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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 12 — Jun. 14, 2004
  • pp: 2603–2609
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Finite-difference time-domain analysis of self-focusing in a nonlinear Kerr film

Hyun-Ho Lee, Kyu-Min Chae, Sang-Youp Yim, and Seung-Han Park  »View Author Affiliations


Optics Express, Vol. 12, Issue 12, pp. 2603-2609 (2004)
http://dx.doi.org/10.1364/OPEX.12.002603


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Abstract

By using a finite-difference time-domain method, we analyze self-focusing effects in a nonlinear Kerr film and demonstrate that the near-field intensity distribution at the film surface can reach a stable state at only a few hundred femtoseconds after the incidence of the beam. Our simulations also show that the formation of multiple filamentations in the near-field is quite sensitive to the thickness of the nonlinear film and the power of the laser beam, strongly indicating the existence of nonlinear Fabry-Perot interference effects of the linearly polarized incident light.

© 2004 Optical Society of America

1. Introduction

Self-focusing, a well-known nonlinear phenomenon, has been extensively investigated, since the intensity-dependent refractive-index gradient may effectively reduce the diffraction limited spot-size [1

1. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479 (1964). [CrossRef]

,2

2. V. I. Talanov, “Self-focusing of waves in nonlinear media,” JETP Lett. 2, 138 (1965).

]. Theoretically, the beam propagation method (BPM)[3

3. J. A. Fleck Jr. and P. L. Kelley, “Temporal aspect of the self-focusing of optical beams,” Appl. Phys. Lett. 15, 313 (1969). [CrossRef]

,4

4. M. D. Feit and J. A. Fleck Jr., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633 (1988). [CrossRef]

], nonlinear Schrödinger equation (NLSE) [5

5. K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the townes profile,” Phys. Rev. Lett. 90, 203902-1 (2003). [CrossRef]

9

9. G. Fibich, W. Ren, and X.-P. Wang, “Numerical simulations of self-focusing of ultrafast laser pulses.” Phys. Rev. E 67, 056603 (2003). [CrossRef]

], and finite-difference time-domain (FDTD) method [10

10. R. M. Joseph and A. Taflove, “Spatial soliton deflection mechanism indicated by FDTD Maxwell’s equations modeling,” IEEE Photon. Tech. Lett. 6, 1251 (1994). [CrossRef]

,11

11. R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” J. Opt. Soc. Am. B 10, 186 (1993). [CrossRef]

] etc. have been employed to understand the temporal and spatial electromagnetic-field variations in nonlinear bulk media, including the self-focusing effect. By using a non-paraxial BPM based on the Helmholtz equation, Feit and Fleck [4

4. M. D. Feit and J. A. Fleck Jr., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633 (1988). [CrossRef]

] reported that the non-paraxial treatment of the nonlinear process limits the decrease in beam size to the order of one optical wavelength, leading to multiple foci, beam breakup and filament formation. Fibich and Ilan [7

7. G. Fibich and B. Ilan, “Multiple filamentation of circularly polarized beams,” Phys. Rev. Lett. 89, 013901-1 (2002). [CrossRef]

] showed that multiple filamentation can be suppressed for the circularly polarized beams in a Kerr medium using a scalar equation to describe self-focusing in the presence of vectorial and non-paraxial effects. The finite-difference time-domain (FDTD) method, which represents the electromagnetic-field vectors in the time domain without paraxial approximation, has also been utilized to study the complex nonlinear phenomenon. Especially, the time-dependent features of nonlinear material, such as the femtosecond optical pulse propagation in a nonlinear Kerr medium [11

11. R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” J. Opt. Soc. Am. B 10, 186 (1993). [CrossRef]

] and the formation of temporal solitons [10

10. R. M. Joseph and A. Taflove, “Spatial soliton deflection mechanism indicated by FDTD Maxwell’s equations modeling,” IEEE Photon. Tech. Lett. 6, 1251 (1994). [CrossRef]

], have been investigated with the FDTD method, since it is a convenient tool to describe the temporal dynamics of nonlinear materials.

