMed]

2013 (3)

2012 (2)

2011 (2)

M. Beresna, M. Gecevi, and P. G. Kazansky, “Polarization sensitive elements fabricated by femtosecond laser nanostructuring of glass,” Opt. Mater. Express 1(4), 783–795 (2011).
[Crossref]

J.-K. Park, S.-H. Cho, K.-H. Kim, and M.-C. Kang, “Optical diffraction gratings embedded in BK-7 glass by low-density plasma formation using femtosecond laser,” Trans. Nonferrous Met. Soc. China. 21, s165–s169 (2011).
[Crossref]

2006 (1)

D. Liu, Y. Li, R. An, Y. Dou, H. Yang, and Q. Gong, “Influence of focusing depth on the microfabrication of waveguides inside silica glass by femtosecond laser direct writing,” Appl. Phys., A Mater. Sci. Process. 84(3), 257–260 (2006).
[Crossref]

2005 (1)

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98(1), 013517 (2005).
[Crossref]

2004 (1)

A. Saliminia, N. T. Nguyen, S. L. Chin, and R. Vallée, “The influence of self-focusing and filamentation on refractive index modifications in fused silica using intense femtosecond pulses,” Opt. Commun. 241(4-6), 529–538 (2004).
[Crossref]

2001 (1)

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. 191(3-6), 333–339 (2001).
[Crossref]

1999 (1)

J. A. Arns, W. S. Colburn, S. C. Barden, K. O. Systems, P. O. Box, and A. Arbor, “Volume phase gratings for spectroscopy, ultrafast laser compressors, and wavelength division multiplexing,” Proc. SPIE 1999, 3779 (1999).

1989 (1)

K. Patorski, “The Self-Imaging Phenomenon and its Applications,” Prog. Opt. 27, 1–108 (1989).
[Crossref]

1988 (1)

T. Jinhong, “The Diffraction Near Fields and Lau Effect of a Square-wave Modulated Phase Grating,” J. Mod. Opt. 35(8), 1399–1408 (1988).
[Crossref]

1980 (1)

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32(1), 14–18 (1980).
[Crossref]

An, R.

D. Liu, Y. Li, R. An, Y. Dou, H. Yang, and Q. Gong, “Influence of focusing depth on the microfabrication of waveguides inside silica glass by femtosecond laser direct writing,” Appl. Phys., A Mater. Sci. Process. 84(3), 257–260 (2006).
[Crossref]

Arbor, A.

J. A. Arns, W. S. Colburn, S. C. Barden, K. O. Systems, P. O. Box, and A. Arbor, “Volume phase gratings for spectroscopy, ultrafast laser compressors, and wavelength division multiplexing,” Proc. SPIE 1999, 3779 (1999).

Arns, J. A.

J. A. Arns, W. S. Colburn, S. C. Barden, K. O. Systems, P. O. Box, and A. Arbor, “Volume phase gratings for spectroscopy, ultrafast laser compressors, and wavelength division multiplexing,” Proc. SPIE 1999, 3779 (1999).

Arriola, A.

Barden, S. C.

J. A. Arns, W. S. Colburn, S. C. Barden, K. O. Systems, P. O. Box, and A. Arbor, “Volume phase gratings for spectroscopy, ultrafast laser compressors, and wavelength division multiplexing,” Proc. SPIE 1999, 3779 (1999).

Beresna, M.

Bhardwaj, V. R.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98(1), 013517 (2005).
[Crossref]

Bolognini, N.

Box, P. O.

J. A. Arns, W. S. Colburn, S. C. Barden, K. O. Systems, P. O. Box, and A. Arbor, “Volume phase gratings for spectroscopy, ultrafast laser compressors, and wavelength division multiplexing,” Proc. SPIE 1999, 3779 (1999).

Brisset, F.

Chanda, D.

Chin, S. L.

A. Saliminia, N. T. Nguyen, S. L. Chin, and R. Vallée, “The influence of self-focusing and filamentation on refractive index modifications in fused silica using intense femtosecond pulses,” Opt. Commun. 241(4-6), 529–538 (2004).
[Crossref]

Cho, S.-H.

J.-K. Park, S.-H. Cho, K.-H. Kim, and M.-C. Kang, “Optical diffraction gratings embedded in BK-7 glass by low-density plasma formation using femtosecond laser,” Trans. Nonferrous Met. Soc. China. 21, s165–s169 (2011).
[Crossref]

Choudhury, D.

Colburn, W. S.

J. A. Arns, W. S. Colburn, S. C. Barden, K. O. Systems, P. O. Box, and A. Arbor, “Volume phase gratings for spectroscopy, ultrafast laser compressors, and wavelength division multiplexing,” Proc. SPIE 1999, 3779 (1999).

Corkum, P. B.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98(1), 013517 (2005).
[Crossref]

Cunningham, C. R.

