## Highly-dispersive dielectric transmission gratings with 100% diffraction efficiency

Optics Express, Vol. 16, Issue 8, pp. 5577-5584 (2008)

http://dx.doi.org/10.1364/OE.16.005577

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### Abstract

A new approach for the realization of highly dispersive dielectric transmission gratings is presented, which enables the suppression of any reflection losses and, thus, 100% diffraction efficiency. By applying a simple two-mode-model a comprehensible explanation as well as a theoretical design of such a reflection-free transmission grating is presented.

© 2008 Optical Society of America

## 1. Introduction

^{st}order. Usually, the highest diffraction efficiency can be achieved, if the grating is illuminated in the Littrow configuration at an angle

*φ*which fulfills the condition

_{in}*λ*is the vacuum wavelength and p the grating period [1

1. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. **72**, 1385–1391 (1982). [CrossRef]

^{st}transmitted order propagates symmetrically to the 0

^{th}order (Fig. 1).

*p*fulfills the condition

*n*is the refractive index of the substrate material) it is theoretically possible to deflect 100% of the transmitted light to the −1

^{st}order. In an earlier paper [2

2. T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, U. Peschel, A. V. Tishchenko, and O. Parriaux, “An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings,” Opt. Express **13**, 10448–10456 (2005). [CrossRef] [PubMed]

*Gsolver*, which is based upon the rigorous Fourier modal method [3]. Depending on the groove depth and the fill factor (ratio between grating ridge width and period) the reflection of a rectangular surface relief grating can vary within the values that are illustrated by the grey area. For comparison, the dashed blue line shows the Fresnel-reflection at a plane interface by illuminating it under an angle that corresponds to the Littrow angle of a particular grating period. The reflection at the grating is mostly less than the reflection at a plane interface, but similar to the Fresnel-reflection, it increases significantly if the grating period approaches λ/2. As an example, the reflectivity of a rectangular fused silica transmission grating with a period of 600nm cannot be less than 7% for a wavelength of 1064nm (Littrow angle 62.45°). The diffraction efficiency of the −1

^{st}transmitted order can, thus, theoretically not exceed 93%.

4. J. Nishii, K. Kintaka, and T. Nakazawa, “High-Efficiency Transmission Gratings Buried in a Fused-SiO2 Glass Plate,” Appl. Opt. **43**, 1327–1330 (2004). [CrossRef] [PubMed]

2. T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, U. Peschel, A. V. Tishchenko, and O. Parriaux, “An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings,” Opt. Express **13**, 10448–10456 (2005). [CrossRef] [PubMed]

## 2. Investigation of the grating reflection

5. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta. **28**, 413–428 (1981). [CrossRef]

2. T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, U. Peschel, A. V. Tishchenko, and O. Parriaux, “An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings,” Opt. Express **13**, 10448–10456 (2005). [CrossRef] [PubMed]

6. A. V. Tishchenko, “Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. **37**, 309–330 (2005). [CrossRef]

7. P. Lalanne, J. P. Hugonin, and P. Chavel, “Optical properties of deep lamellar Gratings: A coupled Blochmode insight,” J. Lightwave Technol. **24**, 2442–2449 (2006). [CrossRef]

1. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. **72**, 1385–1391 (1982). [CrossRef]

^{th}and the 1

^{st}, are equally excited by the incident wave. They propagate through the grating and couple to the diffraction orders at the grating bottom (Fig. 3). If the reflection at the air-grating and grating-substrate interface is neglected, the intensities of the two transmitted diffraction orders (the 0

^{th}and the −1

^{st}) are a result of a two-beam-interference mechanism. Their intensities are determined by the phase differences between the two grating modes at the grating bottom. The propagation of the two modes through the grating (in

*z*-direction) is described by their propagation constants

*k*=

^{m}_{z}*k*·

_{0}*n*, or rather their effective refractive indices

^{m}_{eff}*n*, where

^{m}_{eff}*m*is the number of the mode. According to this simple two-mode-model, the efficiencies

*η*and

_{0T}*η*-

_{1T}of the 0

^{th}and the −1

^{st}transmitted diffraction order can be calculated by

*h*describes the groove depth, where the two modes exhibit a phase difference of π and a maximum diffraction to the −1

_{max}^{st}transmitted order is achieved. In this model the groove depth

*h*just represents a propagation distance and the diffraction is determined by the effective indices of the two modes. These

*n*in turn depend on the grating period, the fill factor, the wavelength and the polarization of light incident upon the grating, and so does the diffraction behavior. For the following investigation a 600nm grating period will be assumed, which is pretty close to λ/2 in the case of illumination with a 1064nm wavelength. Figure 4 illustrates the dependence of both effective indices on the fill factor for this grating. From this dependency, which can easily be found by applying the Modal Method (see e.g. [8

_{eff}8. J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numerique des efficacités de certains réseaux diélectriques profonds,” J. Opt. **14**, 273–288 (1983). [CrossRef]

^{st}order would be 100%, if the two modes accumulated a phase difference of π (and odd-numbered multiples of π) at the grating bottom. Exact numerical calculations for a conventional surface relief grating show a maximum diffraction efficiency of 93%. The missing 7% are due to reflection effects that have been neglected in this simple model.

