# Use of subwavelength gratings in TIR incidence as achromatic phase shifters

Dimitri Mawet, Cédric Lenaerts, Pierre Riaud, Jean Surdej, Serge Habraken,

Denis Vandormael

► To cite this version:

Dimitri Mawet, Cédric Lenaerts, Pierre Riaud, Jean Surdej, Serge Habraken, et al.. Use of subwavelength gratings in TIR incidence as achromatic phase shifters. Optics Express, 2005, 13, pp.8686. hal-03801532

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# Use of subwavelength gratings in TIR incidence as achromatic phase shifters

Dimitri Mawet, Cédric Lenaerts, Pierre Riaud and Jean Surdej

University of Liège, 17 allée du 6 Août,
B-4000, Sart Tilman, Belgium

mawet@astro.ulg.ac.be

Serge Habraken and Denis Vandormael

Centre Spatial de Liège, Avenue du Pré-Aily,
B-4031, Liège-Angleur, Belgium

**Abstract:** Nulling interferometry constitutes a very promising technique in observational astrophysics. This method consists in attenuating the signal of a bright astrophysical object in order to detect much fainter nearby features, e.g. exoplanets around their host star. An on-axis destructive interference is created by adjusting the phases of the beams coming from various telescopes. The huge flux ratio between the parent star and the planet (10⁶ in the thermal infrared) requires unprecedented high performance broadband phase shifters. We present a new design for these key components called Achromatic Phase Shifters (APS). We propose to use subwavelength diffractive optical elements under total internal reflection (TIR) incidence. Our component can be seen as an evolution of the Fresnel Rhomb technology.

© 2005 Optical Society of America

OCIS codes: (120.3180) Interferometry; (050.5080) Phase Shift.

## References and links

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# 1. Introduction

Direct detection and characterization of faint sources around bright astrophysical objects like stars or Active Galactic Nuclei is very difficult due to the large flux ratio between them. For example, an Earth-like exoplanet is typically $10^9$ times fainter than its host star in the visible spectrum, $10^6$ in the thermal infrared. Infrared nulling interferometry proposed by R. Bracewell in 1978 [1], appears to be the most promising technique to achieve the high angular resolution and high dynamic range required to allow the ambitious detection of the first exobiological tracers. The nulling interferometry technique consists in adjusting the phases of the beams coming from various telescopes (two in the most simple configuration) to produce a fully destructive interference on the optical axis. The quality of the destructive interference, or the so-called Null Depth (ND) relies on the optical components ability to induce a very precise phase shift (e.g. $\pi$) and a very low amplitude mismatch over the considered wavelength range. Unfortunately, searching biomarkers in the exoplanet atmospheres requires spectroscopic characterization over large spectral bands. For example, the Darwin Infrared Space Interferometer [2] considered by the European Space Agency will operate in a wavelength band between 6 and 18 microns. High performance Achromatic Phase Shifters (APS) are therefore needed. For such broadband interferometers, we propose to use subwavelength gratings in a total internal reflection (TIR) configuration. These original components fit in the technological evolution of the well-known Fresnel Rhomb technology [3] [4]. Our theoretical calculations using the Rigorous Coupled Wave Analysis point to very promising results.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Schematic representation of a subwavelength grating. The main parameters of the structure are: the grating vector $|K| = 2\pi/\Lambda$, perpendicular to the grating lines, with $\Lambda$ being the spatial period, the grating depth $h$ and the filling factor $f$, such that $f\Lambda$ is the width of the grating ridges. TE and TM are the vectorial orthogonal polarization components of the $\theta$-incident light. $n_i$ and $n_t$ are the refractive indices of the incident (substrate) and emergent (transmitting) media, respectively.</figcaption></figure>

