## Vector diffraction from subwavelength optical disk structures: Two-dimensional near-field profiles

Optics Express, Vol. 2, Issue 5, pp. 191-197 (1998)

http://dx.doi.org/10.1364/OE.2.000191

Acrobat PDF (242 KB)

### Abstract

An efficient finite-difference frequency-domain method is developed for calculating electromagnetic fields in the neighborhood of subwavelength dielectric and metallic structures. The method is used to investigate two-dimensional near-field and far-field patterns of a focused beam diffracted from an optical disk, specifically from a DVD (Digital Versatile Disk). It is shown that the polarization of illumination has a significant impact on diffraction patterns as expected and that scalar theory does not provide an accurate analysis of diffraction from a DVD.

© Optical Society of America

## 1. Introduction

1. J. B. Judkins, C. W. Haggans, and R. W. Ziolkowski, “Two-dimensional finite-difference time-domain simulation for rewritable optical disk surface structure design,” Appl. Opt. **35**, 2477–2487 (1996). [CrossRef] [PubMed]

2. M. Ogawa, M. Nakada, R. Katayama, M. Okada, and M. Itoh, “Analysis of scattering light from magnetic material with land/groove by three-dimensional boundary element method,” Jpn. J. Appl. Phys. **35**, 336–341 (1996). [CrossRef]

3. D. S. Marx and D. Psaltis, “Optical diffraction of focused spots and subwavelength structures,” J. Opt. Soc. Am. A **14**, 1268–1278 (1997). [CrossRef]

## 2. Numerical methods

*E*⃗ is the electric field,

*H*⃗ is the magnetic field,

*∊*is the complex electric permittivity,

*ω*is the angular frequency, and the time dependence

*e*

^{-iωt}has been assumed. The information about material properties is included in

*∊*(

*y*,

*z*). Eliminating the magnetic field, the Maxwell equations become

4. B.-N. Jiang, J. Wu, and L. A. Povinelli, “The origin of spurious solutions in computational electromagnetics,” J. Comput. Phys. **125**, 104–123 (1996). [CrossRef]

### 2.1 Concus-Golub iteration

5. P. Concus and G. H. Golub, “Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations,” SIAM J. Numer. Anal. **10**, 1103–1120 (1973). [CrossRef]

6. B. L. Buzbee, G. H. Golub, and C. W. Nielson, “On direct methods for solving Poisson’s equations,” SIAM J. Numer. Anal. **7**, 627–656 (1970). [CrossRef]

7. R. A. Sweet, “A cyclic reduction algorithm for solving block tridiagonal systems of arbitrary dimension,” SIAM J. Numer. Anal. **14**, 706–720 (1977). [CrossRef]

*O*(3

*n*

^{2}log

*n*) for an

*n*×

*n*matrix and has a small memory requirement. In equation (2), each component has the form of a general Helmholtz equation

*k*-th iteration, we use the expression

5. P. Concus and G. H. Golub, “Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations,” SIAM J. Numer. Anal. **10**, 1103–1120 (1973). [CrossRef]

*∊*(

*y*,

*z*). Usually, the smoother

*∊*(

*y*,

*z*) is, the faster the rate of convergence. However, the Concus-Golub iteration method does not always converge in all cases. For nonconverging cases, we use the conjugate gradient method.

### 2.2 Conjugate gradient method

*A*∙

*x*⃗ =

*b*⃗ by minimizing quadratic functionals in Krylov subspaces, which are spanned by a series of vectors generated by repeated multiplication by

*A*.

*general minimal residual*(GMRES) conjugate gradient algorithm [8

8. Y. Saad and M. H. Schultz, “GMRES: a general minimal residual algorithm for solving nonsym-metric linear systems,” SIAM J. Sci. Stat. Comput. **7**, 856–869 (1986). [CrossRef]

9. C. D. Dimitropoulos and A. N. Beris, “An efficient and robust spectral solver for nonseparable elliptic equations,” J. Comput. Phys. **133**, 186–191 (1997). [CrossRef]

9. C. D. Dimitropoulos and A. N. Beris, “An efficient and robust spectral solver for nonseparable elliptic equations,” J. Comput. Phys. **133**, 186–191 (1997). [CrossRef]

### 2.3 Radiation boundary condition

10. B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. **31**, 629–651 (1977). [CrossRef]

11. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. **EMC-23**, 377–382 (1981). [CrossRef]

*α*

_{1}and

*α*

_{2}are the optimal angles for minimal reflection. We choose the

*z*̂ direction to be the primary direction of wave propagation and the angles α

_{1}and

*α*

_{2}to be zero in order to minimize the reflection from the

*z*̂ direction. This boundary condition is equivalent to the 2nd-order Engquist-Majda radiation boundary condition [12], but it is better suited for the cyclic-reduction method because it only requires the derivative in one direction.

## 3. Configuration

*n*= 1.6 for the polycarbonate substrate and

*n*= 1.5 + 7.8

*i*for the aluminum reflection layer. The wavelength in vacuum of the incident laser light is 650 nm, which corresponds to a reduced wavelength in the polycarbonate substrate of 406 nm. The numerical aperture is

*NA*= 0.6. The incident field is a 2D Gaussian beam that has a full width at half maximum (FWHM) of approximately 600 nm.

*μ*m) in the y direction and 5 wavelengths (≈ 2.03

*μ*m) in the

*z*direction, with 40 grid points in each wavelength. The grid points are located in a Yee cell [13] in two-dimensional space. The unit of length in all figures is the reduced wavelength in the polycarbonate substrate, λ

_{s}= 406 nm.

