3 Analytic expressions of the PSF and corresponding apodization function

In this section, we show that the apodization functions obtained with the interferometric technique correspond to the cosine to the power of N apodizer and calculate the analytical expressions of the PSF for these apodization functions. For more generality and later use, we consider the case where the shift may be b times smaller than the optimal 1/L considered in the previous section.

The amplitude corresponding to the addition of two PSF images distant one another of $\frac{1}{b\;L}$ can be written as:

 

$$\Psi_b^{(1)}(x)=\frac{1}{2}\;\frac{\sin \pi L x}{\pi L x}\ast\left(\delta\left(x-\frac{1}{2 b L}\right)+\delta\left(x+\frac{1}{2 b L}\right)\right)$$ (2)

where for convenience, we use the Dirac delta function $\delta(x)$ to write the displacements between the PSF's and where the symbol $\ast$ denotes the convolution product. For reasons of symmetry, the two PSF's are shifted of $\frac{1}{2 b L}$ in opposite directions. This parameter b is simply the factor by which the angle between the mirrors of the interferometer must be reduced. A second apodization can be obtained by again dividing $\Psi_b^{(1)}(x)$ into two terms and shifting them by the same amount $\pm \frac{1}{2 b L}$ as above (see Figs. 1 and 2 for illustration):

 

$$\Psi_b^{(2)}(x)=\frac{1}{4}\;\frac{\sin\pi L x}{\pi L x}\ast\left... ...elta\left(x-\frac{1}{2 b L}\right)+\delta\left(x+\frac{1}{2 b L}\right)\right),$$ (3)

and we simply obtain the addition of three terms of the form:

$$\Psi_b^{(2)}(x)=\frac{1}{4}\,\left(\frac{\sin\pi L x}{\pi L x}\ri... ...-\frac{1}{b L}\right)+2\delta(x)+\delta\left(x+\frac{1}{b L}\right)\right)\cdot$$ (4)

We can write the resulting impulse response for the Nth order apodization, as follows:

 

$$\Psi_b^{(N)}(x)=\frac{1}{2^N}\, \left(\frac{\sin \pi L x}{\pi L x... ... \ast\left(\delta(x-\frac{1}{2 b L})+\delta(x+\frac{1}{2 b L})\right)^{\ast N},$$ (5)

where the symbol $\ast N$ means that the convolution must be done N times. Alternatively, this equation can be rewritten using the binomial relation:

$$\Psi_b^{(N)}(x)=\frac{1}{2^N}\, \left(\frac{\sin \pi L x}{\pi L x... ...t. \right) \; \delta\left(x-\left(k-\frac{N}{2}\right)\frac{1}{b L}\right)\cdot$$ (6)

For the particular case where b=1, these expressions find a simple form. Depending on the parity of N, we have:

• For N even:
  
   

  $$\Psi_1^{(N)}(x) \, \,\frac{1}{2^N}\displaystyle\frac{\sin \pi L x}{\pi L x \displaystyle\prod_{k=1}^{N/2}\left(1-x^2L^2/k^2\right)}$$ =

  $$\, \,\frac{1}{2^N}\displaystyle\prod_{k=N/2+1}^{\infty}\left(1-\frac{x^2L^2}{k^2}\right)\cdot$$ = (7)

  
  

• For N odd:
  
   

  $$\Psi_1^{(N)}(x) \, \,\frac{1}{2^N}\displaystyle\frac{\cos \pi L x}{\displaystyle\prod_{k=1}^{(N+1)/2}\left(1-4x^2L^2/(2k-1)^2\right)}$$ =

  $$\, \,\frac{1}{2^N}\displaystyle\prod_{k=(N+3)/2}^{\infty}\left(1-\frac{4x^2L^2}{(2k-1)^2}\right),$$ = (8)

  
  

Figure 3: From top to bottom: unapodized PSF, and successive order of N given in Table 1. Dashed plots corresponds to a cut of the PSF along the diagonal direction, as considered in the ASA Concept.

where the last part of Eqs. (7) and (8) are obtained by writing the sine and cosine functions as infinite products. The interest of this last form is mainly to show that the resulting function $\Psi_1^{(N)}(x)$are even functions, with a central width (defined as the distance between first zeros) equal to (N+2)/L. The numerical computation is easily done using the first part of Eqs. (7) and (8) and the three first orders are given in Table 1; illustrations of the intensity PSF are given in Fig. 3 for N=0(no apodization) to N=3.

 

 

Table 1: Apodized impulse responses, for a one-dimensional aperture of width L.

Apodization	Impulse responses (normalized)
unapodized:	$\frac{\sin(\pi L x)}{\pi L x}$
1st order, b=1	$\frac{\cos(\pi L x)}{(1-4x^2L^2)}$
2nd order, b=1	$\frac{\sin(\pi L x)}{(\pi L x)\left(1-x^2L^2\right)}$
3rd order, b=1	$\frac{\cos(\pi L x)}{(1-4x^2L^2)(1-4x^2L^2/9)}$

The corresponding amplitude in the pupil plane can be obtained directly as the Fourier transform of Eq. (5). Using basic properties of the Fourier transform of a convolution product, we obtain the simple expression:

 

$$P_X^{N}(x)= \cos^N\left(\frac{\pi x}{b L}\right)\times\Pi\left(\frac{x}{L}\right)\cdot$$ (9)

For b=1, we recognize the cosine and cosine squared apodization functions proposed by Nisenson & Papaliolios (2001). This result was expected since the apodization corresponds to a wedge fringe in the pupil plane.

The proposed interferometric technique may be a solution to produce these apodization functions for the ASA concept. To obtain the two dimensional apodizer, two successive devices must be successively used in orthogonal directions. Making b larger than one corresponds to the use of a partial cosine arch apodization: only the central part is used. This apodization will be of interest for coronagraphy (Sect. 4). The above expressions for the amplitude impulse response can be used for an analytical study of the ASA concept even if the interferometric technique is not used to produce the aperture transmission: the diffracted light intensity along the diagonal for a square aperture is simply written as the fourth power of the amplitudes given in Eqs. (7), (8) and Table 1. Illustrations of the diagonal diffracted light for the ASA are given in Fig. 3. These curves were already numerically given by Nisenson & Papaliolios (2001).