Skip to main content

Some problems of optimum focusing in the process of second harmonic generation in nonlinear crystals. Part 1. Mathematical apparatus

Abstract

The problem of the second harmonic generation (SHG) of monochromatic spatially coherent paraxial radiation in homogeneous uniaxial quadratically nonlinear crystals upon focusing of a beam into a crystal by two crossed cylindrical lenses with different focal lengths is considered theoretically. The aim of the investigations was to determine optimum focusing parameters as functions of the main radiation power. The first part of this paper presents a mathematical apparatus necessary for carrying out the planned investigations. A variant of transforming a system of nonlinear wave equations to a form more convenient for further use is proposed. Some important particular cases admitting simpler solutions are considered. An asymptotically exact analytical solution of the SHG problem is presented.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics. (Electromagnetic Wvaes in Nonlinear Dispersive Media (VINITI, Moscow, 1965) [in Russian].

    Google Scholar 

  2. 2.

    A. A. Gulamov, E. A. Ibragimov, and V. I. Redkorechev, Transformation of Laser Radiation Frequency with Limit Efficiency (FAN, Tashkent, 1990) [in Russian].

    Google Scholar 

  3. 3.

    V. G. Dmitriev and L. V. Tarasov, Applied Nonlinear Optics, 2nd ed. (Fizmatlit, Moscow, 2004) [in Russian].

    Google Scholar 

  4. 4.

    G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused gaussian light beams,” J. Appl. Phys. 39 (8), 3597–3639 (1968).

    ADS  Article  Google Scholar 

  5. 5.

    V. V. Kolosov and V. O. Troitskii, “Optimal beam focusing at the second harmonic generation in an uniaxial crystal. Approximation of a preset field,” Atmos. Oceana Opt. 20 (2), 94–100 (2007).

    Google Scholar 

  6. 6.

    S. D. Tvorogov and V. O. Troitskii, “Exact and approximate representations for a laser beam in a uniaxial homogeneous medium,” Atmos. Ocean. Opt. 18 (9), 664–673 (2005).

    Google Scholar 

  7. 7.

    V. O. Troitskii, “Second harmonic generation of laser radiation in uniaxial crystals. Alternate problem solutions in the assumed field approximation,” Atmos. Ocean. Opt. 23 (4), 278–283 (2010).

    Article  Google Scholar 

  8. 8.

    V. V. Kolosov and V. O. Troitskii, “Paraxial approximation for the problem of light beam propagation through a planar-cross stratified medium,” Atmos. Ocean. Opt. 18 (9), 674–679 (2005).

    Google Scholar 

  9. 9.

    V. O. Troitskii, “Second harmonic generation by focusing a beam into a uniaxial crystal with crossed cylindrical lenses. Approximation of a preset field,” Atmos. Ocean. Opt. 19 (8), 668–674 (2006).

    Google Scholar 

  10. 10.

    M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).

    Google Scholar 

  11. 11.

    V. O. Troitskii, “Scalar approximation for second harmonic generation in a uniaxial crystal,” Atmos. Ocean. Opt. 20 (9), 741–751 (2007).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to V. O. Troitskii.

Additional information

Original Russian Text © V.O. Troitskii, 2015, published in Optika Atmosfery i Okeana.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Troitskii, V.O. Some problems of optimum focusing in the process of second harmonic generation in nonlinear crystals. Part 1. Mathematical apparatus. Atmos Ocean Opt 29, 191–198 (2016). https://doi.org/10.1134/S1024856016020123

Download citation

Keywords

  • second harmonics generation
  • uniaxial nonlinear crystal
  • crossed cylindrical lenses
  • optimal focusing