Double groove broadband gratings

doi:10.1364/OE.16.013824

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Double groove broadband gratings

Juha Pietarinen and Tuomas Vallius »View Author Affiliations

Optics Express, Vol. 16, Issue 18, pp. 13824-13830 (2008)
http://dx.doi.org/10.1364/OE.16.013824

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▸ Article Outline
– Abstract
1. Introduction
2. Grating configuration and notations
3. Waveguiding analysis
4. Results
5. Conclusions
– References and links

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Abstract

Waveguiding in periodical structures of the size of the wavelength is applied to increase the functional spectral band of diffractive optics. The deviation of the effective refractive index between waveguides as a function of the wavelength is utilized to compensate the strong wavelength dependence of the efficiency of diffraction gratings.

1. Introduction

Since the efficiency of diffraction gratings, in general, depends crucially on the wavelength, wideband illumination usually presents serious problems with gratings [1

B. Braam, J. Okkonen, M. Aikio, K. Makisara, and J. Bolton, “Design and first test results of the Finnish airborne imaging spectrometer for different applications, AISA”, in Imaging Spectrometry of the Terrestial Environment, G. Vane, ed., Proc. SPIE 1937, 142–151 (1993). [CrossRef]

–3

J. Pietarinen, T. Vallius, and J. Turunen, “Wideband four-level transmission gratings with flattened spectral efficiency,” Optics Express , 14, No. 7 2583–2588 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-7-2583 [PubMed]

]. This decreases the resolution in miniature spectral detectors when making digitalization of signal with varying spectral response. Therefore the accuracy of the spectral analyzer can be enhanced by avoiding the wavelength dependence of the grating response [1

The rapid decay of the efficiency when altering the wavelength is a serious problem yet to be solved. In the geometrical-optics picture, the phase delay of light caused by propagation through a dielectric structure of height h in air is proportional to nh/λ, when n is the refractive index of the medium and λ is the optical wavelength [4

R. Petit, ed., Electromagnetic Theory of Gratings (Springer, Berlin, 1980). [CrossRef]

, 5

J. Turunen, M. Kuittinen, and F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, E. Wolf, ed., chap. V (Elsevier, Amsterdam, 2000) Vol. XL.

]. Therefore the phase of the field after passing through the grating depends strongly on the incident wavelength. Traditionally this cannot be avoided but the changes in the phase degrade the performance of the grating in polychromatic illumination [6

H. P. Herzig, ed., Micro-optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).

–8

M. C. Hutley, Diffraction Gratings (Academic Press, Orlando, 1982).

]. Metamaterials comprising photonic crystals have been applied to this problem to obtain broadband behavior [9

C. Sauvan, P. Lalanne, and M.-S. L. Lee, “Broadband blazing with artificial dielectrics,” Opt. Lett. 29, 1593–1595 (2004). [CrossRef] [PubMed]

]. We address this subject by considering novel double-groove resonance-domain structures instead of classical binary, stair-step or continuous surface profiles. Structures that consist of pillars of different heights are found to solve the problem. Interpretations based on wave guiding effects and effective refractive indices are provided.

2. Grating configuration and notations

The parameters of the double groove gratings to be considered are shown in Fig. 1a. The period, incident angle and effective indices of the pillars are denoted by d, θ, and n _eff, respectively (the reason why effective indices instead of real refractive indices are used will become apparent below). Parameters x_i and z _j denote the transition point locations and profile depths with i=1,2,3 and j=1, 2. The configuration is assumed to be invariant in the y-direction and the incident field propagates in the xz-plane. The refractive indices of the half-spaces in incident and transmitted sides are n ₁ and n ₂. The wave vector of the incident field is denoted by k=2π/λ

When light propagates in structures that consist of adjacent pillars close to each other, light is confined to the regions of higher refractive index. This is shown in Fig. 1b, which illustrates the electric energy density of light in such a structure. At least in first approximation, the regions between the pillars can be considered ‘dark’ even though the pillar widths and separations are in the wavelength scale. Thus the response of the grating can be governed by considering light propagation within the pillars alone. Considering each pillar separately, light can be assumed to propagate as in a planar waveguide, infinite in the z-direction. Generally, in waveguides, light propagates in the form of waveguide modes with different lateral distributions and effective propagation speeds. If the thickness of the waveguide is of the order λ, the lowest-order mode is dominant. Thus its effective refractive index n _eff defines the local phase delay of light passing through the structure. It should be emphasized that, although the wider pillar in Fig. 1b supports several upward- and downward-propagating modes, our calculations shows that approximately 95% of energy flows (upward) in the lowest-order mode, which justifies the effective-index approximation. Nevertheless, because of coherent interference, the higher-order modes cause strong intensity modulation inside the pillars.

