The Fresnel imager: instrument numerical model
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Abstract
The methodology used in the endtoend numerical model of the Fresnel Interferometric Imager is presented. This Instrument Numerical Model (INM) performs planetoplane Fresnel propagation, starting from the Fresnel array and ending at the achromatic focal plane, and has been written in c so that it can handle various instrument configurations (sizes of Fresnel arrays from cm to m, from a few Fresnel zones to a few hundred, and for various wavelengths) with a standard desktop computer (a few GHz processor(s) speed, a few GB of memory, execution time per wavelength spanning from few minutes to few hours in the most extreme cases). The INM is used to estimate the performances of the Fresnel Imager: angular resolution, photometric dynamic range, transmission, for on and offaxis sources.
Keywords
Numerical simulation Fresnel PSF modeling Dynamic range1 Introduction
The Fresnel Imager is a concept of spacebased telescope yielding high angular resolution and on sparse sources high photometric dynamic range. It involves two satellites flying in formation: one holds a “Fresnel Array”, the second one is the “focal module”. A Fresnel array is a foil punched with thousands of holes disposed so that if some light comes from a target, a constructive interference can occur at the socalled ‘focus’ of the Fresnel array, yielding to an image with high angular resolution and potentially high dynamic range, but also highly (longitudinally) chromatically dispersed. The “focal module” holds a pupil optical element, an optical device which will correct the chromatism, and the focal instrument. Detailed presentations of the Fresnel Imager concept can be found in Koechlin et al. [1, 2].
The Instrument Numerical Model (INM) computes the Point Spread Function (PSF) of the Fresnel Imager by proceeding to planetoplane Fresnel propagation, taking into account the optical elements from the Fresnel array to the achromatised focal plane (Section 2). It computes the minimal sizes of the optical elements, takes care of the sampling issues, and has been written to be able to handle various Fresnel Imager dimensionings (Section 3). Finally, its outcomes have been compared to some of the results obtained by the socalled “Generation I” prototype, which features all the elements consituting a Fresnel Imager but in reduced size (8 cm aperture), and is used to estimate the performances on various instrument configurations (Section 4).
2 Principles

five optical elements: the Fresnel Array, a field optical element (considered achromatic), a mask which blocks the order zero of interference, the blazed Fresnel zone lens which corrects for the chromatism (introduced in the first article of this Experimental Astronomy “Fresnel Imager” special issue and described in Serre et al. [3]), and an optical element making the beam converge (considered achromatic);

four propagations.
2.1 Fresnel propagation

a quadratic phase addition: \(e^{i\frac{2\pi}{\lambda} \frac{{x_1}^2 + {y_1}^2}{2d_{12}}}\)

a Fourier transform: \(\int \int { e^{i\frac{2\pi}{\lambda} \frac{x_2\,x_1 + y_2\,y_1}{d_{12}}} dx_1 \, dy_1}\)

and another quadratic phase addition: \(e^{i\frac{2\pi}{\lambda} \frac{{x_2}^2 + {y_2}^2}{2d_{12}}}\)
2.2 Optical elements
The optical elements are considered as phase and/or amplitude masks.
Achromatic optical element
Chromatic optical element
Binary masks
A binary mask, whether it is the Fresnel array in itself, the zeroorder mask, or the physically finite dimension of the optical elements, is simply simulated by setting the amplitude of \(\underline{\psi}_{x,y}\) of the concerned regions to 0.
2.3 Additional remarks
Apart from setting some amplitudes to 0 where there are some masks in the beam, the INM doesn’t deal with the transmissions of the optical elements: i.e. a lens will transmit 100% of the energy that will fall on it.
Also, we emphasize that we assume a sequential optical system: back and multireflections, and the associated straylight, cannot be seen by the INM.
Finally, the use of simple Fourier transform (Fraunhofer propagation) is not sufficient, even for the Fresnel Array itself: as there are a number of Fresnel zones involved, the (chromatic) focal plane cannot be considered at an ‘infinite’ distance from the Fresnel array.
3 Implementation
What is presented in Section 2 is applied in the INM, but using sampled versions the wavefront (matrices), FFT algorithms, and computers with finite performances, there are a certain number of issues that must be considered.
3.1 FFT algorithm, quantity of RAM issues, and oversampling factor
 (a)
the size of the (square) matrix nbpts _{user} is set by the user under two constraints: on the one hand, sampling constraints (Section 3.3) which impose a minimal size, and on the other hand RAM capacity of the computer that is used. The linear matrix size doesn’t have to be a power of two (simply must be even), and the smallest it will be, the fastest the computation will be done;
 (b)
the 2D FFT is split into two steps: first step: FFT of the original matrix along the rows, second step: FFT of the computed matrix along the columns;
 (c)in each of those two steps, the FFT are performed on some vectors and not matrices:

the nbpts _{user} pixels are copied into a vector of length nbpts (nbpts = one of the power of 2 greater than nbpts _{user}), and the (nbpts − nbpts _{user}) other pixels are set to zero (zero padding);

the FFT is performed on this vector;

after recentering of the obtained vector, only the nbpts _{user} central pixels are copied back in the original 2D matrix;

