Abstract

Gradient-index (GRIN) media offer advantages over thin optical elements for beam shaping of strongly diffracting fields. A numerical GRIN design method is presented, where diffraction effects are considered in solving for the refractive index profile. The index profile is found by specifying a desired beam transformation throughout the GRIN volume and solving a series of phase retrieval problems. A Gaussian to flat-top beam shaper and a coherent beam combiner are shown as examples. Reduced beam distortion is demonstrated in comparison to a purely geometric design method.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Design and manufacture of a gradient-index axicon

David J. Fischer, Curtis J. Harkrider, and Duncan T. Moore
Appl. Opt. 39(16) 2687-2694 (2000)

Design and fabrication of a metamaterial gradient index diffraction grating at infrared wavelengths

Yu-Ju Tsai, Stéphane Larouche, Talmage Tyler, Guy Lipworth, Nan M. Jokerst, and David R. Smith
Opt. Express 19(24) 24411-24423 (2011)

Geometrical model for the design of gradient-index-rod lens coupling devices

Riccardo Falciai and Tania Pascucci
Appl. Opt. 31(25) 5211-5215 (1992)

References

  • View by:
  • |
  • |
  • |

  1. P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19(20), 3545–3553 (1980).
    [Crossref] [PubMed]
  2. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4(11), 1400–1403 (1965).
    [Crossref]
  3. C. Wang and D. L. Shealy, “Design of gradient-index lens systems for laser beam reshaping,” Appl. Opt. 32(25), 4763–4769 (1993).
    [Crossref] [PubMed]
  4. L. A. Romero and F. M. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A 13(4), 751–760 (1996).
    [Crossref]
  5. F. M. Dickey and S. C. Holswade, “Beam shaping: a review,” in Laser Beam Shaping Applications, F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., 269–303 (CRC, 2006).
  6. W.-H. Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36(6), 469–471 (1981).
    [Crossref]
  7. D. Lin and J. R. Leger, “Numerical gradient-index design for coherent mode conversion,” Adv. Opt. Technol. 1, 195–202 (2012).
  8. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
    [Crossref]
  9. A. Žukauskas, I. Matulaitienė, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
    [Crossref]
  10. A. C. Urness, E. D. Moore, K. K. Kamysiak, M. C. Cole, and R. R. McLeod, “Liquid deposition photolithography for submicrometer resolution three-dimensional index structuring with large throughput,” Light Sci. Appl. 2(3), e56 (2013).
    [Crossref]
  11. A. C. Urness, K. Anderson, C. Ye, W. L. Wilson, and R. R. McLeod, “Arbitrary GRIN component fabrication in optically driven diffusive photopolymers,” Opt. Express 23(1), 264–273 (2015).
    [Crossref] [PubMed]
  12. H. R. Wang, M. J. Cima, B. D. Kernan, and E. M. Sachs, “Alumina-doped silica gradient-index (GRIN) lenses by slurry-based three-dimensional printing (S-3DP),” J. Non-Cryst. Solids 349, 360–367 (2004).
    [Crossref]
  13. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
    [Crossref]
  14. T.-C. Poon and T. Kim, Engineering Optics with MATLAB (World Scientific, 2006).
  15. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).
  16. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3(1), 27–29 (1978).
    [Crossref] [PubMed]
  17. A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE 3666, 40–44 (1999).
    [Crossref]
  18. J. R. Leger, “External methods of phase locking and coherent beam addition of diode lasers,” in Surface Emitting Semiconductor Lasers and Arrays, G. A. Evans and J. M. Hammer, eds. (Academic, 1993).
  19. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6(1), 150–162 (2000).
    [Crossref]

2015 (2)

A. Žukauskas, I. Matulaitienė, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

A. C. Urness, K. Anderson, C. Ye, W. L. Wilson, and R. R. McLeod, “Arbitrary GRIN component fabrication in optically driven diffusive photopolymers,” Opt. Express 23(1), 264–273 (2015).
[Crossref] [PubMed]

2013 (1)

A. C. Urness, E. D. Moore, K. K. Kamysiak, M. C. Cole, and R. R. McLeod, “Liquid deposition photolithography for submicrometer resolution three-dimensional index structuring with large throughput,” Light Sci. Appl. 2(3), e56 (2013).
[Crossref]

2012 (1)

D. Lin and J. R. Leger, “Numerical gradient-index design for coherent mode conversion,” Adv. Opt. Technol. 1, 195–202 (2012).

