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Subsections

  
2 Interferometric apodization of rectangular apertures

The purpose of apodization is to obtain a redistribution of the energy in the diffraction pattern, reducing the energy in the wings of the PSF. In Signal Processing, the problem has been extensively studied for the spectral analysis of temporal signals (Papoulis 1981; Harris 1978). In Optics, this effect can be obtained with appropriate modifications of the pupil function. A large review of apodization techniques was made by Jacquinot & Roizen-Dossier (1964). These authors also suggested to illuminate the pupil with an interference pattern in a spectroscopic system for better slit illumination, but they abandoned the idea. We reconsider this idea herein and give below the principle of the technique.

For the ASA concept, and more generally for an apodized rectangular aperture, the pupil transmission and its corresponding focal plane diffraction can be written as separate functions of x and y. This property can be used to describe the technique for a one-dimensional telescope. The generalization to a two-dimensional rectangular or square aperture is straightforward.

  
2.1 Principle of the technique

For a one-dimensional telescope of width L, the pupil P(x) may be simply written as the window function $\Pi(x/L)$ equal to 1 for $-\frac{L}{2}<x<\frac{L}{2}$ and 0 otherwise.

In the focal plane, the normalized monochromatic amplitude impulse response $\Psi(x)$ is proportional to its Fourier transform: 0pt

$\displaystyle \Psi(x)=\frac{\sin \pi L x}{\pi L x},$     (1)

where x is the angular position in the focal plane and L must be expressed in units of wavelength $\lambda$.

The proposed apodization technique is very simple. Let us assume that by some means, we are able to split this amplitude $\Psi(x)$ into two identical waves and shift one of them by $\frac{1}{L}$. The minima of the first wave then correspond to the maxima of the second and provided that we maintain the coherence, we obtain the addition of the two amplitudes. Outside the central part of the PSF, positive and negative values approximately cancel each other (Fig. 1). The amplitude of the wings of the PSF is then reduced with, of course, the drawback of an enlarged central peak.


  \begin{figure}
\par\includegraphics[width=8.8cm,clip]{MS1697f1.eps} \end{figure} Figure 1: Illustration of the apodization technique in the focal plane. Top left: unapodized amplitude split into two shifted amplitude (bottom left). The result of this coherent addition is the first-apodized amplitude (top center). This approach can be iteratively reproduced: the split and shifted amplitudes (bottom center) and second-apodized amplitude (top right).

The resulting amplitude remains oscillatory with the same distance 1/L between zeros. We can iteratively reproduce this approach: several identical optical systems can be used successively to produce the shifts and obtain the different degrees of apodization. The procedure is schematized in Fig. 1.

These successive apodizations can be considered as a direct superposition of several shifted amplitudes with appropriate coefficients. The second-order apodization is equivalent to the addition of three shifted amplitudes with coefficients (1, 2, 1). The third apodization is a four-amplitude addition, with coefficients (1, 3, 3, 1). It can be easily shown from the illustration that the amplitude coefficients are given by Pascal's triangle (Fig. 2).


  \begin{figure}
\par\includegraphics[width=8.8cm,clip]{MS1697f2.eps} \end{figure} Figure 2: Illustration of the amplitude coefficients for successive apodizations. The encircled numbers corresponds to the amplitude coefficients. The first apodization corresponds to a 2 beam interference, the second apodization to a 3 beam interference with coefficients 1, 2, 1 for the amplitude. Amplitude coefficients are given by Pascal's Triangle.

2.2 Possible optical realization

A shift of the two identical amplitudes in the focal plane corresponds to a slight tilt of the wavefronts at the pupil plane. This is equivalent to considering localized fringes of equal thickness on the pupil.

This can be easily achieved with an interferometer by division of amplitude, such as the Michelson or the Mach-Zendher interferometer, setting images of the entrance aperture on the interferometer mirrors of the interferometer and slightly tilting a mirror to introduce a thin wedge-shaped film.

The required shift of the focal amplitude corresponds to an optical path difference of $\lambda/2$ at the edges of the pupil (destructive interference): the center of the pupil must be bright and the pupil edges dark. The main drawback of a Michelson interferometer is that half of the flux is lost backwards to the source. However, it can be used for an easy laboratory test experiment, which will be reported in a future work. A better solution seems to use a Mach-Zendher interferometer, which has two outputs.

Obtaining two apodized pupils with a Mach-Zendher interferometer appears to be impossible for conservation of energy reasons. The relative phase shift between the reflected and transmitted beams in a thin or symmetric beam-splitter is $\pi/2$; this imposes complementary fringes at the two outputs of the interferometer. Thus we can obtain an apodized output (with same transmission as a perfect classical apodizer) and an "anti-apodized" output which pupil amplitude is brighter at the edges and leads to the opposite effect to apodization. This output cannot directly be used for planet detection, but remains usable for other purposes, such as tracking.

The successive apodizations can be produced with several successive interferometers, but with the problem of multiplying optical surfaces. An unique multiple-beam, specifically designed interferometer may also be possible and has to be studied. Mach-Zendher interferometers made of two glass blocks associated together with total internal reflections may be preferred to mirrors to preserve a high throughput and minimize the number of optical surfaces. However, for coronagraphy, classical interferometers with mirrors would permit the user to tune the wedge fringes for practical optimization, as we shall see later.

Of course, the main drawback of such an interferometric technique is its chromatism. This point will be discussed in Sect. 5. For an actual two-dimensional aperture and if the apodization is required in two dimensions, the use of two interferometers at an orientation of $90^\circ$ will be needed.


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