3. Waveguiding analysis

The geometry of Fig. 1(a) can be used to determine (approximately) the local optical path through the structure if we assume that the two pillars support only the lowest-order waveguide modes and no light is transmitted through the gaps between the pillars. Denoting the effective refractive indices of the lowest-order modes in the two pillars by n _eff1 and n _eff2, the phase difference between the modes guided through these pillars is

Δ ϕ = (h_{2} n_{{eff}_{2}} - h_{1} n_{{eff}_{1}} - h_{1} n_{1}) \frac{2 π}{λ} .

(1)

Though not shown explicitly, the wavelength dependence of the phase difference given by Eq. (1) is influenced by the chromaticity of the effective refractive indices, which we proceed to consider in more detail.

Fig. 1. (a) The double groove type of transmission-grating profile. (b) The electric energy density in and near the grating region.

The parameters of the pillars are designed to simulate a broadband grating for broadband unpolarized illumination over the visible region (the details are presented below in Table 1 on the row VIS-UNPOL and in Fig. 4). For brevity, the difference of the effective refractive indices is denoted by Δn _eff=n _eff2-n _eff1. The wavelength dependence of the effective indices as well as their difference are presented in Figs. 2(a) and 2(b). Although the values of the effective indices decrease towards longer wavelengths, their difference Δn _eff increases (for illustrative purposes, also the dependence on 1/λ is plotted). In first approximation, the wavelength dependence of Δn _eff can be assumed to be linear and therefore the product between Δn _eff and 1/λ is nearly constant. Indeed, as seen from Fig. 2(c), the wavelength dependence of the phase difference Δϕ calculated from Eq. (1) nearly disappears. Here Δϕ is normalized by π to illustrate the magnitude of the deviation from the desired value 1 and the grating height in the analysis is h ₁=h ₂=1220 nm. The error in the phase shift is seen to be less than 15% over the whole wavelength range. Hence, using the double-pillar approach to construct grating profiles, the strong wavelength dependence of diffraction efficiency can be greatly reduced.

Fig. 2. (a) The lowest-order effective refractive indices of two pillars, (b) their difference n _eff2-n _eff1 on _λ and 1/_λ, and (c) the wavelength dependence of the phase difference caused by the pillars.

Since the feature sizes of the structure are of the order the wavelength and the method used to calculate effective indices is valid only for a single pillar [10

T. Tamir, Integrated Optics (Springer-Verlag; 2nd edition, 1979).

], the effective-medium approach presented above can only give on approximation of the optimal profile. Consequently, we must resort to rigorous electromagnetic diffraction theory [4

R. Petit, ed., Electromagnetic Theory of Gratings (Springer, Berlin, 1980). [CrossRef]

, 5

J. Turunen, M. Kuittinen, and F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, E. Wolf, ed., chap. V (Elsevier, Amsterdam, 2000) Vol. XL.

, 7

J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Wiley-VCH, Berlin, 1997).

, 8

M. C. Hutley, Diffraction Gratings (Academic Press, Orlando, 1982).

, 11

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]

] to obtain final designs. The structure given by the effective index approach is used as a starting point of optimization. Then the grating-profile parameters are adjusted by nonlinear optimization as in Refs. [3

,12

E. Noponen, A. Vasara, and J. Turunen, “Parametric optimization of multilevel diffractive optical elements by electromagnetic theory,” Appl. Opt. 31, 5910–5912 (1992). [CrossRef] [PubMed]

,13

K. Blomstedt, E. Noponen, and J. Turunen, “Surface-profile optimization of diffractive imaging lenses,” J. Opt. Soc. Am. A 18, 521–525 (2001). [CrossRef]

]. The goal is to obtain a high and uniform diffraction efficiency in the first diffraction order (-1) over the desired spectral range.

4. Results

The design procedure is applied to gratings for linearly polarized and unpolarized illumination in the visible and infrared regions. The main results are collected in Figs. 3 and 4 and in Table 1. The refractive index of the substrate, n ₁=n _SiO1 (λ), contains the wavelength dependence of fused silica, and n ₂=1.

The first design (Fig. 3(a) and Table 1, IR-TM) is for linearly TM-polarized illumination in the infrared (IR) region over the wavelength range 1000 nm–2000 nm. The diffraction efficiency of order -1 exhibits only small variation in the interval 74%–81% over the entire wavelength band. The second structure is designed to function over the visible wavelength range 300 nm–800 nm, also in TM-polarization (Fig. 3(b) and Table 1, VIS-TM). Now the efficiency is close to 70%, and the result clearly outperforms conventional diffraction grating designs in the sense that the chromatic variation is notably small over more than an octave in wavelength scale.

