Abstract

We analyze the second-harmonic generation in two-dimensional photonic structures with radially periodic domains created by poling of a nonlinear quadratic crystal. We demonstrate that the parametric conversion of the Gaussian fundamental beam propagating along the axis of the annular structure leads to the axial emission of the second-harmonic field in the form of the radially polarized first-order Bessel beam.

© 2007 Optical Society of America

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References

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  1. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second-harmonic generation -Tuning and tolerances," IEEE J. Quant. Electron. QE-28, 2631-2654 (1992).
    [CrossRef]
  2. T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, "Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3," Opt. Commun. 252, 397-401 (2005).
    [CrossRef]
  3. D. Kasimov, A. Arie, E. Winebrand, G. Rosenman, A. Bruner, P. Shaier, and D. Eger, "Annular symmetry nonlinear frequency converters," Opt. Express 14, 9371-9376 (2006); http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-20-9371.
    [CrossRef] [PubMed]
  4. J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, "Nonlinear physical optics with transversely patternedquasi-phase-matching gratings," IEEE J. Sel. Top. Quantum Electron. 8, 660-664 (2002).
    [CrossRef]
  5. A. R. Tunyagi, M. Ulex, and K. Betzler, "Noncollinear optical frequency doubling in strontium barium niobate," Phys. Rev. Lett. 90, 243901 (2003).
    [CrossRef] [PubMed]
  6. R. Fischer, D. Neshev, S. Saltiel, W. Krolikowski, and Yu. S. Kivshar, "Broadband femtosecond frequency doubling in random media," Appl. Phys. Lett. 89, 191105 (2006).
    [CrossRef]
  7. P. Molina, M. O. Ramirez, J. García-Sol, and L. E. Baus, "Wavelength tunability of non-collinear second harmonic generation processes in Strontium Barium Niobate crystal," European Optical Society Meeting, Paris (2006).
  8. A. R. Tunyagi, "Non-collinear second-harmonic generation in Strontium Barium Niobate," PhD thesis (University of Osnabrück, 2004).
  9. V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
    [CrossRef]
  10. S. P. Tewari, H. Huang, and R. W. Boyd, "Theory of third-harmonic generation using Bessel beams, and selfphase-matching," Phys. Rev. A 54, 2314-2325 (1996).
    [CrossRef] [PubMed]
  11. C. Canalias, V. Pasiskevicius, R. Clemens, and F. Laurell, "Submicron periodically poled flux-grown KTiOPO4," Appl. Phys. Lett. 82, 4233-4235 (2003).
    [CrossRef]
  12. S. Moscovich, A. Arie, R. Urneski, A. Agronin, G. Rosenman, and Y. Rosenwaks, "Noncollinear secondharmonic generation in sub-micrometer-poled RbTiOPO4," Opt. Express 12, 2236-2242 (2004).
    [CrossRef] [PubMed]
  13. M. Houe and P. D. Townsend, "An introduction to methods of periodic poling for second-harmonic generation," J. Phys. D: Appl. Phys. 28, 1747-1763 (1995).
    [CrossRef]
  14. B. F. Johnston and M. J. Withford, "Dynamics of domain inversion in LiNbO3 poled using topographic electrode geometries," Appl. Phys. Lett. 86, 262901 (2005).
    [CrossRef]
  15. J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  16. Z. Bouchal and M. Olivik, "Non-diffractive vector Bessel beams," J. Mod. Optics 42, 1555-1556 (1995).
    [CrossRef]
  17. See, e.g., F. P. Schafer, "On some properties of axicons," Appl. Phys. B 39, 1-8 (1985).
    [CrossRef]
  18. Y. Kozawa and S. Sato, "Generation of a radially polarized laser beam by use of a conical Brewster prism," Opt. Lett. 30, 3063-3065 (2005).
    [CrossRef] [PubMed]
  19. S. P. Tewari, H. Huang, and R. W. Boyd, "Theory of self-phase matching,"Phys. Rev. A 51, R2707-R2710 (1995).
    [CrossRef] [PubMed]
  20. A. Piskarskas, V. Smilgevicius, A. Stabinis, V. Jarutis, V. Pasiskevicius, S. Wang, J. Tellefsen, and F. Laurell, "Noncollinear second-harmonic generation in periodically poled KTiOPO4 excited by the Bessel beam," Opt. Lett. 24, 1053-1055 (1999).
    [CrossRef]
  21. V. García-Chavez, D. McGloin, H. Melville, W. Sibbett and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
    [CrossRef]