2008 (1)

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[Crossref]

2004 (1)

H. R. Wang, M. J. Cima, B. D. Kernan, and E. M. Sachs, “Alumina-doped silica gradient-index (GRIN) lenses by slurry-based three-dimensional printing (S-3DP),” J. Non-Cryst. Solids 349, 360–367 (2004).
[Crossref]

2000 (1)

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6(1), 150–162 (2000).
[Crossref]

1999 (1)

A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE 3666, 40–44 (1999).
[Crossref]

1996 (1)

1993 (1)

1981 (1)

W.-H. Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36(6), 469–471 (1981).
[Crossref]

1980 (1)

1978 (1)

1976 (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

1965 (1)

Anderson, K.

Cima, M. J.

H. R. Wang, M. J. Cima, B. D. Kernan, and E. M. Sachs, “Alumina-doped silica gradient-index (GRIN) lenses by slurry-based three-dimensional printing (S-3DP),” J. Non-Cryst. Solids 349, 360–367 (2004).
[Crossref]

Cole, M. C.

A. C. Urness, E. D. Moore, K. K. Kamysiak, M. C. Cole, and R. R. McLeod, “Liquid deposition photolithography for submicrometer resolution three-dimensional index structuring with large throughput,” Light Sci. Appl. 2(3), e56 (2013).
[Crossref]

Dickey, F. M.

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

Fienup, J. R.

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

Frieden, B. R.

Gadonas, R.

A. Žukauskas, I. Matulaitienė, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

Gattass, R. R.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

Ghatak, A. K.

A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE 3666, 40–44 (1999).
[Crossref]

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6(1), 150–162 (2000).
[Crossref]

Goyal, I. C.

A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE 3666, 40–44 (1999).
[Crossref]

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6(1), 150–162 (2000).
[Crossref]

Jindal, R.

A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE 3666, 40–44 (1999).
[Crossref]

Kamysiak, K. K.

A. C. Urness, E. D. Moore, K. K. Kamysiak, M. C. Cole, and R. R. McLeod, “Liquid deposition photolithography for submicrometer resolution three-dimensional index structuring with large throughput,” Light Sci. Appl. 2(3), e56 (2013).
[Crossref]

Kernan, B. D.

H. R. Wang, M. J. Cima, B. D. Kernan, and E. M. Sachs, “Alumina-doped silica gradient-index (GRIN) lenses by slurry-based three-dimensional printing (S-3DP),” J. Non-Cryst. Solids 349, 360–367 (2004).
[Crossref]

Lee, W.-H.

W.-H. Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36(6), 469–471 (1981).
[Crossref]

Leger, J. R.

D. Lin and J. R. Leger, “Numerical gradient-index design for coherent mode conversion,” Adv. Opt. Technol. 1, 195–202 (2012).

Lin, D.

D. Lin and J. R. Leger, “Numerical gradient-index design for coherent mode conversion,” Adv. Opt. Technol. 1, 195–202 (2012).

Malinauskas, M.

A. Žukauskas, I. Matulaitienė, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

Matulaitiene, I.

A. Žukauskas, I. Matulaitienė, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

Mazur, E.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[Crossref]

McLeod, R. R.

A. C. Urness, K. Anderson, C. Ye, W. L. Wilson, and R. R. McLeod, “Arbitrary GRIN component fabrication in optically driven diffusive photopolymers,” Opt. Express 23(1), 264–273 (2015).
[Crossref] [PubMed]

A. C. Urness, E. D. Moore, K. K. Kamysiak, M. C. Cole, and R. R. McLeod, “Liquid deposition photolithography for submicrometer resolution three-dimensional index structuring with large throughput,” Light Sci. Appl. 2(3), e56 (2013).
[Crossref]

Moore, E. D.

A. C. Urness, E. D. Moore, K. K. Kamysiak, M. C. Cole, and R. R. McLeod, “Liquid deposition photolithography for submicrometer resolution three-dimensional index structuring with large throughput,” Light Sci. Appl. 2(3), e56 (2013).
[Crossref]

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

Niaura, G.

A. Žukauskas, I. Matulaitienė, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

Paipulas, D.

A. Žukauskas, I. Matulaitienė, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6(1), 150–162 (2000).
[Crossref]

Rhodes, P. W.

Romero, L. A.

Sachs, E. M.

H. R. Wang, M. J. Cima, B. D. Kernan, and E. M. Sachs, “Alumina-doped silica gradient-index (GRIN) lenses by slurry-based three-dimensional printing (S-3DP),” J. Non-Cryst. Solids 349, 360–367 (2004).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6(1), 150–162 (2000).
[Crossref]

Shealy, D. L.

Urness, A. C.

