## Near-field localization of ultrashort optical pulses in transverse 1-D periodic nanostructures

Optics Express, Vol. 7, Issue 3, pp. 123-128 (2000)

http://dx.doi.org/10.1364/OE.7.000123

Acrobat PDF (148 KB)

### Abstract

We present a transverse 1-D periodic nanostructure which exhibits lateral internal field localization for normally incident ultrashort pulses, and which may be applied to the enhancement of nonlinear optical phenomena. The peak intensity of an optical pulse propagating in the nanostructure is approximately 12 times that of an identical incident pulse propagating in a bulk material of the same refractive index. For second harmonic generation, an overall enhancement factor of approximately 10.8 is predicted. Modeling of pulse propagation is performed using Fourier spectrum decomposition and Rigorous Coupled-Wave Analysis (RCWA).

© Optical Society of America

## 1. Introduction

1. P. Lalanne and J.-P. Hugonin, “High-order effective-medium theory of subwavelength gratings in classical mounting: application to volume holograms,” J. Opt. Soc. Am. A **15**, 1843–1851 (1998). [CrossRef]

2. J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Design of binary subwavelength diffractive lenses by use of zeroth-order effective-medium theory,” J. Opt. Soc. Am. A **16**, 1157–1167 (1999). [CrossRef]

3. F. Xu, R.-C. Tyan, P.-C. Sun, Y. Fainman, C.-C. Cheng, and A. Scherer, “Form-birefringent computer-generated holograms,” Opt. Lett. **21**, 1513–1515 (1996). [CrossRef] [PubMed]

4. R.-C. Tyan, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, and Y. Fainman, “Design, fabrication and characterization of form-birefringent multilayer polarizing beam splitter,” J. Opt. Soc. Am. A **14**, 1627–1636 (1997). [CrossRef]

5. J. E. Sipe and R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A **46**, 1614–1629 (1992). [CrossRef] [PubMed]

10. H. Ma, R. Xiao, and P. Sheng, “Third-order optical nonlinearity enhancement through composite microstructures,” J. Opt. Soc. Am. B **15**, 1022–1029 (1998). [CrossRef]

11. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. **72**, 1385–1392 (1982) [CrossRef]

## 2. Modeling method

11. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. **72**, 1385–1392 (1982) [CrossRef]

11. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. **72**, 1385–1392 (1982) [CrossRef]

12. N. Chateau and J.-P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A **11**, 1321–1331 (1994). [CrossRef]

15. M. Schmitz, R. Br. uer, and O. Bryngdahl, “Comment on numerical stability of rigorous differential methods of diffraction,” Opt. Comm. **124**, 1–8 (1996). [CrossRef]

*r̚*and

*t*are space and time coordinates, respectively,

*a*̂

_{0}indicates the polarization of the incident pulse, ω

_{0}is the center frequency of the pulse, k̚

_{0}is the wave vector corresponding to the center frequency of the pulse,

*k*̂

_{0}is a unit vector in the direction of, k̚

_{0},

*ν*is the group velocity of the pulse,

_{g}*τ*is the width parameter of the Gaussian pulse envelope,

*t*

_{0}is the time at which the pulse peak arrives at

*r*̚=0, Δ

*T*is the temporal separation between pulses in the incident pulse train, ⊗ indicates the convolution operation, and

*n*is an integer. Taking the Fourier transform of Eq. (1) and imposing a finite truncated bandwidth Δω=2

*M*·δω centered at ω

_{0}, we obtain

_{0}and δω=2π/Δ

*T*. Eq. (2) is a finite, discrete frequency-domain representation of the incident fields having 2

*M*+1 discrete frequency components over index n (where

*n*={-

*M*,…, 0, …,

*M*}) at frequencies

*ω*=ω

_{n}_{0}+

*n*δω. Thus, for each component, the RCWA method can be applied to solve for the diffracted fields.

^{-13}sec. The temporal separation between pulses in the incident pulse train is assumed to be Δ

*T*=50 ps, corresponding to a frequency sampling interval of δω=4π×10

^{10}rad. We also choose a truncated bandwidth of Δω=3π×10

^{13}rad, corresponding to 2

*M*+1=751 discrete spectral components. Although a frequency-dependent material refractive index can easily be incorporated due to the spectral decomposition, in this manuscript the effects of material dispersion are omitted for simplicity.

