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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 8 — Apr. 14, 2008
  • pp: 5577–5584
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Highly-dispersive dielectric transmission gratings with 100% diffraction efficiency

T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, A. V. Tishchenko, and O. Parriaux  »View Author Affiliations


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

Dielectric transmission gratings have emerged as the key components in many present-day optical setups. One reason for this is their high resistance against laser induced damage, but also the possibility to realize various optical functions simply by optimizing their profile parameters. Many applications such as high-power femtosecond laser setups require a highly-efficient diffraction into the dispersive −1st order. Usually, the highest diffraction efficiency can be achieved, if the grating is illuminated in the Littrow configuration at an angle φin which fulfills the condition

sinφin=λ2p,
(1)

where 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]

]. In this configuration, the −1st transmitted order propagates symmetrically to the 0th order (Fig. 1).

Fig. 1. Grating and illumination parameters.

If, furthermore, the grating period p fulfills the condition

λ2<p<3λ2n
(2)

Fig. 2. Reflection of a rectangular surface relief fused silica transmission grating as a function of the grating period, and the corresponding Fresnel-reflection at a plane air-fused silica interface (wavelength 1.064µm, TE-polarized illumination).
Fig. 3. The two-mode-diffraction model.

One route to overcome this limitation is to use triangular instead of rectangular grooves. By optimizing such a profile, a diffraction efficiency exceeding 99% is theoretically possible even for a grating period which is nearly half of the wavelength. Unfortunately, such profiles can hardly be fabricated with the required profile accuracy. Another approach, which is the subject of this paper, is to embed a rectangular grating into the dielectric substrate material. By optimizing the profile parameters (groove depth and width) of the buried grating a complete suppression of any reflection and, thus, a diffraction efficiency of 100% is theoretically possible. Previously, it was proposed that a dielectric diffraction grating could be protected against mechanical damage and dust by burying it [4

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]

]. It was demonstrated that the diffraction efficiency barely changes by applying an appropriate coating procedure. In the following, it will be shown how this approach can even be exploited in order to significantly increase the diffraction efficiency.

In Ref. 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]

the diffraction of a rectangular surface relief transmission grating was explained on the basis of a simple two-mode-model. In the present paper this model will be expanded by reflection effects in order to investigate the origin of the reflection losses. This consideration easily explains why the reflection losses can completely be suppressed in case of a buried grating.

2. Investigation of the grating reflection

According to Ref. 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]

, the diffraction of a deep dielectric transmission grating can be described by only two modes, when it is illuminated in the Littrow-configuration and its period fulfills the condition (2). These two modes, namely the 0th and the 1st, 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 0th and the −1st) 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 kmz=k0·nmeff, or rather their effective refractive indices nmeff, where m is the number of the mode. According to this simple two-mode-model, the efficiencies η0T and η-1T of the 0th and the −1st transmitted diffraction order can be calculated by

η1T(f,h)=sin2(π2hhmax(f))
(3a)

and

η0T(f,h)=cos2(π2hhmax(f)),
(3b)

with

hmax(f)=λ2Δneff(f)=λ2neff0(f)neff1(f).
(4)

Fig. 4. Effective indices of the two propagating modes against the fill factor.
Fig. 5. Reflection of the two modes at the grating-air (blue) and the grating-substrate (green) interface.

However, the two-mode-model can easily be expanded to investigate the reflection at a grating. In Fig. 4 the effective indices approach the one of the incident wave in air neffair=cosφin or the refracted wave within the substrate material neffsub=ncosφ’in for fill factors 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 0th and −1st order) both transmitted and reflected diffraction orders possess the same effective index neffair in air or neffsub within the substrate, respectively. Therefore, in analogy to the Fresnel equations, the reflection of the grating modes at the interface to the air superstrate Rmair or to the substrate Rmsub, can be calculated by:

Rmairsub(f)=(neffm(f)neffairsubneffm(f)+neffairsub)2,
(5)

where 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 0th mode is higher at the interface to air, while the 1st 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.

A more detailed analysis of Fig. 5 reveals that for fill factors larger than 0.5 only the 1st 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

The considerations above showed that the reflection behavior of a grating with a fill factor larger than 0.5, which is embedded into fused silica, is almost completely determined by the 1st mode. If the fill factor of the grating is less than 0.5 the reflectivity of the 0th mode will increase, and this assumption becomes less precise. However, even for a fill factor of 0.2 the reflectivity of the 1st mode is much higher than the one of the 0th mode. Even there, the reflection behavior is mainly dominated by the 1st mode. Since the buried grating adjoins to homogeneous fused silica at its top as well as at its bottom, the 1st 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 neff1. If the length of this resonator or rather the groove depth is a whole-number multiple of

hR(f)=λ2neff1(f),
(6)

the reflection will be completely suppressed. In general, the reflectivity R can be described by

where

R(f,h)=12·F(f)·sin2(πhhR(f))1+F(f)·sin2(πhhR(f)),whereF(f)=4R1sub(f)(1R1sub(f))2.
(7)

R1sub is the reflectivity of the 1st 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 0th 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·hR (where k is a positive whole number). On the contrary, the maximum is

Rmax(f)=111+F(f)2,
(8)

for h=(m+1/2hR, where F and, thus, Rmax 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 hR 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 Rmax 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.