The self-focusing phenomenon, including beam breakups and multiple filamentations, has been experimentally observed in bulk medium only for sufficiently intense laser beams. Recently, however, it has been shown that self-focusing effects can be directly observed in nonlinear thin films with a low-power continuous-wave (CW) laser by using a near-field scanning optical microscope [12

12. K. B. Song, J. Lee, J. H. Kim, K. Cho, and S. K. Kim, “Direct observation of self-focusing with subdiffraction limited resolution using near-field scanning optical microscope,” Phy. Rev. Lett. 85, 3842 (2000) [CrossRef]

14

14. J. Tominaga, T. Nakano, and N. Atoda, “An approach for recording and readout beyond the diffraction limit with an Sb thin film,” Appl. Phys. Lett. 73, 2078 (1998). [CrossRef]

]. Song et al. [12

12. K. B. Song, J. Lee, J. H. Kim, K. Cho, and S. K. Kim, “Direct observation of self-focusing with subdiffraction limited resolution using near-field scanning optical microscope,” Phy. Rev. Lett. 85, 3842 (2000) [CrossRef]

] showed that filaments with a minimum size of 0.3 µ m were observed in 1.7 µ m thick arsenic trisulfide (As2S3) films with a 1.6 mW laser diode. Choi et al. [13

13. Y. Choi, J. H. Park, M. R. Kim, W. Jhe, and B. K. Rhee, “Direct observation of self-focusing near the diffraction limit in polycrystalline silicon film,” Appl. Phys. Lett. 78, 856 (2001) [CrossRef]

] reported that the optically focused beam size can be decreased below the diffraction-limit in a 0.3 µ m thick polycrystalline silicon (p-Si) film in the near-field region. Since self-focusing in thin films have more complex features than that in bulk media due to multiple reflections, few theoretical approaches for the nonlinear processes in the thin films have been discussed. In this letter, we present a three-dimensional (3D) FDTD analysis of the self-focusing effects in the nonlinear Kerr thin-films at various configurations. Our simulations show that the nonlinear Kerr film exhibits quite different filamentation features than that of bulk materials due to nonlinear Fabry-Perot interference of linearly polarized incident light.

2. FDTD analysis of self-focusing in a nonlinear Kerr film

The interaction of a beam of light and a nonlinear optical medium can be expressed in terms of the nonlinear polarization. In the time domain, the nonlinear electric polarization (in MKS units) is given by [11

11. R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” J. Opt. Soc. Am. B 10, 186 (1993). [CrossRef]

]

PNL(x,t)=ε0χ(3)(tτ)E(x,τ)2E(x,t)dτ
(1)

where χ (3)(t) is the time-dependent third-order nonlinear susceptibility. The third-order nonlinear susceptibility of Kerr-type materials can be assumed as an instantaneous-response model: χ (3)(t)≅χ0(3) δ(t) since the relaxation time of Kerr-type materials is of the order of or less than 10 fs [10

10. R. M. Joseph and A. Taflove, “Spatial soliton deflection mechanism indicated by FDTD Maxwell’s equations modeling,” IEEE Photon. Tech. Lett. 6, 1251 (1994). [CrossRef]

,11

11. R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” J. Opt. Soc. Am. B 10, 186 (1993). [CrossRef]

]. Then, the nonlinear electric polarization can be obtained by

PNL(x,t)ε0χ0(3)E(t)2E(t)
(2)

where χ0(3) is the third-order electric susceptibility. In our simulation, we assume that the first and third-order electric susceptibilities are constant. In this model, the self-induced refractive index gradients grow linearly with the laser intensity, n=n 0+n 2ε|E|2/2=n 0+n 2 I, where n 0, n 2, and I are the linear refractive index, nonlinear refractive index, and laser beam intensity, respectively. Finally, the electric field in the nonlinear Kerr material is assumed to be as follows [8