## 3. Transverse field localization

*F*=0.09, as shown in Fig. 1. For clarity, the depth of the structure is chosen to be

*d*=100 µm to avoid the introduction of interference effects in the propagation direction. The refractive indices of the grating materials are assumed to be

*n*

_{1}=3.5 in the high index region (corresponding to the properties of GaAs) and

*n*

_{2}=1.0 in the air gap. The incident pulse train is assumed to be normally incident.

*E*|

^{2}) inside the nanostructure occurs for the TE polarization, and is approximately 2.4 times that of the incident pulse. Despite the localization inside the structure, the transmitted and reflected pulses have uniform transverse profiles due to the subwavelength scale of the nanostructure.

*y*̂-direction as shown in Fig. 1), the boundary conditions require continuity of the tangential electric fields, imposing no particular transverse profile on the field. However, the mode structure of the coupled waveguide array results in transverse localization of the field energy in the high refractive index material, in a similar fashion to the mode profile of a single-mode slab waveguide. Since most commonly used bulk nonlinear optical materials tend to have relatively high indices of refraction, it is the TE polarization case—where the pulse energy is localized in the high refractive index region of the grating—that is of interest.

## 4. Enhancement of nonlinear optical phenomena

*E*|

^{2}at the pulse peak for five nanostructures having the same period (Λ=0.65 µm) but differing fill factors: 1%, 3%, 6%, 9%, and 12%, as well as the bulk case (100%). For very small fill factors (e.g. the 1% case), the fields cannot vary significantly on a substantially subwavelength scale, resulting in a nearly uniform transverse field profile (the width of the high refractive index region is ~λ/150 for 1% fill factor). As the fill factor increases, the localization effect strengthens, reaching its maximum value of approximately 2.5 times that of the incident pulse at a fill factor of 6%. As the fill factor continues to increase, however, the increasing volume fraction of the high refractive index material results in a diminishing peak |

*E*|

^{2}. The peak|

*E*|

^{2}values for fill factors varying from 1% to 12% are shown in Fig. 4. For fill factors larger than 12%, multiple transverse modes exist, significantly reducing the peak |

*E*|

^{2}. For the 9% fill factor case (corresponding to the results of Fig. 2), the peak value of |

*E*|

^{2}is approximately 2.4 times that of the incident pulse, and over 12 times that of the bulk material case.

*c*to roughly 0.3

*c*, where

*c*is the speed of light. Thus, as the fraction of the pulse energy contained within the high refractive index material increases, the group velocity of the pulse in the nanostructure decreases. This behavior is similar to the dependence of the mode propagation speed on the guide thickness in a single-mode slab waveguide.

*E*|

^{4}. By integrating |

*E*|

^{4}across the fraction of the grating period corresponding to the high refractive index material and comparing with the bulk case, we can obtain an effective SHG enhancement factor. The effective SHG enhancement factor for fill factors varying from 1% to 12% is shown in Fig. 5. A fill factor

*F*= 0.09, corresponding to the results of Figs. 1 and 2, yields the maximum value of the SHG enhancement factor: approximately 10.8.

## 5. Conclusions

## References and links

1. | P. Lalanne and J.-P. Hugonin, “High-order effective-medium theory of subwavelength gratings in classical mounting: application to volume holograms,” J. Opt. Soc. Am. A |

2. | J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Design of binary subwavelength diffractive lenses by use of zeroth-order effective-medium theory,” J. Opt. Soc. Am. A |

3. | F. Xu, R.-C. Tyan, P.-C. Sun, Y. Fainman, C.-C. Cheng, and A. Scherer, “Form-birefringent computer-generated holograms,” Opt. Lett. |

4. | R.-C. Tyan, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, and Y. Fainman, “Design, fabrication and characterization of form-birefringent multilayer polarizing beam splitter,” J. Opt. Soc. Am. A |

5. | J. E. Sipe and R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A |

6. | R. W. Boyd and J. E. Sipe, “Nonlinear optical susceptibilities of layered composite materials,” J. Opt. Soc. Am. B |

7. | G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett. |

8. | R. S. Bennink, Y.-K. Yoon, R. W. Boyd, and J. E. Sipe, “Accessing the optical nonlinearity of metals with metal-dielectric photonic bandgap structures,” Opt. Lett. |

9. | K. P. Yuen, M. F. Law, K. W. Yu, and P. Sheng, “Enhancement of optical nonlinearity through anisotropic microstructures,” Opt. Comm. |

10. | H. Ma, R. Xiao, and P. Sheng, “Third-order optical nonlinearity enhancement through composite microstructures,” J. Opt. Soc. Am. B |

