## High efficiency half-wave retardation in diffracted light by coupled waves |

Optics Express, Vol. 20, Issue 4, pp. 4681-4689 (2012)

http://dx.doi.org/10.1364/OE.20.004681

Acrobat PDF (1091 KB)

### Abstract

We demonstrate high-efficiency half-wave retardation in diffracted light in the 2nd order Littrow mounting. The diffracting structure is a slanted crossed grating with subwavelength period in the direction of the second grating vector, which makes it possible to mix the polarization states of the input light inside the grating layer, and hence to create the half-wave retardation. We present an experimental result with 58.9 % diffraction efficiency and a near perfect half-wave retardation. We explain the effect qualitatively using the classical coupled-wave approach.

© 2012 OSA

## 1. Introduction

2. D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. **42**, 492–494 (1983). [CrossRef]

6. N. Passilly, K. Ventola, P. Karvinen, P. Laakkonen, J. Turunen, and J. Tervo, “Polarization conversion in conical diffraction by metallic and dielectric subwavelength gratings,” Appl. Opt. **46**, 4258–4265 (2007). [CrossRef] [PubMed]

7. N. Bonod, E. Popov, L. Li, and B. Chernov, “Unidirectional excitation of surface plasmons by slanted gratings,” Opt. Express **15**, 11427–11432 (2007). [CrossRef] [PubMed]

10. K. Ventola, J. Tervo, P. Laakkonen, and M. Kuittinen, “High phase retardation by waveguiding in slanted photonic nanostructures,” Opt. Express **19**, 241–246 (2011). [CrossRef] [PubMed]

## 2. Geometry and numerical designs

*x*,

*z*) plane and the two orthogonal grating vectors

**g**

*and*

_{x}**g**

*are parallel to the*

_{y}*x*and

*y*axes. The unit cell of the grating consists of a slanted cuboid pillar with dimensions

*c*

*and*

_{x}*c*

*, as depicted in the figure. We assume that the grating period*

_{y}*d*

*in the*

_{y}*y*direction is smaller than the wavelength in the substrate and hence the wave vectors of all diffraction orders are restricted to the (

*x*,

*z*) plane. Thus, the propagation directions of the orders are obtained from the grating equation

*k*

_{x}_{,}

*=*

_{m}*k*

_{x}_{,in}+

*m*2

*π*/

*d*

*, where*

_{x}*k*

_{x}_{,}

*and*

_{m}*k*

_{x}_{,in}are the

*x*components of the wave vector of the order

*m*and the incident wave, respectively, and

*d*

*is the grating period in the*

_{x}*x*direction. Since our goal is to obtain phase retardation in the direction opposite to the incidence, we choose the input angle and the period to fulfill the 2nd order Littrow condition

*k*

_{x}_{,–2}= −

*k*

_{x}_{,in}which yields at once

*k*

_{x}_{,in}= 2

*π*/

*d*

*. Even though we could choose any other Littrow configuration as well, the 2nd order condition was found to provide the best efficiency. Furthermore, the period*

_{x}*d*

*in this particular Littrow condition was suitable for reasonable fabrication.*

_{x}*y*direction, the structure behaves essentially as one-dimensional grating, with its grating vector parallel to the

*x*direction. One grating period thus consists of an alternating air–anisotropic material pair, such that one of the principal axes of the (effective) anisotropic material is parallel to the

*x*axis. The other two are rotated around the

*x*axis by the angle Θ, and are defined by unit vectors

**ŷ**sin Θ+

**ẑ**cos Θ and

**ŷ**cos Θ–

**ẑ**sin Θ, where

**ŷ**and

**ẑ**are unit vectors in the

*y*and

*z*directions, respectively. Owing to the rotated anisotropic structure, it is possible to change the polarization inside the grating layer and therefore polarization conversion of some degree is always attainable, at least in principle.