Fig. 6. Simulation of the reflection according to Eq. (7). The dashed white lines represent the groove depths where a complete suppression of the reflection is achieved (Eq. (6)).

In order to realize a buried transmission grating with 100% diffraction efficiency it is necessary to fulfill two conditions. On the one hand, the two propagating grating modes have to accumulate a phase difference of π in order to deflect all transmitted light to the −1st 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)

η1(f,h)=η1T(f,h)·(1R(f,h))
=sin2(π2hhmax(f))·1+12F(f)·sin2(πhhR(f))1+F(f)·sin2(πhhR(f)).
(9)

In Figs. 7(a) and (b) diffraction efficiencies of higher than 98% are illustrated by the white areas. The groove depths for which Eq. (4) and (6) are fulfilled are represented by dashed lines in Fig. 7(a). The intersections of these lines represent the grating profile parameters, which are necessary to achieve 100% diffraction efficiency. Exemplarily, a grating with a fill factor of 0.57 (corresponding to a groove width of 260nm) and a groove depth of 1.44µm theoretically exhibits a 99.9% diffraction efficiency. An efficiency higher than 98% would still be achieved, if the fill factor varies by ±0.03 (groove width variation of ±20nm). For fill factors larger than 0.68 the parameter tolerance increases significantly, but the groove parameters become more challenging for the fabrication. For a grating period of 600nm a fill factor of 0.7 corresponds to a groove width of 180nm and the necessary groove depth would be 1.9µm. Even though it is possible to fabricate gratings with comparable parameters [9

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 3879, S. H. Lee and J. A. Cox, Eds., 71–78 (1999).

], a lower fill factor is much more convenient for the fabrication process.

Fig. 7. (a) Simulation of the diffraction efficiency of a buried grating with a 600nm period as a function of the profile parameters, according to the two-mode-model Eq. (9) (wavelength 1064nm, Littrow-configuration). (b) Corresponding numerical calculation by rigorous Fourier modal method.
Fig. 8. Numerical simulation of the diffraction efficiency of the corresponding surface relief grating. The maximum diffraction efficiency is 93%.

In Fig. 8 the numerically simulated diffraction efficiency is illustrated for a corresponding surface relief grating. It has a maximum of 93%, which is not sufficient for many optical setups.

In the discussion above TE-polarized illumination was assumed, which naturally exhibits higher reflection losses than TM-polarized light. One could ask if the problem of high reflectivity of highly-dispersive gratings could be solved by changing the polarization. However, it can be shown that for a small period to wavelength ratio, the TM-polarization requires a very large groove depth to obtain a significant diffraction to the −1st 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]

]. Therefore, it is hardly possible to deflect the transmitted light to the −1st order, even though the reflectivity could be reduced.

4. Discussion and conclusion

In this paper a new approach for the realization of transmission gratings that overcome reflection losses was presented. These gratings are embedded into their substrate material (fused silica), resulting in a symmetrical layout, and, thus, enabling the suppression of any reflection. As an example a grating period of 600nm (~1670 grooves/mm) was discussed for illumination at a 1064nm wavelength. In order to understand the origin of reflection losses, a two-mode-model has been applied, offering a very comprehensible explanation of the diffraction process. This physical investigation gives rise to the idea of embedded gratings. Based on the analogy to a Fabry-Perot-Resonator it explains why and how this approach physically works.

The concept of buried gratings is still valid if the gratings are embedded into air. In this case not the 1st but the 0th 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.

A promising route to fabricate a grating embedded in fused silica is to etch a rectangular profile with the parameters selected from Fig. 7 into the surface of a plane fused silica substrate and afterwards bond a second plane substrate on top of the first one. The fabrication and characterization of samples is currently in progress.

Acknowledgment

This work was funded by the 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. 72, 1385–1391 (1982). [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]

3.

J. Turunen, “Diffraction theory of dielectric surface relief gratings,” in Micro-optics, H.P. Herzig ed. (Taylor&Francis Inc., 1997).

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]

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]

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]

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]

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 3879, S. H. Lee and J. A. Cox, Eds., 71–78 (1999).

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]

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

  1. M. G. Moharam and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1391 (1982). [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]
  3. J. Turunen, "Diffraction theory of dielectric surface relief gratings," in Micro-optics, H.P. Herzig ed. (Taylor&Francis Inc., 1997).
  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]
  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]
  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 Bloch-mode insight," J. Lightwave Technol. 24, 2442- 2449 (2006). [CrossRef]
  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]
  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 3879, S. H. Lee, J. A. Cox, Eds., 71-78 (1999).
  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]

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