Fig. 3. Spectral efficiency curves of diffraction order -1 for (a) IR-grating and (b) Visible band grating. Dashed lines indicate the efficiency in TE polarization and the solid line in TM polarization for which the grating is designed.

The designs presented above aim at maximal and uniform efficiency in TM polarization. Nevertheless, also in TE polarization the spectral efficiency curves are reasonably uniform even though the efficiencies are lower. The design method can, however, applied also to design gratings for unpolarized light. In that case the merit function in parametric optimization is the average value of the efficiencies in TE and TM polarization. The achieved spectral efficiency curves are presented in Fig. 4, with the grating parameters in Table 1 (VIS-UNPOL). Now the efficiency is somewhat lower than in the case of single polarization, but still remarkably uniform over the entire visible range. The structure is not optimal for either state of linear polarization but works sufficiently well for both polarizations simultaneously and these pillars are used in the waveguide mode computations for Fig. 2. This profile, designed for the visible wavelength band, was chosen to be fabricated for experimental verification.

Table 1. Quantitative characterization of the designed and fabricated grating profiles.

Design	d [nm]	[x ₁,x ₂,x ₃] [nm]	[z ₁, z ₂] [nm]	θ [°]
IR-TM	3572	[929,1464,2072]	[367,3851]	-6
VIS-TM	1287	[309,553,695]	[160,1539]	-4
VIS-UNPOL	1204	[325,506,698]	[140,1347]	-5
Experimental
VIS-UNPOL	1200	[336,533,739]	[123,1297]	-5

Usually the fabrication would require a double exposure process: the first exposure is for reactive ion etching (RIE) mask fabrication of both grooves in a SiO₂ substrate. The chromium mask is fabricated by a standard lift-off process. In the second exposure the masking is done for the other groove etching by the resist mask when only the depth difference of two grooves is etched. Consequently, after the resist mask removal the final depth of pillars can be etched with the metallic mask.

In some cases, such as now when the wider groove is only a few percent deeper than the other, the limitations of the RIE etching process can actually simplify the fabrication process of the double-groove structures considered here. This is because the RIE speed of high-aspect-ratio structures is proportional to the dimension of the groove: the narrower the groove, the slower the etching speed. This feature can enable the etching of the whole structure in one process step. Now only one metal mask is needed and the grooves automatically have different depths as the design requires.

Fig. 4. Comparison of the theoretical spectral efficiency (curves) and experimental results (marks). Dotted line TM, dashed line TE and solid line for the unpolarized light.+indicates the measurement results of TM polarization, ×TE and ∘ the unpolarized light.

The example structure shown in Fig. 5 was fabricated by exposing a 180 nm thick (positive) PMMA electron-beam resist layer (AR-661) with Leica LION LV-1 e-beam writer. After development of the resist, an 80 nm layer of chromium was evaporated in high vacuum at relatively slow (less than 2 nm/s) evaporation speed for the lift-off process to create the mask for RIE of SiO₂. The structure was etched by Plasmalab 80 dry etching equipment with CHF₃ and Ar as the etching gases. The measured parameters of the fabricated grating are also presented in in the last row of Table 1.

Fig. 5. SEM image of the cross-section of the fabricated fused silica grating.

The optical function of the grating was verified by measuring the transmission efficiency of the -1 diffraction order (η-1) at several laser wavelengths under TM and TE polarizations. The beam arrived at the angle of -5° and the intensity of the -1 diffraction order (I_t _-1) was measured. The Fresnel reflection in the first interface were taken into account by using the average of measured transmission of the clear substrate and beam intensity as the intensity of the incident beam (I_beam +I_TSiO ₂)/2=I ₀. Hence, the total diffraction efficiency was calculated from the relation I_t -1/I ₀=η-1. The results of the measurements are shown in Fig. 4, where the +and x marks indicate the measurements of TM and TE polarization, respectively, and the ° marks represents the calculated average of both polarizations. Typically, the experimental efficiencies of resonance-domain gratings are about 10% lower than the predictions of theoretical designs, but here the experimental and theoretical results are almost identical. This is true even in the edges of the desired spectrum, a noteworthy feature with diffractive structures under wideband illumination.

5. Conclusions

The use periodic structure of grooves and waveguide pillars of different heights enables accurate control over the spectral response of diffraction gratings. Depending on the application, we can obtain nearly wavelength-independent efficiency curves for either polarized or unpolarized light by utilization of the waveguide mode dispersion to compensate both grating and material dispersion. The profile designed for visible wavelength range band was successfully fabricated by e-beam lithography and RIE processes of binary gratings. The functionality of the grating was verified by optical transmission measurements and the results strongly support the theoretical calculations.