Desmarchelier, R.

Dias, A.

A. Pan, A. Dias, M. Gomez-Aranzadi, S. M. Olaizola, and A. Rodriguez, “Formation of laser-induced periodic surface structures on niobium by femtosecond laser irradiation,” J. Appl. Phys. 115(17), 173101 (2014).
[Crossref]

Dias-ponte, A.

A. Rodríguez, M. C. Morant-mi, A. Dias-ponte, M. Martínez-Calderón, M. Gómez-aranzadi, and S. M. Olaizola, “Femtosecond laser-induced periodic surface nanostructuring of sputtered platinum thin films,” Appl. Surf. Sci. 351, 135–139 (2015).
[Crossref]

Dou, Y.

D. Liu, Y. Li, R. An, Y. Dou, H. Yang, and Q. Gong, “Influence of focusing depth on the microfabrication of waveguides inside silica glass by femtosecond laser direct writing,” Appl. Phys., A Mater. Sci. Process. 84(3), 257–260 (2006).
[Crossref]

Forte, G.

Franco, M.

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. 191(3-6), 333–339 (2001).
[Crossref]

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32(1), 14–18 (1980).
[Crossref]

Gecevi, M.

Gomez-Aranzadi, M.

A. Pan, A. Dias, M. Gomez-Aranzadi, S. M. Olaizola, and A. Rodriguez, “Formation of laser-induced periodic surface structures on niobium by femtosecond laser irradiation,” J. Appl. Phys. 115(17), 173101 (2014).
[Crossref]

Gómez-aranzadi, M.

A. Rodríguez, M. C. Morant-mi, A. Dias-ponte, M. Martínez-Calderón, M. Gómez-aranzadi, and S. M. Olaizola, “Femtosecond laser-induced periodic surface nanostructuring of sputtered platinum thin films,” Appl. Surf. Sci. 351, 135–139 (2015).
[Crossref]

Gong, Q.

D. Liu, Y. Li, R. An, Y. Dou, H. Yang, and Q. Gong, “Influence of focusing depth on the microfabrication of waveguides inside silica glass by femtosecond laser direct writing,” Appl. Phys., A Mater. Sci. Process. 84(3), 257–260 (2006).
[Crossref]

Gross, S.

I. Spaleniak, S. Gross, N. Jovanovic, R. J. Williams, J. S. Lawrence, M. J. Ireland, and M. J. Withford, “Multiband processing of multimode light: Combining 3D photonic lanterns with waveguide Bragg gratings,” Laser Photonics Rev. 8(1), L1–L5 (2014).
[Crossref]

Herman, P. R.

Hervé, E.

Hnatovsky, C.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98(1), 013517 (2005).
[Crossref]

Ireland, M. J.

I. Spaleniak, S. Gross, N. Jovanovic, R. J. Williams, J. S. Lawrence, M. J. Ireland, and M. J. Withford, “Multiband processing of multimode light: Combining 3D photonic lanterns with waveguide Bragg gratings,” Laser Photonics Rev. 8(1), L1–L5 (2014).
[Crossref]

Jinhong, T.

T. Jinhong, “The Diffraction Near Fields and Lau Effect of a Square-wave Modulated Phase Grating,” J. Mod. Opt. 35(8), 1399–1408 (1988).
[Crossref]

Jovanovic, N.

I. Spaleniak, S. Gross, N. Jovanovic, R. J. Williams, J. S. Lawrence, M. J. Ireland, and M. J. Withford, “Multiband processing of multimode light: Combining 3D photonic lanterns with waveguide Bragg gratings,” Laser Photonics Rev. 8(1), L1–L5 (2014).
[Crossref]

Juodkazis, S.

Kang, M.-C.

J.-K. Park, S.-H. Cho, K.-H. Kim, and M.-C. Kang, “Optical diffraction gratings embedded in BK-7 glass by low-density plasma formation using femtosecond laser,” Trans. Nonferrous Met. Soc. China. 21, s165–s169 (2011).
[Crossref]

Kazansky, P. G.

Kim, K.-H.

J.-K. Park, S.-H. Cho, K.-H. Kim, and M.-C. Kang, “Optical diffraction gratings embedded in BK-7 glass by low-density plasma formation using femtosecond laser,” Trans. Nonferrous Met. Soc. China. 21, s165–s169 (2011).
[Crossref]

Kudrius, T.

Labadie, L.

Lancry, M.

Lawrence, J. S.

I. Spaleniak, S. Gross, N. Jovanovic, R. J. Williams, J. S. Lawrence, M. J. Ireland, and M. J. Withford, “Multiband processing of multimode light: Combining 3D photonic lanterns with waveguide Bragg gratings,” Laser Photonics Rev. 8(1), L1–L5 (2014).
[Crossref]

Lee, D.

