Abstract

The phenomena of beam-steering and negative ray refraction in a graded 2D photonic crystal structure interface with plane tapered transitions to homogeneous waveguide regions is used for efficient far-field refocusing of highly divergent Gaussian beams. It is shown by numerical simulation that a 14μm long photonic crystal structure is able to refocus a small-spot Gaussian beam with an efficiency of about 90%.

©2005 Optical Society of America

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References

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  1. A. Schuster, An Introduction to the Theory of Optics, Edward Arnold, London (1904).
  2. V.G. Veselago, “The electrodynamics of substances with simultaneous negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [Crossref]
  3. R.A. Silin, “Optical properties of artificial dielectrics,” Radiophysics and Quantum Electronics 15, 615–624 (1972).
    [Crossref]
  4. R. A. Silin, “Possibility of creating plane-parallel lenses,” Opt. Spektrosk. 44, 189–191 (1978).
  5. R. Zengerle, PhD thesis, University of Stuttgart, Germany (1979).
  6. R. Ulrich and R. Zengerle, “Optical Bloch waves in periodic planar waveguides,” Integrated and Guided-Wave Optics, Incline Village, NV, USA 1980, TuB1/1-4 (1980).
  7. R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Optics 34, 1589–1617 (1987).
    [Crossref]
  8. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059 (1987).
    [Crossref] [PubMed]
  9. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
    [Crossref]
  10. P. St. J. Russell, “Interference of integrated Floquet-Bloch waves,” Phys. Rev. A 33, 3232–3242 (1986).
    [Crossref] [PubMed]
  11. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic bad gap,” Phys. Rev. B 62, 10696–10705 (2000).
    [Crossref]
  12. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
    [Crossref]
  13. A. Martinez and J. Marti, “Analysis of wave focusing inside a negative-index photonic-crystal slab,” Opt. Express 13, 2858–2868 (2005).
    [Crossref] [PubMed]
  14. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [Crossref] [PubMed]
  15. A. Berrier, M. Mulot, M. Swillo, M. Qiu, L Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
    [Crossref] [PubMed]
  16. R. Zengerle, “Polarization splitter based on beam steering in periodic planar optical waveguides,” Electron. Lett. 24, 11–12 (1988).
    [Crossref]
  17. X. Wang, Z. F. Ren, and K. Kempa, “Unrestricted superlensing in a triangular two-dimensional photonic crystal,” Opt. Express 12, 2919–2924 (2004).
    [Crossref] [PubMed]
  18. S. Xiao, M. Qiu, Z. Ruan, and S. He, “Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction”, Appl. Phys. Lett 85, 4269–4271 (2004).
    [Crossref]
  19. Z.-Y. Li and L.-L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,” Phys. Rev. B 68245110 (2003).
    [Crossref]
  20. A. Martinez, H. Miguez, A. Griol, and J. Marti, “Experimental and theoretical analysis of self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004).
    [Crossref]
  21. O. Leminger, “Wave-vector diagrams for two-dimensional photonic crystals,” Opt. and Quant. Electron. 34, 435–443 (2002).
    [Crossref]

2005 (1)

2004 (4)

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[Crossref] [PubMed]

X. Wang, Z. F. Ren, and K. Kempa, “Unrestricted superlensing in a triangular two-dimensional photonic crystal,” Opt. Express 12, 2919–2924 (2004).
[Crossref] [PubMed]

S. Xiao, M. Qiu, Z. Ruan, and S. He, “Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction”, Appl. Phys. Lett 85, 4269–4271 (2004).
[Crossref]

A. Martinez, H. Miguez, A. Griol, and J. Marti, “Experimental and theoretical analysis of self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004).
[Crossref]

2003 (1)

Z.-Y. Li and L.-L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,” Phys. Rev. B 68245110 (2003).
[Crossref]

2002 (2)

O. Leminger, “Wave-vector diagrams for two-dimensional photonic crystals,” Opt. and Quant. Electron. 34, 435–443 (2002).
[Crossref]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[Crossref]

2000 (2)

J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref] [PubMed]

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic bad gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

1998 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

1988 (1)

R. Zengerle, “Polarization splitter based on beam steering in periodic planar optical waveguides,” Electron. Lett. 24, 11–12 (1988).
[Crossref]

1987 (2)

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Optics 34, 1589–1617 (1987).
[Crossref]

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

1986 (1)

P. St. J. Russell, “Interference of integrated Floquet-Bloch waves,” Phys. Rev. A 33, 3232–3242 (1986).
[Crossref] [PubMed]

1978 (1)

R. A. Silin, “Possibility of creating plane-parallel lenses,” Opt. Spektrosk. 44, 189–191 (1978).

1972 (1)

R.A. Silin, “Optical properties of artificial dielectrics,” Radiophysics and Quantum Electronics 15, 615–624 (1972).
[Crossref]

1968 (1)

V.G. Veselago, “The electrodynamics of substances with simultaneous negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

Anand, S.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[Crossref] [PubMed]

Berrier, A.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[Crossref] [PubMed]

Griol, A.