8. G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” PHYSICA D 157, 112 (2001). [CrossRef]

,10

10. R. M. Joseph and A. Taflove, “Spatial soliton deflection mechanism indicated by FDTD Maxwell’s equations modeling,” IEEE Photon. Tech. Lett. 6, 1251 (1994). [CrossRef]

,11

11. R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” J. Opt. Soc. Am. B 10, 186 (1993). [CrossRef]

]:

E(t)=D(t)ε0(1+χ(1)+χ0(3)E(t)2)
(3)

where χ (1) is are the first-order electric susceptibility

In each iteration step of the FDTD method, the latest value of E can be calculated from the latest value of D and the old value of E. Therefore, Eq. (3) gives us a complete description of the computational FDTD model for the Kerr nonlinearity. Since the electric field components (Ex , Ey , Ez ) are separated spatially by Yee algorithm in the 3D FDTD method [15

15. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302 (1966). [CrossRef]

], in addition, the electric fields are interpolated on the grid for the calculation of |E(t)|2 in Eq. (3).

For FDTD simulations, it is also assumed that the nonlinear Kerr thin film is deposited on a glass substrate and the linearly-polarized CW laser light of wavelength (λ 0) 632.8 nm is incident on the film after passing through the grass substrate. In addition, it is assumed that the incident light of radius 1 µ m has a Gaussian intensity distribution and the refractive index (n 0) and the nonlinear refractive index (n 2) of the film are 2.61 and 4×10-10 m 2/W (χ0(3)≅9.244×10-21 m 2/V 2), respectively. Therefore, in our configuration the average intensity of a Gaussian beam produced from a 5 mW laser provides approximately Iave =power/2π (beam radius)2≅0.796×109 (W/m 2), giving the effective refractive index and wavelength of n eff=n 0+n 2 I ave≅2.928 and λ eff=λ 0/n eff≅216 nm, respectively. With all these assumptions, the near-field intensity distributions at the film surface are calculated as functions of total power of the incident beam and film thickness by using the 3D FDTD method.

Figure 1 shows the temporal dynamics of the near-field intensity distribution at the film surface for a time period of 2 picoseconds. The stabilization process of the filamentation in a nonlinear Kerr film is clearly revealed. When the intense light is incident on the film surface, one can expect that the two time-dependent elements, originating from the change of the refractive index and the momentum vector of focused beam, produce much more complex features compared to bulk materials because of the multiple reflections on both sides of the nonlinear film. As can be seen in Fig. 1, however, it is found that the near-field intensity distribution at the film surface reaches a stable state at around 300 femtoseconds after the beam incidence. In addition, our simulations show that the time to reach a stable filament formation is quite sensitive to the film thickness, beam power, and wavelength, strongly indicating that nonlinear Fabry-Perot interference is playing a significant role in the Kerr film.

In order to check the existence of multiple interference effects in the Kerr film, we compare the FDTD simulation result of the thin film with that of the bulk medium. We calculate the intensity distribution in the bulk medium after the same propagation distance without reflections from the second boundary, as shown in Fig. 2. It is clearly found that multiple filamentation does not appear in the nonlinear Kerr bulk medium in contrast to the case of the thin film.

Fig. 1. Temporal dynamics of intensity distribution at 336nm-thick film surface obtained by 3D FDTD method.
Fig. 2. Intensity distribution in a nonlinear Kerr bulk medium after propagating 336nm from the surface.

Figure 3 displays the intensity distribution at the film surface after the beam from the 5.0 mW laser passes through films of various thicknesses, where the direction of polarization is along the vertical axis. The results show that the self-focusing effect is maximized for films of thickness 0.5λeff , 1λeff , 1.5λeff , and 2λeff . Note that the electric fields in the film can be maximized due to the nonlinear Fabry-Perot interference. As shown in Fig. 3, the near-field intensity distribution of 1.5λeff and 2λeff thick films display beam breakups into two filaments of sub-diffraction-limited spot sizes, while others do not show filamentations clearly. This strongly suggests that filamentation is dominantly determined by the condition of the nonlinear Fabry-Perot interference. In addition, we found that the film of 0.5λeff and 1λeff thickness do not generate the filamentation, indicating the existence of a threshold thickness for the filamentation.