# 2. Principle of the TIRG APS

It is a well-known fact that the TIR phenomenon comes with a differential phase shift between the vectorial TE and TM polarization components. This vectorial phase shift takes the following form [5]

$$ \Delta\phi_{TE-TM} = 2\arctan\left[\frac{\sqrt{\sin^2\theta - n_{ti}^2}}{n_{ti}^2 \cos\theta}\right] - 2\arctan\left[\frac{\sqrt{\sin^2\theta - n_{ti}^2}}{\cos\theta}\right] \quad (1) $$

where $\theta$ is the angle of incidence, greater or equal to $\theta_c$, the critical angle defined as $\sin\theta_c = n_{ti} = n_i/n_t$ and where $n_i$ and $n_t$ are the refractive indices of the incident and emergent media, respectively. This property is exploited in the well-known APS component called Fresnel Rhomb. We demonstrate that engraving a subwavelength grating on the TIR interface leads to a significant improvement over the Fresnel Rhomb technology which is limited by the intrinsic index
dispersion of the material used. When the period of a grating becomes smaller than the wave-
length of the incident light, it does not diffract light as usual in the sense that only the zeroth
transmitted and reflected orders are allowed to propagate outside the modulated regions, leav-
ing wavefronts free from any further aberrations. Furthermore, interaction between the grating
and the vectorial electromagnetic field leads to interesting effects on the phase and amplitude of
the external propagating fields. Indeed, one dimensional subwavelength gratings, i.e., gratings
only modulated along one dimension, turn out to be artificially birefringent. It means that the
structure can be associated with two so-called effective indices, one for each polarization com-
ponent *TE* and *TM*. These effective indices, *n*<sub>TE</sub> and *n*<sub>TM</sub> are totally dependent on the grating
and incidence geometries (see Fig. 1). One can really speak of refractive index engineering.

The principle of the Total Internal Reflection Grating Achromatic Phase Shifter (TIRG APS)
is to use a subwavelength grating in the TIR incidence condition (see Fig. 2). We will show in
the next section that by carefully controlling the grating parameters, the induced vectorial phase
shift can be made as nearly achromatic as possible. Achromatic means that the phase shift value
has to remain constant over the considered wavelength range. As the leading application of
the TIRG APS component concerns nulling interferometry (the TIRG APS high performances
could as well be used in other applications like polarimetry), we have chosen to optimize the
grating design with a figure of merit called the Null Depth. The Null Depth somehow is the
darkness of the destructive interference. The TIR configuration ensures a one hundred percent
efficiency for the back reflected light whatever the polarization, therefore no amplitude mis-
matches. The figure of merit to be minimized consequently resumes to $ND(\lambda) = \sigma^2/4$ where
$\sigma(\lambda)$ is the phase shift error standard deviation with respect to the nominal value of $\pi$ (other
values are possible). This relation, derived from the interference between two plane waves the-
ory, simply means that the Null Depth is directly proportional to the variance of the phase shift
error, i.e., the achromaticity.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Schematic of the TIRG APS component. The TIRG APS is analog to a Fresnel rhomb which TIR interfaces are engraved with an optimized subwavelength grating. A TIRG APS component calculated for a $\pi$ phase shift possesses two TIR interfaces, each providing a $\pi/2$ phase shift such that the resultant is $\Delta\Phi_{TE-TM,1+2} = \Delta\Phi_{TE-TM,1} + \Delta\Phi_{TE-TM,2} = \pi/2 + \pi/2 = \pi$. Such a component is to be inserted in each interferometer arm and orthogonally from one another.</figcaption></figure>