A. Martinez, H. Miguez, A. Griol, and J. Marti, “Experimental and theoretical analysis of self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004).
[Crossref]

He, S.

S. Xiao, M. Qiu, Z. Ruan, and S. He, “Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction”, Appl. Phys. Lett 85, 4269–4271 (2004).
[Crossref]

Joannopoulos, J. D.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[Crossref]

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[Crossref]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Kempa, K.

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Leminger, O.

O. Leminger, “Wave-vector diagrams for two-dimensional photonic crystals,” Opt. and Quant. Electron. 34, 435–443 (2002).
[Crossref]

Li, Z.-Y.

Z.-Y. Li and L.-L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,” Phys. Rev. B 68245110 (2003).
[Crossref]

Lin, L.-L.

Z.-Y. Li and L.-L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,” Phys. Rev. B 68245110 (2003).
[Crossref]

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[Crossref]

Marti, J.

A. Martinez and J. Marti, “Analysis of wave focusing inside a negative-index photonic-crystal slab,” Opt. Express 13, 2858–2868 (2005).
[Crossref] [PubMed]

A. Martinez, H. Miguez, A. Griol, and J. Marti, “Experimental and theoretical analysis of self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004).
[Crossref]

Martinez, A.

A. Martinez and J. Marti, “Analysis of wave focusing inside a negative-index photonic-crystal slab,” Opt. Express 13, 2858–2868 (2005).
[Crossref] [PubMed]

A. Martinez, H. Miguez, A. Griol, and J. Marti, “Experimental and theoretical analysis of self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004).
[Crossref]

Miguez, H.

A. Martinez, H. Miguez, A. Griol, and J. Marti, “Experimental and theoretical analysis of self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004).
[Crossref]

Mulot, M.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[Crossref] [PubMed]

Notomi, M.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic bad gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Pendry, J. B.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[Crossref]

J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref] [PubMed]

Qiu, M.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[Crossref] [PubMed]

S. Xiao, M. Qiu, Z. Ruan, and S. He, “Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction”, Appl. Phys. Lett 85, 4269–4271 (2004).
[Crossref]

Ren, Z. F.

Ruan, Z.

S. Xiao, M. Qiu, Z. Ruan, and S. He, “Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction”, Appl. Phys. Lett 85, 4269–4271 (2004).
[Crossref]

Russell, P. St. J.

P. St. J. Russell, “Interference of integrated Floquet-Bloch waves,” Phys. Rev. A 33, 3232–3242 (1986).
[Crossref] [PubMed]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Schuster, A.

A. Schuster, An Introduction to the Theory of Optics, Edward Arnold, London (1904).

Silin, R. A.

R. A. Silin, “Possibility of creating plane-parallel lenses,” Opt. Spektrosk. 44, 189–191 (1978).

Silin, R.A.

R.A. Silin, “Optical properties of artificial dielectrics,” Radiophysics and Quantum Electronics 15, 615–624 (1972).
[Crossref]

Swillo, M.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[Crossref] [PubMed]

Talneau, A.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[Crossref] [PubMed]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Thylén, L

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[Crossref] [PubMed]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Ulrich, R.

R. Ulrich and R. Zengerle, “Optical Bloch waves in periodic planar waveguides,” Integrated and Guided-Wave Optics, Incline Village, NV, USA 1980, TuB1/1-4 (1980).

Veselago, V.G.

V.G. Veselago, “The electrodynamics of substances with simultaneous negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

Wang, X.

Xiao, S.

S. Xiao, M. Qiu, Z. Ruan, and S. He, “Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction”, Appl. Phys. Lett 85, 4269–4271 (2004).
[Crossref]

Yablonovitch, E.

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

Zengerle, R.

R. Zengerle, “Polarization splitter based on beam steering in periodic planar optical waveguides,” Electron. Lett. 24, 11–12 (1988).
[Crossref]

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Optics 34, 1589–1617 (1987).
[Crossref]

R. Ulrich and R. Zengerle, “Optical Bloch waves in periodic planar waveguides,” Integrated and Guided-Wave Optics, Incline Village, NV, USA 1980, TuB1/1-4 (1980).