*d*

*, fill factors*

_{y}*f*

*and*

_{x}*f*

*, depth*

_{y}*h*, and slant angle Θ) that leads to the desired effect with maximum performance. In the numerical analysis, we used the rigorous Fourier modal method in slanted three-dimensional geometry, that is obtained as a special case of the more general approach [11

11. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A: Pure appl. Opt. **5**, 345–355 (2003). [CrossRef]

12. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder-Mead simplex algorithm in low dimensions”, SIAM Optim J. **9**, 112–147 (1999). [CrossRef]

*d*

*for an easier prototype fabrication prospects, although with slightly diminished optical performance. For a better insight, the output polarization states presented in table 1 are illustrated also in Fig. 2.*

_{y}## 3. Physical interpretation of the effect

11. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A: Pure appl. Opt. **5**, 345–355 (2003). [CrossRef]

*y*direction are propagating, and hence the grating can be understood to behave, at least qualitatively, like a common linear grating but made with anisotropic material. However, in-depth investigation of the field reveals that the field consists essentially of only two Fourier components in the

*x*-direction. Further, two of the grating modes, with almost identical propagation constants in the

*z*direction, are dominating, i.e., their amplitudes are larger than those of the other modes. Thus, the examined geometry leads to a situation in which the field inside the grating consists mainly of two plane waves. Figure 3 illustrates the situation immediately after the input-boundary. We emphasize that e.g. for the TM-polarized input, only the TM-part of the forward-propagating plane-wave components are shown in the figure. However, the amplitudes of the waves propagating in the opposite directions are almost negligible, and hence we have not included them in the figure for clarity.

*ψ*= ±45°, as well as the diffraction efficiency, of the desired reflected −2nd order as a function of the structure depth. The results, illustrated in Fig. 5, clearly show that the two-wave coupling discussed above is gradually increased as a function of the thickness, which is natural if we recall the classical coupled-wave approach [13]. A similar issue happens also to the phase difference between the components, i.e., the behavior of the components resembles that of the standard anisotropic half-wave plate, with fast and slow axes oriented at ±45° with respect to the

*x*-axis. The total effect is thus a mixture of these two phenomena – the actual combination of parameters leading to both high diffraction efficiency and desired phase retardation naturally requires careful optimization.

## 4. Grating fabrication

_{3}atmosphere. Subsequently, the residual Cr mask was removed. Critical issue with this type of deep slanted structure is the optimization of the Cr mask shape, since both mask linewidth and thickness affect the linewidth of the final slanted profile. For this reason, a design with a larger period

*d*

*was chosen for this prototype fabrication, despite the smaller expected efficiency.*

_{y}*f*

*is approximately 0.57.*

_{y}## 5. Optical measurements

*ψ*= 0° (TM), 90° (TE), 45°, and 25°. The intensities are normalized such that the input intensity (before the prism surface) equals unity. The figure also illustrates the polarization ellipses in the analyzer plane, corresponding to the measured intensities. We see that, for all four inputs, the polarization direction of the output is flipped around the 45° axis. In other words, a half-wave retardation (with fast- and slow axis oriented at 45° and −45°) occurs in the −2nd reflected diffraction order. The measured diffraction efficiencies are 60.3% (TM), 59.1% (TE), 57.3% (45°), and 58.7% (25°) (average 58.9 %). The wave-plate effect is not perfect, especially with TE-polarized input we see that the output is not purely linear. However, the ellipticity is only

*β*= 3.1°. Nevertheless, the experimental results show good agreement with the theory, and the differences are mainly due to the fabrication errors. With careful optimization of the fabrication process, it is expected that also the design leading to the higher diffraction efficiency can be realized experimentally.

## 6. Conclusions and discussion

8. T. Levola and P. Laakkonen, “Replicated slanted gratings with a high refractive index material for in and outcoupling of light,” Opt. Express **15**, 2067–2074 (2007). [CrossRef] [PubMed]

14. S. Siitonen, P. Laakkonen, J. Tervo, and M. Kuittinen, “A double-sided grating coupler for thin light guides,” Opt. Express **15**, 2008–2018 (2007). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | M. Born and E. Wolf, |

2. | D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. |

3. | F. Xu, R.-C. Tyan, P.-C. Sun, Y. Fainman, C.-C. Cheng, and A. Scherer, “Fabrication, modeling, and characterization of form-birefringent nanostructures,” Opt. Lett. |

4. | D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si-Li, and G.-F. Jin, “Broadband achromatic phase retarder by subwavelength grating,” Opt. Commun. |

5. | T. Isano, Y. Kaneda, N. Iwakami, K. Ishizuka, and N. Susuki, “Fabrication of half-wave plates with subwave-length structures,” Jpn. J. Appl. Phys. |

