L’Imageur diffractif de Fresnel

Why a Fresnel Imager ? 

The Fresnel Diffractive Imager is an instrument with a new focussing concept for future space observation. Since Fresnel diffractive arrays are made of thin foil, this quasi-weightless focussing instrument can be built in meters large dimensions. The larger aperture means the better angular resolution, and more deep sky objects. Therefore, Fresnel imager is proposed as an alternative for future space observation missions. This concept of diffractive focussing has been applied in optics; but diffractive focussing with arrays for space application is a new concept. Then, this is a pioneer for the next generation of diffractive focussing instruments. 

How it helps science 

This thesis presents some of the first results of a Fresnel diffractive array Imager for astrophysics. It demonstrates the conditions and functions of the instrument for sky observation. With larger size of arrays up to 20-m in UV or 30-meter in the visible, it can resolve deep sky objects at respectively 1.5 to 3.5 mas resolution, see chapter 6. This is expected to have an impact on discoveries made with direct imaging in space. 

 How the Fresnel imager compares with other focussing devices

 Fresnel Imaging systems can be compared to other telescopes for angular resolution. Our 200 mm aperture Fresnel imager is equal to classical telescopes having the same aperture. However, the weight of 200 mm Fresnel array itself is only a few grams. Another benefit of Fresnel Imaging system is high dynamic range imaging. The dynamic range has reached 10−8 in numerical simulations (no air turbulence). With our prototypes of 80 mm and 200 mm, it has reached 10−6 in laboratory on artificial sources, and 2.5 10−6 on the sky. From the above comparisons, the Fresnel imager can be seen as a lightweight optical instrument which, when it is used at large aperture, will provide a high angular resolution and high dynamic range.

The difficulties encountered, and resolved, in this thesis work

 The difficulties that I have encountered concern the character and peculiar behavior of this instrument, and the observation of the astrophysical targets. Since Fresnel imaging involves a new concept of astrophysical instrument: it is the first Fresnel focussing instrument used for such sky observation, many problems have had to be detected, understood, and resolved. Some examples are: the alinements, being given the long focal length of the prototype (18 meters), the residual aberrations (coma) of the chromatic correction blazed Fresnel lens, the underestimated stiffness requirement of the mechanical structure (not only the 18m telescope tube, but also the 2m optical rail of the receptor module). 

 Principle of light focalisation

 Focalisation converts incoming light from a plane wave to a focal point. It is required to construct images. Focalisation of light usually means refraction by an optical lens or reflection by a concave mirror. 

 Principle of refraction

 Focalization by refraction is a classical method using an optical element, such as a lens, to focus light onto a focal plane. Light travels as a plane wave before reaching the medium. The curved surface refracts light into a convergent wavefront. Figure 1.1 shows the wavefront travelling through an optical lens and converging to a focal point. (Perez J. P., 2004)[14] Refraction is used in many imaging systems. 

 Fresnel diffractive focalisation 

 Figure 1.1: Focalization by refraction on a curved surface optics changes a plane wave into a spherical wave to form an image at the focal point. 1.2.2 Principle of diffraction. Diffraction in light waves was discovered by Huygen in 1678 and developed further by Augusten Fresnel in 1829. Diffraction occurs when light wave is limited in cross-section by a diaphragm. Diffraction patterns are controlled by the dimensions and the shape of the aperture. For example, the diffraction pattern of a circular aperture is shown in figure 1.3 and that of a rectangular aperture is shown in figure 1.4. Figure 1.2 describes the basic experiment where diffracted waves are created by a double slits aperture. In all the above cases, the amplitude of the light wave is added in in-phase positions and cancelled out in out-phase positions. 