Lencina, A.

Li, Y.

D. Liu, Y. Li, R. An, Y. Dou, H. Yang, and Q. Gong, “Influence of focusing depth on the microfabrication of waveguides inside silica glass by femtosecond laser direct writing,” Appl. Phys., A Mater. Sci. Process. 84(3), 257–260 (2006).
[Crossref]

Liu, D.

D. Liu, Y. Li, R. An, Y. Dou, H. Yang, and Q. Gong, “Influence of focusing depth on the microfabrication of waveguides inside silica glass by femtosecond laser direct writing,” Appl. Phys., A Mater. Sci. Process. 84(3), 257–260 (2006).
[Crossref]

MacLachlan, D. G.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32(1), 14–18 (1980).
[Crossref]

Martínez-Calderón, M.

A. Rodríguez, M. C. Morant-mi, A. Dias-ponte, M. Martínez-Calderón, M. Gómez-aranzadi, and S. M. Olaizola, “Femtosecond laser-induced periodic surface nanostructuring of sputtered platinum thin films,” Appl. Surf. Sci. 351, 135–139 (2015).
[Crossref]

Mikutis, M.

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32(1), 14–18 (1980).
[Crossref]

Morant-mi, M. C.

A. Rodríguez, M. C. Morant-mi, A. Dias-ponte, M. Martínez-Calderón, M. Gómez-aranzadi, and S. M. Olaizola, “Femtosecond laser-induced periodic surface nanostructuring of sputtered platinum thin films,” Appl. Surf. Sci. 351, 135–139 (2015).
[Crossref]

Mukherjee, S.

Mysyrowicz, A.

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. 191(3-6), 333–339 (2001).
[Crossref]

Ng, M. L.

Nguyen, N. T.

A. Saliminia, N. T. Nguyen, S. L. Chin, and R. Vallée, “The influence of self-focusing and filamentation on refractive index modifications in fused silica using intense femtosecond pulses,” Opt. Commun. 241(4-6), 529–538 (2004).
[Crossref]

Olaizola, S. M.

A. Rodríguez, M. C. Morant-mi, A. Dias-ponte, M. Martínez-Calderón, M. Gómez-aranzadi, and S. M. Olaizola, “Femtosecond laser-induced periodic surface nanostructuring of sputtered platinum thin films,” Appl. Surf. Sci. 351, 135–139 (2015).
[Crossref]

A. Pan, A. Dias, M. Gomez-Aranzadi, S. M. Olaizola, and A. Rodriguez, “Formation of laser-induced periodic surface structures on niobium by femtosecond laser irradiation,” J. Appl. Phys. 115(17), 173101 (2014).
[Crossref]

Paipulas, D.

Pan, A.

A. Pan, A. Dias, M. Gomez-Aranzadi, S. M. Olaizola, and A. Rodriguez, “Formation of laser-induced periodic surface structures on niobium by femtosecond laser irradiation,” J. Appl. Phys. 115(17), 173101 (2014).
[Crossref]

Park, J.-K.

J.-K. Park, S.-H. Cho, K.-H. Kim, and M.-C. Kang, “Optical diffraction gratings embedded in BK-7 glass by low-density plasma formation using femtosecond laser,” Trans. Nonferrous Met. Soc. China. 21, s165–s169 (2011).
[Crossref]

Patorski, K.

K. Patorski, “The Self-Imaging Phenomenon and its Applications,” Prog. Opt. 27, 1–108 (1989).
[Crossref]

Poulin, J. C.

Poumellec, B.

Prade, B.

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. 191(3-6), 333–339 (2001).
[Crossref]

Rayner, D. M.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98(1), 013517 (2005).
[Crossref]

Rodriguez, A.

A. Pan, A. Dias, M. Gomez-Aranzadi, S. M. Olaizola, and A. Rodriguez, “Formation of laser-induced periodic surface structures on niobium by femtosecond laser irradiation,” J. Appl. Phys. 115(17), 173101 (2014).
[Crossref]

Rodríguez, A.

A. Rodríguez, M. C. Morant-mi, A. Dias-ponte, M. Martínez-Calderón, M. Gómez-aranzadi, and S. M. Olaizola, “Femtosecond laser-induced periodic surface nanostructuring of sputtered platinum thin films,” Appl. Surf. Sci. 351, 135–139 (2015).
[Crossref]

Saliminia, A.

A. Saliminia, N. T. Nguyen, S. L. Chin, and R. Vallée, “The influence of self-focusing and filamentation on refractive index modifications in fused silica using intense femtosecond pulses,” Opt. Commun. 241(4-6), 529–538 (2004).
[Crossref]

Simova, E.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98(1), 013517 (2005).
[Crossref]

Šlekys, G.

Spaleniak, I.