Fig. 3. Near-field intensity distribution of film with various thickness obtained by 3D FDTD method.

Figure 4 displays the near-field intensity distribution of the 1.5λeff thick film for four different laser beam powers. When the beam power increases, the number of filaments due to the beam breakup dramatically grows because of the increase of the effective thickness of the film with increasing intensity. Note that the nonlinear refractive index is proportional to the incident beam power. Much more complicated multiple filamentations are produced for higher input intensities, indicating that the nonlinear Fabry-Perot resonance condition is quite sensitive to the effective film thickness.

Recently, multiple filamentation in bulk media has been known to be induced not only by random noise in the input beam but also by beam polarization [8

8. G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” PHYSICA D 157, 112 (2001). [CrossRef]

]. Due to the gradient of the nonlinear refractive-index in the film of sub-wavelength thickness, the momentum vector of the incident beam is deflected toward the center of the Gaussian beam. The beams which are deflected to the direction parallel to the polarization of the input beam and to the direction perpendicular to the polarization have different reflection conditions. Therefore, by the oblique incidence of a linearly-polarized beam due to focusing, the polarization-dependent symmetry breaking can be occurred even when the input beam is radially symmetric.

Fig. 4. Near-field intensity distribution of a nonlinear Kerr film for various beam powers obtained by 3D FDTD method.

3. Conclusion

In conclusion, we analyze the self-focusing phenomenon of nonlinear Kerr film for several configurations using the FDTD method. The temporal dynamics and the near-field intensity distributions of multiple filamentation in various film thicknesses and input beam powers are investigated theoretically. In particular, we demonstrate that linearly polarized beams can lead to multiple filamentation in nonlinear Kerr film, where filaments are narrowed below the diffraction-limit. We also find that filamentation in film is very sensitive to the nonlinear film thickness and the laser beam power, strongly indicating nonlinear Fabry-Parot interference effects on nonlinear Kerr films. Our simulations of filamentation in the nonlinear Kerr film in the near field region can be applied to optical storage and near field devices, which require a detailed intensity distribution of a sub-diffraction limited spot size.

Acknowledgments

This research was supported by the Ministry of Science and Technology through the National Research Laboratory Program (Contact No. M1-0203-00-0082).

References and links

1.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479 (1964). [CrossRef]

2.

V. I. Talanov, “Self-focusing of waves in nonlinear media,” JETP Lett. 2, 138 (1965).

3.

J. A. Fleck Jr. and P. L. Kelley, “Temporal aspect of the self-focusing of optical beams,” Appl. Phys. Lett. 15, 313 (1969). [CrossRef]

4.

M. D. Feit and J. A. Fleck Jr., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633 (1988). [CrossRef]

5.

K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the townes profile,” Phys. Rev. Lett. 90, 203902-1 (2003). [CrossRef]

6.

G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335 (2000). [CrossRef]

7.

G. Fibich and B. Ilan, “Multiple filamentation of circularly polarized beams,” Phys. Rev. Lett. 89, 013901-1 (2002). [CrossRef]

8.

G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” PHYSICA D 157, 112 (2001). [CrossRef]

9.

G. Fibich, W. Ren, and X.-P. Wang, “Numerical simulations of self-focusing of ultrafast laser pulses.” Phys. Rev. E 67, 056603 (2003). [CrossRef]

10.

R. M. Joseph and A. Taflove, “Spatial soliton deflection mechanism indicated by FDTD Maxwell’s equations modeling,” IEEE Photon. Tech. Lett. 6, 1251 (1994). [CrossRef]

11.

R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” J. Opt. Soc. Am. B 10, 186 (1993). [CrossRef]

12.

K. B. Song, J. Lee, J. H. Kim, K. Cho, and S. K. Kim, “Direct observation of self-focusing with subdiffraction limited resolution using near-field scanning optical microscope,” Phy. Rev. Lett. 85, 3842 (2000) [CrossRef]

13.