11. | M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. |

12. | N. Chateau and J.-P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A |

13. | M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A |

14. | L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A |

15. | M. Schmitz, R. Br. uer, and O. Bryngdahl, “Comment on numerical stability of rigorous differential methods of diffraction,” Opt. Comm. |

16. | R. Tyan, “Design, modeling and characterization of multifunctional diffractive optical elements,” Ph.D. Thesis, University of California, San Diego (1998). |

17. | W. Nakagawa, R.-C. Tyan, P.-C. Sun, F. Xu, and Y. Fainman, “Ultrashort pulse propagation in near-field periodic diffractive structures using Rigorous Coupled-Wave Analysis,” submitted to J. Opt. Soc. Am. A (2000). |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(230.3990) Optical devices : Micro-optical devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 5, 2000

Published: July 31, 2000

**Citation**

Wataru Nakagawa, Rong-Chung Tyan, Pang-Chen Sun, and Yeshaiahu Fainman, "Near-field localization of ultrashort optical pulses in transverse 1-D periodic nanostructures," Opt. Express **7**, 123-128 (2000)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-3-123

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### References

- P. Lalanne and J.-P. Hugonin, "High-order effective-medium theory of subwavelength gratings in classical mounting: application to volume holograms," J. Opt. Soc. Am. A 15, 1843-1851 (1998). [CrossRef]
- J. N. Mait, D. W. Prather, and M. S. Mirotznik, "Design of binary subwavelength diffractive lenses by use of zeroth-order effective-medium theory," J. Opt. Soc. Am. A 16, 1157-1167 (1999). [CrossRef]
- F. Xu, R.-C. Tyan, P.-C. Sun, Y. Fainman, C.-C. Cheng, and A. Scherer, "Form-birefringent computer-generated holograms," Opt. Lett. 21, 1513-1515 (1996). [CrossRef] [PubMed]
- R.-C. Tyan, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, and Y. Fainman, "Design, fabrication and characterization of form-birefringent multilayer polarizing beam splitter," J. Opt. Soc. Am. A 14, 1627-1636 (1997). [CrossRef]
- J. E. Sipe and R. W. Boyd, "Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model," Phys. Rev. A 46, 1614-1629 (1992). [CrossRef] [PubMed]
- R. W. Boyd and J. E. Sipe, "Nonlinear optical susceptibilities of layered composite materials," J. Opt. Soc. Am. B 11, 297-303 (1994). [CrossRef]
- G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A Weller-Brophy, "Enhanced nonlinear optical response of composite materials," Phys. Rev. Lett. 74, 1871-1874 (1995). [CrossRef] [PubMed]
- R. S. Bennink, Y.-K. Yoon, R. W. Boyd, and J. E. Sipe, "Accessing the optical nonlinearity of metals with metal- dielectric photonic bandgap structures," Opt. Lett. 24, 1416-1418 (1999). [CrossRef]
- K. P. Yuen, M. F. Law, K. W. Yu, and P. Sheng, "Enhancement of optical nonlinearity through anisotropic microstructures," Opt. Comm. 148, 197-207 (1998). [CrossRef]
- H. Ma, R. Xiao, and P. Sheng, "Third-order optical nonlinearity enhancement through composite microstructures," J. Opt. Soc. Am. B 15, 1022-1029 (1998). [CrossRef]
- M. G. Moharam and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1392 (1982). [CrossRef]
- N. Chateau and J.-P. Hugonin, "Algorithm for the rigorous coupled-wave analysis of grating diffraction," J. Opt. Soc. Am. A 11, 1321-1331 (1994). [CrossRef]
- M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, "Stable implementation of the rigorous coupled- wave analysis for surface-relief gratings: enhanced transmittance matrix approach," J. Opt. Soc. Am. A 12, 1077- 1086 (1995). [CrossRef]
- L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996). [CrossRef]
- M. Schmitz, R. Brauer, O. Bryngdahl, "Comment on numerical stability of rigorous differential methods of diffraction," Opt. Comm. 124, 1-8 (1996). [CrossRef]
- R. Tyan, "Design, modeling and characterization of multifunctional diffractive optical elements," Ph.D. Thesis, University of California, San Diego (1998).
- W. Nakagawa, R.-C. Tyan, P.-C. Sun, F. Xu, and Y. Fainman, "Ultrashort pulse propagation in near-Field periodic diffractive structures using Rigorous Coupled-Wave Analysis," submitted to J. Opt. Soc. Am. A (2000).

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