*n*=

_{eff}^{air}*cosφ*or the refracted wave within the substrate material

_{in}*n*=

_{eff}^{sub}*ncosφ’*for fill factors

_{in}*f*→0 or

*f*→1, respectively. This is obvious since the grating represents a plane interface in this case. The reflectivity of such a “grating” with a fill factor of 0 or 1 is well known; it corresponds to the Fresnel-reflection at a plane interface. This reflectivity depends on how much the propagation direction changes. In other words: the more the effective index is changed at the interface, the higher is the reflectivity. This description of the reflection at a plane interface can also be used for the reflection of grating modes. At the air-grating interface the effective index of the incident wave is changed to those of the two grating modes. Furthermore, at the grating bottom, both modes couple to the two transmitted diffraction orders, where the effective index is changed again. Due to the Littrow configuration (symmetrical propagation of 0

^{th}and −1

^{st}order) both transmitted and reflected diffraction orders possess the same effective index

*n*in air or

_{eff}^{air}*n*within the substrate, respectively. Therefore, in analogy to the Fresnel equations, the reflection of the grating modes at the interface to the air superstrate

_{eff}^{sub}*R*or to the substrate

_{m}^{air}*R*, can be calculated by:

_{m}^{sub}*m*is the index of the respective mode. Figure 5 shows the calculated reflectivity of both propagating grating modes at the grating-air (blue lines) and the grating-substrate interface (green lines). The reflection of the 0

^{th}mode is higher at the interface to air, while the 1

^{st}mode shows a higher reflection at the substrate interface. The reflection of the whole grating device is a result of the complex interplay of all contributions from both modes and both interfaces. This fact makes it impossible to suppress the reflection losses.

^{st}mode exhibits a significant reflection at the grating-substrate interface. Therefore, the entire reflection process would be much easier to understand and to control, if there was only this kind of interface. More precisely, the complex reflection process can be reduced to a very simple problem by embedding the fused silica – air grating into its substrate material.

## 3. The new approach: a buried grating

^{st}mode. If the fill factor of the grating is less than 0.5 the reflectivity of the 0

^{th}mode will increase, and this assumption becomes less precise. However, even for a fill factor of 0.2 the reflectivity of the 1

^{st}mode is much higher than the one of the 0

^{th}mode. Even there, the reflection behavior is mainly dominated by the 1

^{st}mode. Since the buried grating adjoins to homogeneous fused silica at its top as well as at its bottom, the 1

^{st}mode exhibits the same reflectivity at both interfaces. Therefore, the reflection of the grating can be described by a symmetrical Fabry-Perot-Resonator which is filled with a medium of refractive index

*n*. If the length of this resonator or rather the groove depth is a whole-number multiple of

_{eff}^{1}*R*can be described by

*R*is the reflectivity of the 1

_{1}^{sub}^{st}mode at the grating-substrate interface. The multiplier ½, in contrast to a classical Fabry-Perot-Resonator, is due to the undisturbed transmission of the 0

^{th}mode, which carries half of the energy and does not take part in the reflection. The reflectivity is

*R*=0 for groove depths of

*h*=

*k*·

*h*(where

_{R}*k*is a positive whole number). On the contrary, the maximum is

*h*=(

*m*+

*1/2*)·

*h*, where

_{R}*F*and, thus,

*R*depend on the fill factor of the grating. In Fig. 6 the reflectivity according to Eq. (7) is plotted as a function of the groove depth and the fill factor of the rectangular grating profile. The groove depths

_{max}*h*and their whole-numbered multiples, which effect a complete suppression of reflection, are illustrated by dashed white lines. The plot reveals a decrease of the reflectivity

_{R}*R*with increased grating fill factor. Thus, the tolerance of the grating parameters to obtain a grating with a transmission near 100% is much larger for large fill factors.