A. C. Urness, K. Anderson, C. Ye, W. L. Wilson, and R. R. McLeod, “Arbitrary GRIN component fabrication in optically driven diffusive photopolymers,” Opt. Express 23(1), 264–273 (2015).
[Crossref] [PubMed]

A. C. Urness, E. D. Moore, K. K. Kamysiak, M. C. Cole, and R. R. McLeod, “Liquid deposition photolithography for submicrometer resolution three-dimensional index structuring with large throughput,” Light Sci. Appl. 2(3), e56 (2013).
[Crossref]

Wang, C.

Wang, H. R.

H. R. Wang, M. J. Cima, B. D. Kernan, and E. M. Sachs, “Alumina-doped silica gradient-index (GRIN) lenses by slurry-based three-dimensional printing (S-3DP),” J. Non-Cryst. Solids 349, 360–367 (2004).
[Crossref]

Wilson, W. L.

Ye, C.

Žukauskas, A.

A. Žukauskas, I. Matulaitienė, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

Adv. Opt. Technol. (1)

D. Lin and J. R. Leger, “Numerical gradient-index design for coherent mode conversion,” Adv. Opt. Technol. 1, 195–202 (2012).

Appl. Opt. (3)

Appl. Phys. (Berl.) (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6(1), 150–162 (2000).
[Crossref]

J. Non-Cryst. Solids (1)

H. R. Wang, M. J. Cima, B. D. Kernan, and E. M. Sachs, “Alumina-doped silica gradient-index (GRIN) lenses by slurry-based three-dimensional printing (S-3DP),” J. Non-Cryst. Solids 349, 360–367 (2004).
[Crossref]

J. Opt. Soc. Am. A (1)

Laser Photonics Rev. (1)

A. Žukauskas, I. Matulaitienė, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

Light Sci. Appl. (1)

A. C. Urness, E. D. Moore, K. K. Kamysiak, M. C. Cole, and R. R. McLeod, “Liquid deposition photolithography for submicrometer resolution three-dimensional index structuring with large throughput,” Light Sci. Appl. 2(3), e56 (2013).
[Crossref]

Nat. Photonics (1)

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[Crossref]

Opt. Commun. (1)

W.-H. Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36(6), 469–471 (1981).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optik (Stuttg.) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

Proc. SPIE (1)

A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE 3666, 40–44 (1999).
[Crossref]

Other (3)

J. R. Leger, “External methods of phase locking and coherent beam addition of diode lasers,” in Surface Emitting Semiconductor Lasers and Arrays, G. A. Evans and J. M. Hammer, eds. (Academic, 1993).

F. M. Dickey and S. C. Holswade, “Beam shaping: a review,” in Laser Beam Shaping Applications, F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., 269–303 (CRC, 2006).

T.-C. Poon and T. Kim, Engineering Optics with MATLAB (World Scientific, 2006).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1 Split-step scheme for determining phase modulations from refractive index. (a) An individual split-step sequence and (b) a cascade of consecutive steps.
Fig. 2
Fig. 2 Method for phase interpolation from ray slopes at axial position zi. (a) Wavefront slopes are obtained from ray angles and (b) the phase profile is obtained at zi from the constructed wavefront.
Fig. 3
Fig. 3 Specified ray trajectories (black) for a Gaussian to flat-top conversion. The intensity profiles (red) are implied by the density of rays at each z.
Fig. 4
Fig. 4 Comparison of two designs for a GRIN element meant to convert a Gaussian beam into a supergaussian. (a), (b) phase retrieval method; (c), (d) geometric design method [7]. Plots (a) and (c) show the refractive index minus background (Δn) evaluated at each of the marked z values (white triangles). Both z and Δn(x) run along the horizontal direction. Positive Δn is to the right, and Δn = 0 at each pertinent reference dot. (b), (d) Intensity profile of the field after wave-based propagation through the associated GRIN element.
Fig. 5
Fig. 5 A GRIN element that performs coherent addition of three Gaussian beams. (a) Rays and field magnitude expressing the desired intensity evolution. The vertical dashed line marks the boundary between the two parts of the GRIN element: an aperture-filling stage (left) followed by a reshaping stage (right). (b) Desired intensity profiles at selected axial positions. (c) Index profile of the desired GRIN element found using phase retrieval. Refractive index minus background (Δn) is plotted at each of the marked z values (white triangles). (d) Irradiance profile of the beam after wave-based propagation through the GRIN element.

Equations (5)

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

β=C r i r o ( λ/ n 0 )L ,
φ i ( x, z i )= k 0 ΔzΔ n i ( x, z i )/ n 0
Δ n i ( x, z i )= n i ( x, z i ) n 0 .
dφ( x, z i )= k 0 tanθ( x, z i )dx.
u( x )=exp[ ( x/a ) m ],

Metrics