Fresnel diffractive focalisation 

In the Fresnel Imager arrays, diffraction is used to focus light. When light propagates from a distant source through Fresnel arrays, it passes through the open parts of a given Fresnel arrays zone and diffracts, but it is blocked by the opaque parts of the same zone. Figure 1.5 shows how light travels through the open parts of a Fresnel zone and then to focal point. The electric field (amplitude) in a light wave can be described as E = A exp(iωt − kx) (1.1) while k = 2π/λ From figure 1.5, we define the wave emerging from the opened part of the first zone as E1, and E2 from the second zone and etc. Thus there is an optical path difference of λ between two different apertures in consecutive zones. An optical path difference of λ corresponds to phase shift of 2π. The phase shift allows diffracted light to be coherently added at the focal point. The total electric field at the focal point is a sum of diffracted wave from each Fresnel zones. Thus, we can rewrite the output wave as Ef = X Ek (1.2) 16 Introduction Figure 1.2: Diffraction of wave by a double slit: first experiment by Thomas Young in the early 1800s. The red nodes and lines represent amplitude of in-phase position from a secondary wavelet and the blank blue anti-nodes represent the out-phase position, where the waves cancel each other out. Figure 1.3: Diffraction pattern from a circular aperture Figure 1.4: Diffraction pattern from a rectangular aperture. 1.3 Fresnel diffractive focalisation 17 where E1 = A exp(i(ωt − 2πx λ )) (1.3) Ek = A exp(i(ωt − 2πx λ + 2π(k − 1))) (1.4) Figure 1.5: A plane wave propagates through the apertures in Fresnel zones. The optical paths (E1 and E2) difference between two Fresnel zones is λ. The diffracted light by arrays aperture converges to a focal point (f) This concept has been applied since 1875 as a Soret or Fresnel zone plate (FZP)( Soret J. L., 1875)[22]. It is used in the optical domain as a focusing device, which is shown in fig 1.6. Today, FZP is used in a number of applications such as antenna design for millimetre waves ( Minin I.V. and Minin I.O., 2000)[12]. In classic implementation, a Fresnel zone plate in optics is built the following way: its opaque regions are made from dark material covering the surface of transparent supports. This means drawing the Fresnel zones on the optical surface according to the position of Fresnel zones. The following relation yields to the Fresnel zone radii (a). a 2 + f 2 = (f + kλ) 2 (1.5) a 2 = 2kλ f + (kλ) 2 (1.6) where a is the distance of a Fresnel zone from the associated optical axis to determine the radius « a » of the Frensel zone index k in the array. if kλ  f , then we can write equation 1.6 as a ‘ p 2kλ f (1.7) 18 Introduction where f = focal length k = Fresnel index zone λ = wavelength From eq.1.7, we have two associated parameters; central wavelength and focal distance, which allows us to determine Fresnel zones radii on the FZP. 

 Fresnel array design 

Figure 1.6: Circular geometry form of array (original Fresnel Zone Plate in 1875)(Soret, 1875)[22]. Figure 1.7: Orthogonal geometry form of arrays, by Laurent Koechlin et al., developing Fresnel array in orthogonal geometric form from Soret’s Fresnel zone plate in 2005. The concept of Fresnel arrays , developed by Laurent Koechlin, Denis Serre and Paul Duchon in 2005, is a transformation of Fresnel zone plates from circular geometry to an orthogonal geometry.(Koechlin L. Serre D., Duchon P.)[10] The orthogonal geometry in Fresnel arrays design supports a rigid mechanical structure of array and reinforces it. Another advantage, which is most useful in space, is the use of void subapertures. This void aperture layout in geometrical arrays is made by using orthogonal bars to maintain Fresnel zones structure and to define the opened aperture at the same time. As a result, there is no optical material in transparent zones. In space applications, this void subaperture will permit light to pass through the Fresnel arrays , diffract and converge to a focal point without being affected spectrally nor by phase aberration. These advantages mean less weight and a lower cost in building large apertures for space-borne observations in the future. Fresnel arrays are described by a transmission function having « 1 » or « 0 » value respectively for a void aperture and opaque area. Equation 1.8 is a transmission function of a Fresnel array as a function of a: the position of the optical axis at the center. 1.5 Fresnel Imaging System 19 g(a) = 1 if p a 2 + f 2 ∈ [(k + f mλ + 1 4 )mλ; (k + f mλ + 3 4 )mλ [ g(a) = 0 otherwise h(a) = 1 − g(a) (1.8) where f = focal distance of Fresnel array k = Fresnel zones index λ = central wavelength m = diffraction order; in this case, m = 1 To transform the circular Fresnel zone plate into orthogonal form, a transmission law T(x, y) of 2D arrays has been developed from g and h functions as : Tc(x, y) = h(x)g(y) + g(x)h(y) To(x, y) = h(x)h(y) + g(x)g(y) (1.9) where x and y are orthogonal; Tc and To yield similar arrays in phase opposition (Koechlin L. Serre D., Duchon P.)[10]. One can choose one of these functions to determine the opaque and void areas on Fresnel arrays. We use diffraction at order one, which means that we use the wave front that is obtained by a wavelength shift from a given zone to the next. Figure 1.6 and 1.7 shows the circular and orthogonal geometries. 

 Fresnel Imaging 

System Fresnel Imaging System (FIS) is the actual implementation of the concept either in space or as a ground-based prototype. It is different to other classical Image Systems since it uses diffractive focussing, which is created by thousands of subapertures. An orthogonal geometry array creates a highly confined central lobe in the Point-Spread-Function (PSF): there are a central lobe and four spikes see in figure 1.8. The spikes limit four quadrants in the image field, in which there is a very low background level. Figure 1.8 shows the circular and orthogonal comparative design. The orthogonal and circular geometry have the same opaque and void parts with respect to Fresnel zone index. 

Fresnel Modules 

The Fresnel Imaging System (FIS) is the addition of two main modules that are required to perform the vital tasks of collecting light, focusing light, correcting chromatism and constructing images. The two main modules are classified as : Fresnel arrays module or Primary module, and Fresnel receptor module or secondary module shown in figure 1.9