6. | N. Passilly, K. Ventola, P. Karvinen, P. Laakkonen, J. Turunen, and J. Tervo, “Polarization conversion in conical diffraction by metallic and dielectric subwavelength gratings,” Appl. Opt. |

7. | N. Bonod, E. Popov, L. Li, and B. Chernov, “Unidirectional excitation of surface plasmons by slanted gratings,” Opt. Express |

8. | T. Levola and P. Laakkonen, “Replicated slanted gratings with a high refractive index material for in and outcoupling of light,” Opt. Express |

9. | F. Van Laere, M. V. Kotlyar, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact slanted grating couplers between optical fiber and InP-InGaAsP waveguides,” IEEE Photon. Technol. Lett. |

10. | K. Ventola, J. Tervo, P. Laakkonen, and M. Kuittinen, “High phase retardation by waveguiding in slanted photonic nanostructures,” Opt. Express |

11. | L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A: Pure appl. Opt. |

12. | J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder-Mead simplex algorithm in low dimensions”, SIAM Optim J. |

13. | H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. |

14. | S. Siitonen, P. Laakkonen, J. Tervo, and M. Kuittinen, “A double-sided grating coupler for thin light guides,” Opt. Express |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1960) Diffraction and gratings : Diffraction theory

(260.5430) Physical optics : Polarization

(050.2065) Diffraction and gratings : Effective medium theory

(050.6624) Diffraction and gratings : Subwavelength structures

(050.6875) Diffraction and gratings : Three-dimensional fabrication

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: December 21, 2011

Revised Manuscript: January 20, 2012

Manuscript Accepted: January 20, 2012

Published: February 9, 2012

**Citation**

Kalle Ventola, Jani Tervo, Samuli Siitonen, Hemmo Tuovinen, and Markku Kuittinen, "High efficiency half-wave retardation in diffracted light by coupled waves," Opt. Express **20**, 4681-4689 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4681

Sort: Year | Journal | Reset

### References

- M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
- D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett.42, 492–494 (1983). [CrossRef]
- F. Xu, R.-C. Tyan, P.-C. Sun, Y. Fainman, C.-C. Cheng, and A. Scherer, “Fabrication, modeling, and characterization of form-birefringent nanostructures,” Opt. Lett.20, 2457–2459 (1995). [CrossRef] [PubMed]
- D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si-Li, and G.-F. Jin, “Broadband achromatic phase retarder by subwavelength grating,” Opt. Commun.227, 49–55 (2003). [CrossRef]
- T. Isano, Y. Kaneda, N. Iwakami, K. Ishizuka, and N. Susuki, “Fabrication of half-wave plates with subwave-length structures,” Jpn. J. Appl. Phys.43, 5294–5296 (2004). [CrossRef]
- N. Passilly, K. Ventola, P. Karvinen, P. Laakkonen, J. Turunen, and J. Tervo, “Polarization conversion in conical diffraction by metallic and dielectric subwavelength gratings,” Appl. Opt.46, 4258–4265 (2007). [CrossRef] [PubMed]
- N. Bonod, E. Popov, L. Li, and B. Chernov, “Unidirectional excitation of surface plasmons by slanted gratings,” Opt. Express15, 11427–11432 (2007). [CrossRef] [PubMed]
- T. Levola and P. Laakkonen, “Replicated slanted gratings with a high refractive index material for in and outcoupling of light,” Opt. Express15, 2067–2074 (2007). [CrossRef] [PubMed]
- F. Van Laere, M. V. Kotlyar, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact slanted grating couplers between optical fiber and InP-InGaAsP waveguides,” IEEE Photon. Technol. Lett.19, 396–398 (2007). [CrossRef]
- K. Ventola, J. Tervo, P. Laakkonen, and M. Kuittinen, “High phase retardation by waveguiding in slanted photonic nanostructures,” Opt. Express19, 241–246 (2011). [CrossRef] [PubMed]
- L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A: Pure appl. Opt.5, 345–355 (2003). [CrossRef]
- J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder-Mead simplex algorithm in low dimensions”, SIAM Optim J.9, 112–147 (1999). [CrossRef]
- H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J.48, 2909–2947 (1969).
- S. Siitonen, P. Laakkonen, J. Tervo, and M. Kuittinen, “A double-sided grating coupler for thin light guides,” Opt. Express15, 2008–2018 (2007). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.