 (d)
applying this principle along all the rows, and then along all the columns, permit to have a matrix stored in the computer memory whose size is not a power of 2.
Moreover, to optimize the RAM use, only one matrix is stored and propagated through the whole algorithm: therefore an additional constraint of the INM is that the number of pixels will be the same in all the planes.
3.2 Sizes of fields of view
The physical dimension represented by the matrix in a plane in which the wavefront state is computed is constrained by a minimal value: at least the physical size of the beam must be englobed.
Therefore, the physical size of a given plane is inversely proportional to the physical size of the previous plane, and can be adjusted using an optimal oversampling factor. This oversampling factor doesn’t need to be the same for the four Fresnel propagations presented in Section 2.
3.3 Sampling issues
Each time a phase is added to that of a wavefront, the algorithm must make sure that there are at least two pixels of the matrix each time a phase shift of 2π is added when applying (1) and (2).
Therefore, a minimal number of pixels is the matrix describing a wavefront will be set by the sampling constraints.
3.4 Multithreading
Looking at Section 3.1, it seems natural to implement a multithreaded algorithm, which we have done: each of the p processors involved compute \(\frac{nb{\kern2pt}pts_{\rm user}}{p}\) 1D FFT, first along the rows and then along the columns. Note that the processors therefore need to be able to access the same memory space: in the second step of Section 3.1, when the FFT is done along the columns, each column has been obtained by combining 1 pixel of each row, therefore by combining some results from all the processors. Not also that there is a need to have ‘wait’ functions, so that none of the processor start the computations along the columns before all the FFTs along the rows have been computed.
We have implemented the multithreading by using the pthread (POSIX threads) library. Gains turned out to be more than 45% in execution time when using the two cores of a processor instead of only one, but the improvement should get more and more marginal with the number of processors or cores, because there is still a need for the processors to use the same memory access.
3.5 Summary
Constraints on the matrix size
 Physical size represented by the matrix  Number of pixels in the matrix 

Minimal  Matrix must not diaphragm the beam  Sampling must ensure that a 2π phase addition involves at least 2 pixels 
Maximal  None, but the bigger it is, 1) the smaller the following one will be (Eq. 4), and 2) the smaller the sampling per pixel will be  Limited by the quantity of RAM: 1 pixel coded as complex ‘double’ occupies 16 Bytes in RAM 
3.6 Additional information
The user has to give the dimension of the Fresnel array, the distance between the Fresnel array and the field optical element, the diameter of this field optical element, and the focal length of the image reforming optical element. From those information, the INM computes what are the physical distances between the field optical element and the zeroorder mask, between the zeroorder mask and the FZL, and the image reforming optical element and the focal plane. It also computes what is the physical size of the FZL, and the size of the beam on the zeroorder mask plane.
If the sampling is not sufficient, the focal distance of the image reforming optical element too small, the number of pixels in the matrix not sufficient... the INM stops and indicates it, often providing an indication of by how much it is insufficient.
At the end of each computation for a given wavelength, an additional matrix stores the intensity computed in the ‘central region’ of the final focal plane (plane 6 in Fig. 1). The size (in pixels) of this ‘central region’ is defined by the user, and should be set to the size of the field of view. If the user has asked for several wavelengths, the nbpts _{user}*nbpts _{user} matrix is reused for the ‘new’ wavelength, and therefore any information about the previous wavelength is lost, except for this matrix which saves the intensity only: the values of the pixels of this matrix are in fact incremented each time the computation at a wavelength has been performed (increment potentially scaled if the user has also provided a source spectrum). As a result, after the propagations for all the wavelengths have been computed, this matrix is the ‘broadband’ intensity at the focal plane, provided that the spectral sampling is sufficient 1) with respect to the source spectrum and 2) with respect to the fulfillment of the underlying assumption that in the resulting field of view, and considering PSF(λ _{1}) and PSF(λ _{2}) being computed by the INM, PSF(λ _{1}) and PSF(λ _{2}) are of sufficiently close shape so that any intermediate wavelength would have its PSF which could be linearly interpolated from those two PSFs.
In Section 3.1, we have explained that after each 1D FFT, some of the pixels (those which are not recopied in the main matrix) are simply erased: but note that the INM increments a value each time with the quantity of energy lost during the computation, so at the end of the computation it can know which fraction of the energy has been lost during the whole process.
4 Examples of results
4.1 Validation: PSF shape and dynamic range
4.2 Offaxis source
4.3 Apodisation
5 For the future

implement the ability to take into account nonthin lenses but ‘real’ optical elements, for example coming from Zemax simulations;

implement the ability to add defaults on the surfaces: this could be done by the mean of Zernike coefficients, generating a phase mask which would represent the defaults of the optical elements;

rewrite the code so that the number of planes where the Fresnel propagation must be performed and the number of optical elements can be arbitrary, so that for example a field optical device with two optical elements or a downstream coronographic stage could be included easily; it involves making not only the propagation algorithm be more flexible, but also adapting the whole process of sampling computation and size of fields optimization;

potentially, if an analytical expression of the Fresnel array can be computed, it would on the one hand diminish the computation time, and on the other hand avoid the problems of sampling constraints on the Fresnel array plane;

the problem of non sequential reflections might be implemented, not using Fresnel propagation but statistical considerations on the reflectivity of the different elements and estimation of where the straylight would be confined; but this item is far beyond the original goals of the INM.
6 Conclusion
The principles involved in the Fresnel Imager Instrument Numerical Model and their implementation have been presented, as well as some results illustrating the efficiency of the INM. The INM can be used to simulate the results of the ‘small’ prototypes (socalled Generation I, II, III; see other articles of this EXPA “Fresnel Imager” special edition) as well as the results which would be obtained with spacebased telescopes.
Notes
Acknowledgement
This work was realized during Denis Serre’s PhD thesis, which was cofunded by Thales Alenia Space and Europe.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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