2006 (2)

2005 (3)

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, "Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3," Opt. Commun. 252, 397-401 (2005).
[CrossRef]

B. F. Johnston and M. J. Withford, "Dynamics of domain inversion in LiNbO3 poled using topographic electrode geometries," Appl. Phys. Lett. 86, 262901 (2005).
[CrossRef]

Y. Kozawa and S. Sato, "Generation of a radially polarized laser beam by use of a conical Brewster prism," Opt. Lett. 30, 3063-3065 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (2)

C. Canalias, V. Pasiskevicius, R. Clemens, and F. Laurell, "Submicron periodically poled flux-grown KTiOPO4," Appl. Phys. Lett. 82, 4233-4235 (2003).
[CrossRef]

A. R. Tunyagi, M. Ulex, and K. Betzler, "Noncollinear optical frequency doubling in strontium barium niobate," Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef] [PubMed]

2002 (2)

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, "Nonlinear physical optics with transversely patternedquasi-phase-matching gratings," IEEE J. Sel. Top. Quantum Electron. 8, 660-664 (2002).
[CrossRef]

V. García-Chavez, D. McGloin, H. Melville, W. Sibbett and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef]

1999 (1)

1998 (1)

V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

1996 (1)

S. P. Tewari, H. Huang, and R. W. Boyd, "Theory of third-harmonic generation using Bessel beams, and selfphase-matching," Phys. Rev. A 54, 2314-2325 (1996).
[CrossRef] [PubMed]

1995 (3)

M. Houe and P. D. Townsend, "An introduction to methods of periodic poling for second-harmonic generation," J. Phys. D: Appl. Phys. 28, 1747-1763 (1995).
[CrossRef]

S. P. Tewari, H. Huang, and R. W. Boyd, "Theory of self-phase matching,"Phys. Rev. A 51, R2707-R2710 (1995).
[CrossRef] [PubMed]

Z. Bouchal and M. Olivik, "Non-diffractive vector Bessel beams," J. Mod. Optics 42, 1555-1556 (1995).
[CrossRef]

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second-harmonic generation -Tuning and tolerances," IEEE J. Quant. Electron. QE-28, 2631-2654 (1992).
[CrossRef]

1987 (1)

1985 (1)

See, e.g., F. P. Schafer, "On some properties of axicons," Appl. Phys. B 39, 1-8 (1985).
[CrossRef]

Agronin, A.

Arie, A.

Berger, V.

V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

Betzler, K.

A. R. Tunyagi, M. Ulex, and K. Betzler, "Noncollinear optical frequency doubling in strontium barium niobate," Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef] [PubMed]

Bouchal, Z.

Z. Bouchal and M. Olivik, "Non-diffractive vector Bessel beams," J. Mod. Optics 42, 1555-1556 (1995).
[CrossRef]

Boyd, R. W.

S. P. Tewari, H. Huang, and R. W. Boyd, "Theory of third-harmonic generation using Bessel beams, and selfphase-matching," Phys. Rev. A 54, 2314-2325 (1996).
[CrossRef] [PubMed]

S. P. Tewari, H. Huang, and R. W. Boyd, "Theory of self-phase matching,"Phys. Rev. A 51, R2707-R2710 (1995).
[CrossRef] [PubMed]

Bruner, A.

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second-harmonic generation -Tuning and tolerances," IEEE J. Quant. Electron. QE-28, 2631-2654 (1992).
[CrossRef]

Canalias, C.

C. Canalias, V. Pasiskevicius, R. Clemens, and F. Laurell, "Submicron periodically poled flux-grown KTiOPO4," Appl. Phys. Lett. 82, 4233-4235 (2003).
[CrossRef]

Cheng, B.

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, "Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3," Opt. Commun. 252, 397-401 (2005).
[CrossRef]

Clemens, R.

C. Canalias, V. Pasiskevicius, R. Clemens, and F. Laurell, "Submicron periodically poled flux-grown KTiOPO4," Appl. Phys. Lett. 82, 4233-4235 (2003).
[CrossRef]

Dholakia, K.