Y. Choi, J. H. Park, M. R. Kim, W. Jhe, and B. K. Rhee, “Direct observation of self-focusing near the diffraction limit in polycrystalline silicon film,” Appl. Phys. Lett. 78, 856 (2001) [CrossRef]

14.

J. Tominaga, T. Nakano, and N. Atoda, “An approach for recording and readout beyond the diffraction limit with an Sb thin film,” Appl. Phys. Lett. 73, 2078 (1998). [CrossRef]

15.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302 (1966). [CrossRef]

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.3270) Nonlinear optics : Kerr effect
(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter
(260.5950) Physical optics : Self-focusing

ToC Category:
Research Papers

History
Original Manuscript: April 20, 2004
Revised Manuscript: May 20, 2004
Published: June 14, 2004

Citation
Hyun-Ho Lee, Kyu-Min Chae, Sang-Youp Yim, and Seung-Han Park, "Finite-difference time-domain analysis of self-focusing in a nonlinear Kerr film," Opt. Express 12, 2603-2609 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-12-2603


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References

  1. R. Y. Chiao, E. Garmire, and C. H. Townes, �??Self-trapping of optical beams,�?? Phys. Rev. Lett. 13, 479 (1964). [CrossRef]
  2. V. I. Talanov, �??Self-focusing of waves in nonlinear media,�?? JETP Lett. 2, 138 (1965).
  3. J. A. Fleck, Jr., and P. L. Kelley, �??Temporal aspect of the self-focusing of optical beams,�?? Appl. Phys. Lett. 15, 313 (1969). [CrossRef]
  4. M. D. Feit, and J. A. Fleck, Jr., �??Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,�?? J. Opt. Soc. Am. B 5, 633 (1988). [CrossRef]
  5. K. D. Moll, A. L. Gaeta, and G. Fibich, �??Self-similar optical wave collapse: observation of the townes profile,�?? Phys. Rev. Lett. 90, 203902-1 (2003). [CrossRef]
  6. G. Fibich and A. L. Gaeta, �??Critical power for self-focusing in bulk media and in hollow waveguides,�?? Opt. Lett. 25, 335 (2000). [CrossRef]
  7. G. Fibich and B. Ilan, �??Multiple filamentation of circularly polarized beams,�?? Phys. Rev. Lett. 89, 013901-1 (2002). [CrossRef]
  8. G. Fibich and B. Ilan, �??Vectorial and random effects in self-focusing and in multiple filamentation,�?? PHYSICA D 157, 112 (2001). [CrossRef]
  9. G. Fibich, W. Ren, and X.-P. Wang, �??Numerical simulations of self-focusing of ultrafast laser pulses,�?? Phys. Rev. E 67, 056603 (2003). [CrossRef]
  10. R. M. Joseph, A. Taflove, �??Spatial soliton deflection mechanism indicated by FDTD Maxwell�??s equations modeling,�?? IEEE Photon. Tech. Lett. 6, 1251 (1994). [CrossRef]
  11. R. W. Ziolkowski, and J. B. Judkins, �??Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,�?? J. Opt. Soc. Am. B 10, 186 (1993). [CrossRef]
  12. K. B. Song, J. Lee, J. H. Kim, K. Cho, and S. K. Kim, �??Direct observation of self-focusing with subdiffraction limited resolution using near-field scanning optical microscope,�?? Phy. Rev. Lett. 85, 3842 (2000). [CrossRef]
  13. Y. Choi, J. H. Park, M. R. Kim, W. Jhe, and B. K. Rhee, �??Direct observation of self-focusing near the diffraction limit in polycrystalline silicon film,�?? Appl. Phys. Lett. 78, 856 (2001) [CrossRef]
  14. J. Tominaga, T. Nakano, and N. Atoda, �??An approach for recording and readout beyond the diffraction limit with an Sb thin film,�?? Appl. Phys. Lett. 73, 2078 (1998). [CrossRef]
  15. K. S. Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propagat. 14, 302 (1966). [CrossRef]

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