6. Acknowledgments

The work of T. Vallius was supported by the Academy of Finland (projects 106410 and 207523). Support from the Network of Excellence in Micro-Optics (NEMO, www.micro-optics.org), assistance of Jari Turunen, Heikki J. Hyvärinen and Hemmo Tuovinen, as well as discussions with Pasi Vahimaa and Pasi Laakkonen are appreciated.

References and links

1.	B. Braam, J. Okkonen, M. Aikio, K. Makisara, and J. Bolton, “Design and first test results of the Finnish airborne imaging spectrometer for different applications, AISA”, in Imaging Spectrometry of the Terrestial Environment, G. Vane, ed., Proc. SPIE 1937, 142–151 (1993). [CrossRef]
2.	S. H. Kong, D. D. L. Wijngaards, and R. F. Wolffenbuttel, “Infrared micro-spectrometer based on diffraction gratings,” Sensors and Actuators A 92, 88–95 (2001). [CrossRef]
3.	J. Pietarinen, T. Vallius, and J. Turunen, “Wideband four-level transmission gratings with flattened spectral efficiency,” Optics Express , 14, No. 7 2583–2588 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-7-2583 [PubMed]
4.	R. Petit, ed., Electromagnetic Theory of Gratings (Springer, Berlin, 1980). [CrossRef]
5.	J. Turunen, M. Kuittinen, and F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, E. Wolf, ed., chap. V (Elsevier, Amsterdam, 2000) Vol. XL.
6.	H. P. Herzig, ed., Micro-optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).
7.	J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Wiley-VCH, Berlin, 1997).
8.	M. C. Hutley, Diffraction Gratings (Academic Press, Orlando, 1982).
9.	C. Sauvan, P. Lalanne, and M.-S. L. Lee, “Broadband blazing with artificial dielectrics,” Opt. Lett. 29, 1593–1595 (2004). [CrossRef] [PubMed]
10.	T. Tamir, Integrated Optics (Springer-Verlag; 2nd edition, 1979).
11.	L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]
12.	E. Noponen, A. Vasara, and J. Turunen, “Parametric optimization of multilevel diffractive optical elements by electromagnetic theory,” Appl. Opt. 31, 5910–5912 (1992). [CrossRef] [PubMed]
13.	K. Blomstedt, E. Noponen, and J. Turunen, “Surface-profile optimization of diffractive imaging lenses,” J. Opt. Soc. Am. A 18, 521–525 (2001). [CrossRef]

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(050.1970) Diffraction and gratings : Diffractive optics

ToC Category:
Diffraction and Gratings

History
Original Manuscript: May 19, 2008
Revised Manuscript: June 26, 2008
Manuscript Accepted: July 30, 2008
Published: August 22, 2008

Citation
Juha Pietarinen and Tuomas Vallius, "Double groove broadband gratings," Opt. Express 16, 13824-13830 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13824

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References

B. Braam, J. Okkonen, M. Aikio, K. Makisara, J. Bolton, "Design and first test results of the Finnish airborne imaging spectrometer for different applications, AISA," Proc. SPIE 1937, 142-151 (1993). [CrossRef]
S. H. Kong, D. D. L. Wijngaards, and R. F. Wolffenbuttel, "Infrared micro-spectrometer based on diffraction gratings," Sens. Actuators A 92, 88-95 (2001). [CrossRef]
J. Pietarinen, T. Vallius, and J. Turunen, "Wideband four-level transmission gratings with flattened spectral efficiency," Opt. Express 14, 2583-2588 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14 7-2583 [PubMed]
R. Petit, ed., Electromagnetic Theory of Gratings (Springer, Berlin, 1980). [CrossRef]
J. Turunen, M. Kuittinen, and F. Wyrowski, "Diffractive optics: electromagnetic approach," in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 2000) Vol. XL., Chap. V
H. P. Herzig, ed., Micro-optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).
J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Wiley-VCH, Berlin, 1997).
M. C. Hutley, Diffraction Gratings (Academic Press, Orlando, 1982).
C. Sauvan, P. Lalanne, and M.-S. L. Lee, "Broadband blazing with artificial dielectrics," Opt. Lett. 29, 1593-1595 (2004). [CrossRef] [PubMed]
T. Tamir, Integrated Optics, 2nd ed., (Springer-Verlag; 1979).
L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870- 1876 (1996). [CrossRef]
E. Noponen, A. Vasara, and J. Turunen, "Parametric optimization of multilevel diffractive optical elements by electromagnetic theory," Appl. Opt. 31, 5910-5912 (1992). [CrossRef] [PubMed]
K. Blomstedt, E. Noponen, and J. Turunen, "Surface-profile optimization of diffractive imaging lenses," J. Opt. Soc. Am. A 18, 521-525 (2001). [CrossRef]

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