V. García-Chavez, D. McGloin, H. Melville, W. Sibbett and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef]

Durnin, J.

Eger, D.

Fejer, M. M.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, "Nonlinear physical optics with transversely patternedquasi-phase-matching gratings," IEEE J. Sel. Top. Quantum Electron. 8, 660-664 (2002).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second-harmonic generation -Tuning and tolerances," IEEE J. Quant. Electron. QE-28, 2631-2654 (1992).
[CrossRef]

Fischer, R.

R. Fischer, D. Neshev, S. Saltiel, W. Krolikowski, and Yu. S. Kivshar, "Broadband femtosecond frequency doubling in random media," Appl. Phys. Lett. 89, 191105 (2006).
[CrossRef]

García-Chavez, V.

V. García-Chavez, D. McGloin, H. Melville, W. Sibbett and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef]

Houe, M.

M. Houe and P. D. Townsend, "An introduction to methods of periodic poling for second-harmonic generation," J. Phys. D: Appl. Phys. 28, 1747-1763 (1995).
[CrossRef]

Huang, H.

S. P. Tewari, H. Huang, and R. W. Boyd, "Theory of third-harmonic generation using Bessel beams, and selfphase-matching," Phys. Rev. A 54, 2314-2325 (1996).
[CrossRef] [PubMed]

S. P. Tewari, H. Huang, and R. W. Boyd, "Theory of self-phase matching,"Phys. Rev. A 51, R2707-R2710 (1995).
[CrossRef] [PubMed]

Hum, D. S.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, "Nonlinear physical optics with transversely patternedquasi-phase-matching gratings," IEEE J. Sel. Top. Quantum Electron. 8, 660-664 (2002).
[CrossRef]

Jarutis, V.

Johnston, B. F.

B. F. Johnston and M. J. Withford, "Dynamics of domain inversion in LiNbO3 poled using topographic electrode geometries," Appl. Phys. Lett. 86, 262901 (2005).
[CrossRef]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second-harmonic generation -Tuning and tolerances," IEEE J. Quant. Electron. QE-28, 2631-2654 (1992).
[CrossRef]

Kasimov, D.

Kivshar, Yu. S.

R. Fischer, D. Neshev, S. Saltiel, W. Krolikowski, and Yu. S. Kivshar, "Broadband femtosecond frequency doubling in random media," Appl. Phys. Lett. 89, 191105 (2006).
[CrossRef]

Kozawa, Y.

Krolikowski, W.

R. Fischer, D. Neshev, S. Saltiel, W. Krolikowski, and Yu. S. Kivshar, "Broadband femtosecond frequency doubling in random media," Appl. Phys. Lett. 89, 191105 (2006).
[CrossRef]

Kurz, J. R.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, "Nonlinear physical optics with transversely patternedquasi-phase-matching gratings," IEEE J. Sel. Top. Quantum Electron. 8, 660-664 (2002).
[CrossRef]

Laurell, F.

Ma, B.

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, "Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3," Opt. Commun. 252, 397-401 (2005).
[CrossRef]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second-harmonic generation -Tuning and tolerances," IEEE J. Quant. Electron. QE-28, 2631-2654 (1992).
[CrossRef]

McGloin, D.

V. García-Chavez, D. McGloin, H. Melville, W. Sibbett and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef]

Melville, H.

V. García-Chavez, D. McGloin, H. Melville, W. Sibbett and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef]

Moscovich, S.

Neshev, D.

R. Fischer, D. Neshev, S. Saltiel, W. Krolikowski, and Yu. S. Kivshar, "Broadband femtosecond frequency doubling in random media," Appl. Phys. Lett. 89, 191105 (2006).
[CrossRef]

Ni, P.

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, "Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3," Opt. Commun. 252, 397-401 (2005).
[CrossRef]

Olivik, M.

Z. Bouchal and M. Olivik, "Non-diffractive vector Bessel beams," J. Mod. Optics 42, 1555-1556 (1995).
[CrossRef]

Pasiskevicius, V.

Piskarskas, A.

Rosenman, G.

Rosenwaks, Y.

Saltiel, S.