**3. Theoretical design**

We have carried out extensive numerical simulations focused on the two spectral bands of the
Darwin mission: the first one ranges from 6 to 11 microns, the second one from 11 to 18 mi-
crons. We have now to choose among the restricted list of IR materials keeping in mind that
the ZOG technology is flexible enough to accommodate the majority of them provided that
their transparency is sufficient. We will present simulation results for Diamond, Zinc Selenide
<table><caption>Table 1. Temperature-dependant coefficients for material index representations.</caption><thead><tr><th>Coeff.</th><th>Diamond</th><th>ZnSe</th><th>CdTe</th><th>Ge</th></tr></thead><tbody><tr><td>A</td><td>1</td><td>1</td><td>-2.373 × 10<sup>-4</sup>T + 3.8466</td><td>-6.040 × 10<sup>-3</sup>T + 11.05128</td></tr><tr><td>B</td><td>0.3306</td><td>4.46395</td><td>8.057 × 10<sup>-4</sup>T + 3.2215</td><td>9.295 × 10<sup>-3</sup>T + 4.00536</td></tr><tr><td>C</td><td>0.030625</td><td>0.0404</td><td>-1.10 × 10<sup>-4</sup>T + 0.1866</td><td>-5.392 × 10<sup>-4</sup>T + 0.599034</td></tr><tr><td>D</td><td>4.3356</td><td>0.46132</td><td>-2.160 × 10<sup>-2</sup>T + 12.718</td><td>4.151 × 10<sup>-4</sup>T + 0.09145</td></tr><tr><td>E</td><td>0.011236</td><td>0.1538</td><td>-3.160 × 10<sup>1</sup>T + 18753</td><td>1.51408T + 3426.5</td></tr><tr><td>F</td><td>0</td><td>2.88289</td><td>0</td><td>0</td></tr><tr><td>G</td><td>0</td><td>2213.5</td><td>0</td><td>0</td></tr></tbody></table>

(ZnSe), Cadmium Telluride (CdTe) and Germanium (Ge). This choice is justified in three ways: these selected materials are common in IR applications, they cover a large refractive index spectrum (2-4) and their etching processes are well-known. The indices and corresponding dispersions for Diamond[6], ZnSe[7], CdTe[8], Ge[8] will be taken from the following Sellmeier-type relation

$$n_{\text{Diamond,ZnSe,CdTe,Ge}}(\lambda, T) = \left( A + \frac{B\lambda^2}{\lambda^2 - C} + \frac{D\lambda^2}{\lambda^2 - E} + \frac{F\lambda^2}{\lambda^2 - G} \right)^{1/2} \quad (2)$$

The corresponding coefficients are given in Table 1. We will assume a refrigerant temperature $T = 100K$, keeping in mind that corrections are needed if $T$ is different. Note that $\lambda$ is expressed in microns in all representations. To simulate grating responses in the subwavelength domain, scalar theories of diffraction dramatically fail. The vectorial nature of light must be taken into account implying a resolution of the Maxwell equations. We use an algorithm based on the Rigorous Coupled Wave Analysis [9] (RCWA) for the simulations of the grating responses. This algorithm also allows, for each polarization, the visualization of the electromagnetic field distribution (amplitudes and phases). We have performed simplex optimization of the component parameters for the selected materials and for the wavelength band ranging from 6 to 11 microns. The results are summarized in Table 2. We notice that the improvement of the Null Depth performance between the Fresnel Rhomb and TIRG APS technology depends on the selected material, ranging from $10^2$ for the CdTe case up to $10^4$ for the Diamond one. Let us note that in the particular case of the Diamond TIRG APS, the rhomb bulk material can be, for example, ZnSe on which a thin Diamond layer is deposited by Chemical Vapor Deposition (CVD). In Fig. 3, we explicitly show the CdTe TIRG APS results for the two Darwin spectral bands. These results are very good with deep nulls around $5 \times 10^{-9}$ in average. We have also plotted, for comparison, the Null Depth of the corresponding Fresnel Rhomb.

<table><caption>Table 2. Null Depths for the optimal Fresnel Rhomb configurations and TIRG APS ones and corresponding grating periods for the selected materials.</caption><thead><tr><th></th><th>Diamond</th><th>ZnSe</th><th>CdTe</th><th>Ge</th></tr></thead><tbody><tr><td>Fresnel Rhomb 6 – 11 µm ND</td><td>4 × 10<sup>-4</sup></td><td>1.73 × 10<sup>-5</sup></td><td>6.4 × 10<sup>-7</sup></td><td>7 × 10<sup>-9</sup></td></tr><tr><td>TIRG APS 6 – 11 µm ND</td><td>1.4 × 10<sup>-8</sup></td><td>3.4 × 10<sup>-7</sup></td><td>3.8 × 10<sup>-9</sup></td><td>3.5 × 10<sup>-10</sup></td></tr><tr><td>Period</td><td>980 nm</td><td>1100 nm</td><td>1400 nm</td><td>662 nm</td></tr></tbody></table>
<figure><img src="image_4.png" /><figcaption>Fig. 3. Continuous lines: CdTe TIRG APS performances in terms of Null Depth (logarithmic scale). Dotted lines: CdTe Fresnel Rhomb Null Depths (logarithmic scale). Left: 6-11 microns band, the TIRG APS mean Null Depth is $\mu_{ND} = 3.8 \times 10^{-9}$. Right: 11-18 microns band, the TIRG APS mean Null Depth is $\mu_{ND} = 1.8 \times 10^{-9}$.</figcaption></figure>

