Use of subwavelength gratings in TIR incidence as achromatic phase shifters

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 1. R.N. Bracewell, “Detecting Non Solar Planets by Spinning Infrared Interferometer,” Nature 274,780–781 (1978). 2. A. Léger, J.M. Mariotti, B. Mennesson, M. Ollivier, J.L. Puget, D. Rouan, J. Schneider, “Could We Search for Primitive Life on Extrasolar Planets in the Near Future,” Icarus 123,249–255 (1996). 3. R. Anderson, “Quarterwaveplate and Fresnel rhomb compared in the 10-Mum CO2 laser emission region,” Appl. Opt. 27,13, 2746–2747 (1988). 4. K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Design and performance of a stable linear retarder,” Appl. Opt. 36,25, 6458–6465 (1997). 5. M. Born & E. Wolf, “Reflection and refraction of a plane wave,” in Principles of Optics eds. (Cambridge University press, seventh edition, 1997), pp. 49–53. 6. F. Peter, Z Phys 15, 358-368 (1923). 7. W. J. Tropf, “Temperature-dependent refractive index models for BaF2, CaF2, MgF2, SrF2, LiF, NaF, KCl, ZnS and ZnSe,” Opt. Eng. 34, 1369-1373 (1995). 8. Gary J. Hawkins, Spectral Characterisation of Infrared Optical Materials and Filters (PhD Thesis The University of Reading UK, 1998). 9. M.G. Moharam and T.K Gaylord, “Rigourous Coupled-Wave Analysis of planar grating diffraction,” J. Opt. Soc. Am. 71, 811-818 (1981). 10. P. Lalanne, P. Pichon, P. Chavel, E. Cambril, H. Launois, “Interferometric Characterization of Subwavelength Lamellar Gratings,” Appl. Opt. 38,23, 4980–4984 (1999). #7908 $15.00 USD Received 23 June 2005; revised 28 July 2005; accepted 25 August 2005 (C) 2005 OSA 31 October 2005 / Vol. 13, No. 22 / OPTICS EXPRESS 8686


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.π) 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.
Fig. 1.Schematic representation of a subwavelength grating.The main parameters of the structure are: the grating vector |K| = 2π/Λ, perpendicular to the grating lines, with Λ being the spatial period, the grating depth h and the filling factor f , such that f Λ is the width of the grating ridges.T E and T M are the vectorial orthogonal polarization components of the θ -incident light.n i and n t are the refractive indices of the incident (substrate) and emergent (transmitting) media, respectively.

Principle of the TIRG APS
It is a well-known fact that the TIR phenomenon comes with a differential phase shift between the vectorial T E and T M polarization components.This vectorial phase shift takes the following form [5] ∆φ where θ is the angle of incidence, greater or equal to θ c , the critical angle defined as sin θ c = n ti = n t /n i 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 wavelength 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, leaving 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 component T E and T M.These effective indices, n T E and n T M 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 mismatches.The figure of merit to be minimized consequently resumes to ND(λ ) = σ 2 /4 where σ (λ ) is the phase shift error standard deviation with respect to the nominal value of π (other values are possible).This relation, derived from the interference between two plane waves theory, simply means that the Null Depth is directly proportional to the variance of the phase shift error, i.e., the achromaticity.
ter arm and orthogonally from one another.

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 microns.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 (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 The corresponding coefficients are given in Table 1.We will assume a refrigerant temperature T = 100 K, keeping in mind that corrections are needed if T is different.Note that λ 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 × 10 −9 in average.We have also plotted, for comparison, the Null Depth of the corresponding Fresnel Rhomb.

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 T E and T M 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 T M component penetrates less deeper than the T E one inducing a pseudo-Optical Path Delay responsible for the phase shift.This differential skin effect can be explained by the difference in the T E −T M 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.

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 ∆χ be the misalignement angle.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 × 10 −9 for the CdTe) over wavelength ranges corresponding to a spectral resolution R λ = λ /∆λ ≤ 1.5.A prototype is currently under realization in collaboration with the "Centre Spatial de Liège".

Fig. 2 .
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 π phase shift possesses two TIR interfaces, each providing a π/2 phase shift such that the resultant is∆Φ T E−T M,1+2 = ∆Φ T E−T M,1 + ∆Φ T E−T M,2 = π/2 + π/2 = π.Such a component is to be inserted in each interferome-

Fig. 4 .
Fig. 4. Electromagnetic field RCWA visualization in the CdTe TIRG APS case at 6 microns.Top left: T E component field.Top right: T M component.Bottom left: T E − T M phase shift field visualization.Bottom right: TIRG APS T E − T M phase shift versus wavelength together with the effective indices.

Table 1 .
Temperature-dependant coefficients for material index representations.

Table 2 .
Null Depths for the optimal Fresnel Rhomb configurations and TIRG APS ones and corresponding grating periods for the selected materials.