R. Fischer, D. Neshev, S. Saltiel, W. Krolikowski, and Yu. S. Kivshar, "Broadband femtosecond frequency doubling in random media," Appl. Phys. Lett. 89, 191105 (2006).
[CrossRef]

Saltzman, A. J.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, "Nonlinear physical optics with transversely patternedquasi-phase-matching gratings," IEEE J. Sel. Top. Quantum Electron. 8, 660-664 (2002).
[CrossRef]

Sato, S.

Schafer, F. P.

See, e.g., F. P. Schafer, "On some properties of axicons," Appl. Phys. B 39, 1-8 (1985).
[CrossRef]

Schober, A. M.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, "Nonlinear physical optics with transversely patternedquasi-phase-matching gratings," IEEE J. Sel. Top. Quantum Electron. 8, 660-664 (2002).
[CrossRef]

Shaier, P.

Sheng, Y.

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, "Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3," Opt. Commun. 252, 397-401 (2005).
[CrossRef]

Sibbett, W.

V. García-Chavez, D. McGloin, H. Melville, W. Sibbett and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef]

Smilgevicius, V.

Stabinis, A.

Tellefsen, J.

Tewari, S. P.

S. P. Tewari, H. Huang, and R. W. Boyd, "Theory of third-harmonic generation using Bessel beams, and selfphase-matching," Phys. Rev. A 54, 2314-2325 (1996).
[CrossRef] [PubMed]

S. P. Tewari, H. Huang, and R. W. Boyd, "Theory of self-phase matching,"Phys. Rev. A 51, R2707-R2710 (1995).
[CrossRef] [PubMed]

Townsend, P. D.

M. Houe and P. D. Townsend, "An introduction to methods of periodic poling for second-harmonic generation," J. Phys. D: Appl. Phys. 28, 1747-1763 (1995).
[CrossRef]

Tunyagi, A. R.

A. R. Tunyagi, M. Ulex, and K. Betzler, "Noncollinear optical frequency doubling in strontium barium niobate," Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef] [PubMed]

Ulex, M.

A. R. Tunyagi, M. Ulex, and K. Betzler, "Noncollinear optical frequency doubling in strontium barium niobate," Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef] [PubMed]

Urneski, R.

Wang, S.

Wang, T.

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, "Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3," Opt. Commun. 252, 397-401 (2005).
[CrossRef]

Winebrand, E.

Withford, M. J.

B. F. Johnston and M. J. Withford, "Dynamics of domain inversion in LiNbO3 poled using topographic electrode geometries," Appl. Phys. Lett. 86, 262901 (2005).
[CrossRef]

Zhang, D.

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, "Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3," Opt. Commun. 252, 397-401 (2005).
[CrossRef]

Appl. Phys. B (1)

See, e.g., F. P. Schafer, "On some properties of axicons," Appl. Phys. B 39, 1-8 (1985).
[CrossRef]

Appl. Phys. Lett. (3)

B. F. Johnston and M. J. Withford, "Dynamics of domain inversion in LiNbO3 poled using topographic electrode geometries," Appl. Phys. Lett. 86, 262901 (2005).
[CrossRef]

R. Fischer, D. Neshev, S. Saltiel, W. Krolikowski, and Yu. S. Kivshar, "Broadband femtosecond frequency doubling in random media," Appl. Phys. Lett. 89, 191105 (2006).
[CrossRef]

C. Canalias, V. Pasiskevicius, R. Clemens, and F. Laurell, "Submicron periodically poled flux-grown KTiOPO4," Appl. Phys. Lett. 82, 4233-4235 (2003).
[CrossRef]

IEEE J. Quant. Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second-harmonic generation -Tuning and tolerances," IEEE J. Quant. Electron. QE-28, 2631-2654 (1992).
[CrossRef]

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

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, "Nonlinear physical optics with transversely patternedquasi-phase-matching gratings," IEEE J. Sel. Top. Quantum Electron. 8, 660-664 (2002).
[CrossRef]

J. Mod. Optics (1)

Z. Bouchal and M. Olivik, "Non-diffractive vector Bessel beams," J. Mod. Optics 42, 1555-1556 (1995).
[CrossRef]

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

J. Phys. D: Appl. Phys. (1)

M. Houe and P. D. Townsend, "An introduction to methods of periodic poling for second-harmonic generation," J. Phys. D: Appl. Phys. 28, 1747-1763 (1995).
[CrossRef]