### 4. Phase shift and electromagnetic field visualization

To better understand the exact nature of the phase shift and its achromaticity, we propose to visualize the distribution of the electromagnetic field (amplitudes and phases). For this purpose, the structure is divided in three different regions: the first one is the incident medium, in this case the substrate, the second one is the modulated region, i.e., the grating and finally, the last one is the external medium, usually air or vacuum. Each of these three layers is coded with 384 × 128 (x × z) pixels. The x-axis range corresponds to one period of the grating (Λ) whereas the z-axis range corresponds to 3 units of depth (h). The use of the ZOG in TIR incidence implies that the field interacts with the structure only by means of its evanescent waves. The phase shift between the TE and TM polarization components arises due to the fact that their associated vanishing fields penetrate more or less deeply into the modulated region. Figure 4 shows that the TM component penetrates less deeper than the TE one inducing a pseudo-Optical Path Delay responsible for the phase shift. This differential skin effect can be explained by the difference in the TE–TM zeroth order effective indices. The achromaticity of the subsequent phase shift can be understood from the particular grating-induced artificial dispersions of the effective indices (see Fig. 4 bottom right) but, most of all, by the complex interaction between the higher order vanishing modes.

### 5. Discussion: implementation and manufacturing

Implementation of the vectorial phase shift in a nulling interferometer is straightforward. Considering two identical components belonging to the two distinct interferometer arms, rotated by ninety degrees around the optical axis and from one another, then the potentially-interfering parallel polarization states are two by two in phase opposition. It must be noted that there is a strong constraint on the alignment of the components. Let $\Delta\chi$ be the misalignment angle. $\Delta\chi$ is geometrically related to the Null Depth $ND$ by vectorial additions of the phase shifted polarization components: $ND = (1 + \sin\Delta\chi)(1 - \cos\Delta\chi)/2$. Thus, in order to fulfill the constraint $ND = 10^{-6}$, we must reach $\Delta\chi \le 6 \text{ arcmin}$.

The fabrication of the TIRG APS is based on micro-electronic technologies. The first step
<figure><img src="image_5.png" /><figcaption>Fig. 4. Electromagnetic field RCWA visualization in the CdTe TIRG APS case at 6 microns. Top left: TE component field. Top right: TM component. Bottom left: TE – TM phase shift field visualization. Bottom right: TIRG APS TE – TM phase shift versus wavelength together with the effective indices.</figcaption></figure>

consists in imprinting a grating mask in a resin coated on the chosen substrate material. It
can be realized by laser direct writing or e-beam lithography. The precision of this step is
critical because it defines once and for all the lateral dimensions of the ZOG, i.e., the filling
factor *f*. This pattern is then uniformly transferred in the substrate by an appropriate reactive
plasma beam etching down to the desired depth. The fabrication must be interactive to properly
compensate for process errors, e.g. by using in situ monitoring [10]. To summarize, we have
presented a new concept of APS relying on subwavelength grating technology. The TIRG APS
consists in a grating optimized at TIR incidence. Theoretical results are excellent with deep
nulls around (less than $5 \times 10^{-9}$ for the CdTe) over wavelength ranges corresponding to a
spectral resolution $R_{\lambda} = \lambda/\Delta\lambda \le 1.5$. A prototype is currently under realization in collaboration
with the “Centre Spatial de Liège”.
Acknowledgments

The first and second authors acknowledge the financial support of the Belgian “Fond pour la
formation à la Recherche dans l’Industrie et dans l’Agriculture”.