## 4. Near fields

_{s}(≈ 40–320 nm). The total field is shown here, which includes both incident and diffracted fields. For TE illumination there is only an

*x*̂ component, whereas for TM illumination there are both

*y*̂ and

*z*̂ components that are coupled together. Here we only show the

*y*̂ component of the TM case, which is generally the dominant component for this polarization.

## 5. Far fields

*E*component,

_{x}_{s}(≈ 120 nm). The position of minima and maxima for TM illumination is roughly the same as for scalar theory.

_{s}in the TE case and 0.5 λ

_{s}in the TM case, the far-field intensities return to close to unity and the near-fields in the substrate look similar to those without a pit at all. For pit heights greater than 0.5 λ

_{s}, the intensity predicted by scalar theory is considerably smaller than vector theory. Our initial calculations indicate that this difference is due to a large contribution from evanescent waves in scalar theory.

## 6. Conclusion

## Acknowledgments

## References

1. | J. B. Judkins, C. W. Haggans, and R. W. Ziolkowski, “Two-dimensional finite-difference time-domain simulation for rewritable optical disk surface structure design,” Appl. Opt. |

2. | M. Ogawa, M. Nakada, R. Katayama, M. Okada, and M. Itoh, “Analysis of scattering light from magnetic material with land/groove by three-dimensional boundary element method,” Jpn. J. Appl. Phys. |

3. | D. S. Marx and D. Psaltis, “Optical diffraction of focused spots and subwavelength structures,” J. Opt. Soc. Am. A |

4. | B.-N. Jiang, J. Wu, and L. A. Povinelli, “The origin of spurious solutions in computational electromagnetics,” J. Comput. Phys. |

5. | P. Concus and G. H. Golub, “Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations,” SIAM J. Numer. Anal. |

6. | B. L. Buzbee, G. H. Golub, and C. W. Nielson, “On direct methods for solving Poisson’s equations,” SIAM J. Numer. Anal. |

7. | R. A. Sweet, “A cyclic reduction algorithm for solving block tridiagonal systems of arbitrary dimension,” SIAM J. Numer. Anal. |

8. | Y. Saad and M. H. Schultz, “GMRES: a general minimal residual algorithm for solving nonsym-metric linear systems,” SIAM J. Sci. Stat. Comput. |

9. | C. D. Dimitropoulos and A. N. Beris, “An efficient and robust spectral solver for nonseparable elliptic equations,” J. Comput. Phys. |

10. | B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. |

11. | Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. |

12. | R. L. Higdon, “Absorbing boundary conditions for difference approximations to the multidimensional wave equation,” Math. Comput. |

13. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

14. | R. E. Gerber and M. Mansuripur, “Dependence of the tracking performance of an optical disk on the direction of the incident-light polarization,” Appl. Opt. |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(210.0210) Optical data storage : Optical data storage

(210.4590) Optical data storage : Optical disks

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 12, 1997

Revised Manuscript: December 10, 1997

Published: March 2, 1998

**Citation**

Wei-Chih Liu and Marek Kowarz, "Vector diffraction from subwavelength
optical disk structures: Two-dimensional near-field profiles," Opt. Express **2**, 191-197 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-5-191

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### References

- J. B. Judkins, C. W. Haggans, and R. W. Ziolkowski, "Two-dimensional finite-difference time- domain simulation for rewritable optical disk surface structure design," Appl. Opt. 35, 2477-2487 (1996).<br> [CrossRef] [PubMed]
- M. Ogawa, M. Nakada, R. Katayama, M. Okada, and M. Itoh, "Analysis of scattering light from magnetic material with land/groove by three-dimensional boundary element method," Jpn. J. Appl. Phys. 35, 336-341 (1996).<br> [CrossRef]
- D. S. Marx and D. Psaltis, "Optical diffraction of focused spots and subwavelength structures," J. Opt. Soc. Am. A 14, 1268-1278 (1997).<br> [CrossRef]
- B.-N. Jiang, J. Wu, and L. A. Povinelli, "The origin of spurious solutions in computational electromagnetics," J. Comput. Phys. 125, 104-123 (1996).<br> [CrossRef]
- P. Concus and G. H. Golub, "Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations," SIAM J. Numer. Anal. 10, 1103-1120 (1973).<br> [CrossRef]
- B. L. Buzbee, G. H. Golub, and C. W. Nielson, "On direct methods for solving Poisson's equations," SIAM J. Numer. Anal. 7, 627-656 (1970).<br> [CrossRef]
- R. A. Sweet, "A cyclic reduction algorithm for solving block tridiagonal systems of arbitrary dimension," SIAM J. Numer. Anal. 14, 706-720 (1977).<br> [CrossRef]
- Y. Saad and M. H. Schultz, "GMRES: a general minimal residual algorithm for solving nonsym- metric linear systems," SIAM J. Sci. Stat. Comput. 7, 856-869 (1986).<br> [CrossRef]
- C. D. Dimitropoulos and A. N. Beris, "An efficient and robust spectral solver for nonseparable elliptic equations," J. Comput. Phys. 133, 186-191 (1997).<br> [CrossRef]
- B. Engquist and A. Majda, "Absorbing boundary conditions for the numerical simulation of waves," Math. Comput. 31, 629-651 (1977).<br> [CrossRef]
- Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. EMC-23, 377-382 (1981). [CrossRef]
- R. L. Higdon, "Absorbing boundary conditions for difference approximations to the multi- dimensional wave equation," Math. Comput. 47, 437-459 (1986).<br>
- K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).<br>
- R. E. Gerber and M. Mansuripur, "Dependence of the tracking performance of an optical disk on the direction of the incident-light polarization," Appl. Opt. 34, 8192-8200 (1995). [CrossRef] [PubMed]

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