Nature (1)

V. García-Chavez, D. McGloin, H. Melville, W. Sibbett and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef]

Opt. Commun. (1)

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, "Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3," Opt. Commun. 252, 397-401 (2005).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (2)

S. P. Tewari, H. Huang, and R. W. Boyd, "Theory of self-phase matching,"Phys. Rev. A 51, R2707-R2710 (1995).
[CrossRef] [PubMed]

S. P. Tewari, H. Huang, and R. W. Boyd, "Theory of third-harmonic generation using Bessel beams, and selfphase-matching," Phys. Rev. A 54, 2314-2325 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

A. R. Tunyagi, M. Ulex, and K. Betzler, "Noncollinear optical frequency doubling in strontium barium niobate," Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef] [PubMed]

Other (2)

P. Molina, M. O. Ramirez, J. García-Sol, and L. E. Baus, "Wavelength tunability of non-collinear second harmonic generation processes in Strontium Barium Niobate crystal," European Optical Society Meeting, Paris (2006).

A. R. Tunyagi, "Non-collinear second-harmonic generation in Strontium Barium Niobate," PhD thesis (University of Osnabrück, 2004).

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Figures (4)

Fig. 1.
Fig. 1.

Schematic of the parametric generation of the axial Bessel beam with the double frequency 2ω. NLC denotes a quadratic nonlinear crystal with poled domains of the radially symmetric periodic modulation of the second-order nonlinearity.

Fig. 2.
Fig. 2.

(a). Phase matching diagram for the second-harmonic generation in the medium with the transverse modulation of the second-order nonlinearity. Notations are: Gm - quasi-phase matching vectors, P 2’medium polarization at the doubled frequency. (b) Emitted cone of the radially polarized second-harmonic radiation, presented as a sum of infinite plane waves propagating at the angle α with respect to the propagation axis z.

Fig. 3.
Fig. 3.

Transverse structure of the second-harmonic field described by the first-order Bessel function (blue), and the corresponding annular domain grating (red) for three different values of the phase offset δ defined in Eq. (1): (a) δ = 0, (b) δ = π/4, and (c) δ = π/2.

Fig. 4.
Fig. 4.

Dependence of J TPM on the period Λ of the circular grating in Strontium Barium Niobate crystal for different values of the phase offset δ and fixed angle α defined from the condition 2k 1 = k 2 cosα. The radius of the fundamental beam w 01 = 40μm.

Equations (12)

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

g ( ρ ) = sgn [ cos ( 2 π ρ Λ + δ ) ] ,
E 0 ω ( ρ , z ) = u 2 [ A ( z ) e i k 1 z exp ( ρ 2 w 01 2 ) + c . c ] ,
P z 2 ω = d 31 E x ω E x ω + d 32 E y ω E y ω .
P z 2 ω = d 32 ( E 0 ω ) 2 .
k 2 2 k 1 = G m ,
E 2 ω P z 2 ω sin α = ± d 32 ( E 0 ω ) 2 sin α ,
E 2 ω ( ρ , z ) = S ( z ) e i k 2 z z 0 2 π u ρ ( ϕ ) e i k 2 ρ ρ cos ( ϕ φ ) d ϕ ,
E 2 ω ( ρ , z ) = 2 πS ( z ) e i k 2 z z [ i J 1 ( k 2 ρ sin α ) u ρ tan α J 0 ( k 2 ρ sin α ) u z ] ,
E 2 ω ( ρ , z ) = [ u ρ 2 S ( z ) J 1 ( k 2 ρ ρ ) exp ( i k 2 z z ) + c . c . ] .
I 2 ω ( ρ , L ) = 8 π 2 d 32 2 ( I ω ) 2 L 2 ε 0 c λ 0 2 n 2 n 1 2 tan α J 1 ( k 2 ρ sin α ) J TPM 2 ,
J TPM = 1 T 0 w m J 1 ( r ) exp ( 4 r 2 k 2 ρ 2 w 01 2 ) g ( r k 2 ρ ) dr ,
T = 0 w m r J 1 2 ( r ) exp ( 2 r 2 k 2 ρ 2 w 01 2 ) dr

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