I. Spaleniak, S. Gross, N. Jovanovic, R. J. Williams, J. S. Lawrence, M. J. Ireland, and M. J. Withford, “Multiband processing of multimode light: Combining 3D photonic lanterns with waveguide Bragg gratings,” Laser Photonics Rev. 8(1), L1–L5 (2014).
[Crossref]

Sudrie, L.

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. 191(3-6), 333–339 (2001).
[Crossref]

Systems, K. O.

J. A. Arns, W. S. Colburn, S. C. Barden, K. O. Systems, P. O. Box, and A. Arbor, “Volume phase gratings for spectroscopy, ultrafast laser compressors, and wavelength division multiplexing,” Proc. SPIE 1999, 3779 (1999).

Taylor, R. S.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98(1), 013517 (2005).
[Crossref]

Tebaldi, M.

Thomson, R. R.

Vallée, R.

A. Saliminia, N. T. Nguyen, S. L. Chin, and R. Vallée, “The influence of self-focusing and filamentation on refractive index modifications in fused silica using intense femtosecond pulses,” Opt. Commun. 241(4-6), 529–538 (2004).
[Crossref]

Williams, R. J.

I. Spaleniak, S. Gross, N. Jovanovic, R. J. Williams, J. S. Lawrence, M. J. Ireland, and M. J. Withford, “Multiband processing of multimode light: Combining 3D photonic lanterns with waveguide Bragg gratings,” Laser Photonics Rev. 8(1), L1–L5 (2014).
[Crossref]

Withford, M. J.

I. Spaleniak, S. Gross, N. Jovanovic, R. J. Williams, J. S. Lawrence, M. J. Ireland, and M. J. Withford, “Multiband processing of multimode light: Combining 3D photonic lanterns with waveguide Bragg gratings,” Laser Photonics Rev. 8(1), L1–L5 (2014).
[Crossref]

Yang, H.

D. Liu, Y. Li, R. An, Y. Dou, H. Yang, and Q. Gong, “Influence of focusing depth on the microfabrication of waveguides inside silica glass by femtosecond laser direct writing,” Appl. Phys., A Mater. Sci. Process. 84(3), 257–260 (2006).
[Crossref]

Appl. Opt. (1)

Appl. Phys., A Mater. Sci. Process. (1)

D. Liu, Y. Li, R. An, Y. Dou, H. Yang, and Q. Gong, “Influence of focusing depth on the microfabrication of waveguides inside silica glass by femtosecond laser direct writing,” Appl. Phys., A Mater. Sci. Process. 84(3), 257–260 (2006).
[Crossref]

Appl. Surf. Sci. (1)

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Figures (5)

Fig. 1
Fig. 1 (a) Sketch of a VPG diffracting a green laser beam. n is the refractive index of the glass and n’ is the refractive index of the modified region. (b) Calculated first order diffraction efficiency as a function of thickness by RCWA simulations (solid lines) and Eq. (1). (dashed lines). The illumination wavelength is 532 nm and s-polarized. Incidence is normal and the gratings, with no tilt, were approximated to a rectangular index profile with 10 µm of period.
Fig. 2
Fig. 2 (a) Sketch of the main phenomena involved in the formation of structures with different refractive index inside glass. I, sketch of a self-focused beam, ZSF is the distance for self-focusing. II, longitudinal spherical aberration Δ at a focusing depth Z. III expected modified region in the absence of self-focusing and aberrations (C.P. is the confocal parameter). IV, modified region in a general case. (b) Sketch of the fabrication process of a binary VPG in glass.
Fig. 3
Fig. 3 (a) Top view of a set of 2x2 mm2 gratings made in AF45. (b) Cross-sectional images of groups of five overlapped lines made with phase contrast microscope, written with 1, 1.5, 2.6, 3.6, 4.6 μJ from right to left, and focusing depth Z = 200 μm (top) and Z = 550 μm (bottom).(c) Top view of some grating lines written in glass by the method described in section 3.3. Pulse energy of 3.6 μJ, scanning speed of 200 μm/s and 10 μm of period.
Fig. 4
Fig. 4 Plot of the measured thickness of the structures written in glass from microscope images, left. First order diffraction efficiency vs. focusing depth Z; wavelength of 532 nm and normal incidence, center. Plot of phase difference and optical path difference vs. focusing depth, values were obtained by the application of Eq. (1), right.
Fig. 5
Fig. 5 (a) Self-image of the grating fabricated with pulse energy of 3.6 μJ and Z = 550 μm (top) and its grayscale profile fitted to a sine function (bottom). (b) Plot of the measured visibility of the complete set of gratings vs. the estimated Δφ/π shown in Fig. 4c. Red line represents V = sin(Δφ).

Equations (1)

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η m =sin c 2 ( mπ 2 )co s 2 [ 1 2 ( Δφmπ ) ]

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