R. Zengerle, PhD thesis, University of Stuttgart, Germany (1979).

Appl. Phys. Lett (1)

S. Xiao, M. Qiu, Z. Ruan, and S. He, “Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction”, Appl. Phys. Lett 85, 4269–4271 (2004).
[Crossref]

Electron. Lett. (1)

R. Zengerle, “Polarization splitter based on beam steering in periodic planar optical waveguides,” Electron. Lett. 24, 11–12 (1988).
[Crossref]

J. Mod. Optics (1)

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Optics 34, 1589–1617 (1987).
[Crossref]

Opt. and Quant. Electron. (1)

O. Leminger, “Wave-vector diagrams for two-dimensional photonic crystals,” Opt. and Quant. Electron. 34, 435–443 (2002).
[Crossref]

Opt. Express (2)

Opt. Spektrosk. (1)

R. A. Silin, “Possibility of creating plane-parallel lenses,” Opt. Spektrosk. 44, 189–191 (1978).

Phys. Rev. A (1)

P. St. J. Russell, “Interference of integrated Floquet-Bloch waves,” Phys. Rev. A 33, 3232–3242 (1986).
[Crossref] [PubMed]

Phys. Rev. B (5)

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic bad gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[Crossref]

Z.-Y. Li and L.-L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,” Phys. Rev. B 68245110 (2003).
[Crossref]

A. Martinez, H. Miguez, A. Griol, and J. Marti, “Experimental and theoretical analysis of self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Phys. Rev. Lett. (3)

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref] [PubMed]

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[Crossref] [PubMed]

Radiophysics and Quantum Electronics (1)

R.A. Silin, “Optical properties of artificial dielectrics,” Radiophysics and Quantum Electronics 15, 615–624 (1972).
[Crossref]

Sov. Phys. Usp. (1)

V.G. Veselago, “The electrodynamics of substances with simultaneous negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

Other (3)

A. Schuster, An Introduction to the Theory of Optics, Edward Arnold, London (1904).

R. Zengerle, PhD thesis, University of Stuttgart, Germany (1979).

R. Ulrich and R. Zengerle, “Optical Bloch waves in periodic planar waveguides,” Integrated and Guided-Wave Optics, Incline Village, NV, USA 1980, TuB1/1-4 (1980).

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Figures (10)

Fig. 1.
Fig. 1. Principle of refocusing in real space and in wave-vector space explained in a WVD (only the central and outermost ray paths are drawn). The WVD is normalized to the grating constant. (a, b) homogeneous (c, d) inhomogeneous photonic crystal at a normalized frequency of 0.21. In (c, d) we use three different values of dnback (0.43, 0.45 and 0.5 for the inner contour curve).
Fig. 2.
Fig. 2. (a) Wave-vector diagram: Detail oriented in the launch direction at different background refractive index differences (dnback = 0.0; 0.2; 0.4) at constant wavelength of 1345nm. (b) Band diagram including the used frequency of 0.21. (c) Shift of the (real) beam waist from z = -20 μm to a virtual beam waist at z = -10μm due to “wide-angle self-collimation” at dnback = 0.2. (d) Shift of the image beam waist to the second transition by negative ray refraction.
Fig. 3.
Fig. 3. (a) Negative ray refraction in a homogeneous 2D-photonic crystal, (b) Negative ray refraction and ray redirection in a 2-section inhomogeneous 2D photonic crystal; dashed line: Boundary for different background indices.
Fig. 4.
Fig. 4. (a) Refractive index profile used for modeling the tapered photonic crystal transitions using 9 rows of dielectric columns: Generalized layout with a refractive index of the surrounding medium nhom ≥ 1 (b) Simulation results for the power reflection at different angles of incidence with respect to the x-axis and nhom = 1 . Three cases are drawn: Step-like transition, linear and nonlinear taper functions with respect to the change of the refractive index of the columns.
Fig. 5.
Fig. 5. Schematic layout of an inhomogeneous 2D photonic crystal planar lens with tapered transitions to the surrounding homogeneous medium along the z-axis. The background index in the colored regions can be individually changed by different values of dnback , forming lateral index inhomogeneities in the periodic regions.
Fig. 6.
Fig. 6. Homogeneous photonic crystal lens with straight boundaries and abrupt (step-like) transitions. Focusing efficiency: 19%. (a) Field map (b) Intensity trace along the z-axis at x=0.
Fig. 7.
Fig. 7. Homogeneous photonic crystal lens with tapered transitions. Focusing efficiency: 57.5%
Fig. 8.
Fig. 8. Inhomogeneous photonic crystal with tapered transitions: Focusing efficiency: 90%
Fig. 9.
Fig. 9. Inhomogeneous photonic crystal together with tapered transitions to surrounding media with n = 2.0. Spot-size = 0.8 μm. Focusing efficiency: 85 %
Fig. 10.
Fig. 10. (a) Relevant detail of the WVD oriented in the direction of a grating diagonal. Equivalence between changes in the background index and the radius of the dielectric columns. The solid curves denote changes in the columns radii whereas the dashed curves represent changes in the background index. (b) Relation between changes in background index and variations in the radii of the dielectric columns.

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