_{max}^{st}order. This is achieved for a groove depth according to Eq. (4) and its whole-number multiples. On the other hand, the mode that is reflected, has to accumulate a phase of 2π by propagating twice through the grating. This is achieved for groove depths according to Eq. (6) and whole-number multiples. The diffraction efficiency η

_{-1}of a buried grating can be expressed using Eqs. (7) and (4)

^{st}order. A grating with a period as discussed above even shows almost no diffraction for a fill factor near 0.5. This is caused by the very small difference of the effective indices of the two TM-modes for small periods, which has been discussed in more detail, e.g. in Ref. [10

10. T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, A. V. Tishchenko, and O. Parriaux, “Investigation of the polarization-dependent diffraction of deep dielectric rectangular transmission gratings,” Appl. Opt. **46**, 819–826 (2007). [CrossRef] [PubMed]

^{st}order, even though the reflectivity could be reduced.

## 4. Discussion and conclusion

^{st}but the 0

^{th}mode has to be considered, according to Fig. 5. Small fill factors instead of large ones would be necessary. This concept, however, appears to be much more difficult to fabricate, which is why it has not been discussed here in detail.

## Acknowledgment

*Deutsche Forschungsgemeinschaft*within the Sonderforschungsbereich “Gravitational wave astronomy” and the Gottfried-Wilhelm-Leibniz-Program.

## References and links

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

2. | T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, U. Peschel, A. V. Tishchenko, and O. Parriaux, “An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings,” Opt. Express |

3. | J. Turunen, “Diffraction theory of dielectric surface relief gratings,” in |

4. | J. Nishii, K. Kintaka, and T. Nakazawa, “High-Efficiency Transmission Gratings Buried in a Fused-SiO2 Glass Plate,” Appl. Opt. |

5. | L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta. |

6. | A. V. Tishchenko, “Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. |

7. | P. Lalanne, J. P. Hugonin, and P. Chavel, “Optical properties of deep lamellar Gratings: A coupled Blochmode insight,” J. Lightwave Technol. |

8. | J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numerique des efficacités de certains réseaux diélectriques profonds,” J. Opt. |

9. | E. B. Kley, H. J. Fuchs, and K. Zöllner, “Fabrication technique for high-aspect-ratio gratings,” in Micromachine Technology for Diffractive and Holographic Optics, Proc. SPIE |

10. | T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, A. V. Tishchenko, and O. Parriaux, “Investigation of the polarization-dependent diffraction of deep dielectric rectangular transmission gratings,” Appl. Opt. |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(230.1950) Optical devices : Diffraction gratings

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: February 7, 2008

Revised Manuscript: March 31, 2008

Manuscript Accepted: April 2, 2008

Published: April 7, 2008

**Citation**

T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, A. V. Tishchenko, and O. Parriaux, "Highly-dispersive dielectric transmission gratings with 100% diffraction efficiency," Opt. Express **16**, 5577-5584 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5577

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### References

- M. G. Moharam and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1391 (1982). [CrossRef]
- T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, U. Peschel, A. V. Tishchenko, and O. Parriaux, "An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings," Opt. Express 13, 10448-10456 (2005). [CrossRef] [PubMed]
- J. Turunen, "Diffraction theory of dielectric surface relief gratings," in Micro-optics, H.P. Herzig ed. (Taylor&Francis Inc., 1997).
- J. Nishii, K. Kintaka, and T. Nakazawa, "High-Efficiency Transmission Gratings Buried in a Fused-SiO2 Glass Plate," Appl. Opt. 43, 1327-1330 (2004). [CrossRef] [PubMed]
- L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta. 28, 413-428 (1981). [CrossRef]
- A. V. Tishchenko, "Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method," Opt. Quantum Electron. 37, 309-330 (2005). [CrossRef]
- P. Lalanne, J. P. Hugonin, and P. Chavel, "Optical properties of deep lamellar Gratings: A coupled Bloch-mode insight," J. Lightwave Technol. 24, 2442- 2449 (2006). [CrossRef]
- J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numerique des efficacités de certains réseaux diélectriques profonds," J. Opt. 14, 273-288 (1983). [CrossRef]
- E. B. Kley, H. J. Fuchs, and K. Zöllner, "Fabrication technique for high-aspect-ratio gratings," in Micromachine Technology for Diffractive and Holographic Optics, Proc. SPIE 3879, S. H. Lee, J. A. Cox, Eds., 71-78 (1999).
- T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, A. V. Tishchenko, and O. Parriaux, "Investigation of the polarization-dependent diffraction of deep dielectric rectangular transmission gratings," Appl. Opt. 46, 819-826 (2007). [CrossRef] [PubMed]

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