1.

Description of the instrument

The NISP Instrument is the near-infrared Spectrometer and Photometer operating in the 0.9 – 2.0 micron range. The NISP instrument has two main observing modes: the photometric mode, for the acquisition of images with broad band filters, and the spectroscopic mode, for the acquisition of slitless dispersed images on the detectors. In the photometer mode the telescope light is in the wavelength range from 920 nm to 2000 nm (Y, J, H bands).

In the spectrometer mode the light of the observed target is dispersed by means of grisms covering the wavelength range of 1250 – 2000 nm. In order to provide a flat resolution over the specified wavelength range, four grisms are mounted in a wheel. These four grisms yield three dispersion directions tilted against each other by 90˚ in order to reduce confusion from overlapping. The spatial resolution is required to be 0.3 arcsec per pixel. The Field of View of the instrument is 0.55 deg² having a rectangular shape of 1.25 * 0.722 deg².

The instrument focal plane will support 16 H2RG sensors, of 2k×2k pixels of 18µm each. The temperature of the instrument is around 140K, except for detectors, cooled to ∼95 K or below. A warm electronics has been design for commanding the NISP and the detectors and will be located in the service module, at a temperature around 20˚C.

2.

Optical model

The optical design shown on Figure 2, is composed of:

Figure 1:

Focal plane of the NISP

Figure 2:

Optical design of the NISP

The figure 1 highlight the orientation of the X and Y axis, the Z-axis being oriented in the direction of beams.

An optical model (Figure 2) has been developed based on Zemax (http://www.zemax.com), a modeling and analysis software of optical systems. Zemax works by ray tracing, modeling the propagation of rays through an optical system. It can model the effect of the optical elements such as lenses (spherical, aspherical), reflectors, or other dispersive optical elements. In particular, it allows to generate simulated PSF images of the instrument in different points onto the focal plane for all wavelengths accounting for all the configurations of the instrument. Figure 3 shows an example of one PSF in center of the focal plane at 1 µm. The PSF is very regular (not far to look like a 2D Gaussian).

Figure 3:

Example of PSF observed with Zemax

The PSF images generated by Zemax are prefect, without noise and over sampled (128*128 subpixels with a size of 0.9 µm*0.9 µm).

To reproduce NISP realistic PSF as measured on the detector, we should introduce realistic noises (Poisson noise, detector noise etc..) and we should pixelize them with the H2RG pixel size of 18µm. We expect then a PSF to be imaged on ∼6×6 pixels. The signal has been defined to reach a signal to noise of 560s on the PSF determination.

After pixelisation (with 1 pixel = 18 µm) and addition of a Poisson noise on each images, the PSFs look like the ones on Figure 4. We see that the PSF is strongly under sampled and it will be difficult to determine precisely its position and size because of a sub sampling effect.

Figure 4:

Process of pixelisation and addition of noise

We have then developed a tool, entirely automatized, to build a full database of PSF images, from the Zemax design. We have first simulate the PSFs for wavelengths from 0.9 µm to 1.9 µm, for different Zemax tolerancing.

Figure 5 below summarizes the procedure used to determine the database. This database will be used to simulate PSF images in different configurations, in particular to prepare the on ground test analysis and define the test procedure.

Figure 5:

Procedure of the generation of the database of the PSFs

1.

Description of the method

During the ground test campaign, a full analysis framework will be developed to verify the functionalities of the instrument and test its performance. Flexible analysis tools based on MATLAB libraries are under development to fulfill the test procedure needs.

One of the first test that should be done is to determine the focus of the instrument. This means to reconstruct the object plane of the instrument, which should be placed where the telescope simulator ¹ is focusing (∼ 3m).The telescope simulator rotate around the center of the entrance pupil of the NISP (see Dichroic in Figure 1).

The instrument should be aligned at the good focus compared to the telescope with a precision of 150 µm. The budget for focus measurement is allocated to be less than 20 µm.

To develop the procedure, we propose to use the previous simulated PSF images, as representative of the ground test ones, at several positions.

The PSF images at the center position and at different focus are shown on Figure 6. We can observe the effects of focus/defocus of the instrument by the variation of the size of the PSF on the image:

Figure 6:

PSFs for different focus (-2 mm; -1 mm; BF; + 1 mm; + 2 mm)

We observe that there is one image with a smaller PSF that corresponds to the best focus at z = 0.0 mm, best focus determined by Zemax. This verification is facilitated thanks to a very good sampling in Zemax. In the instrument, the pixels will be 20x coarser.

We need an estimator of the width of the PSF. To do so, there is different way. We can use:

To simulate the procedure, we start by simulating PSF images for 5 positions of the focal plane (center + 4 corners). For each of these positions, a focus pad is recovered: the focus is varied from -2 mm to + 2mm over the best focus (process of focus/defocus) as explain in the previous section.

The position of z = 0.0 mm is defined as the best focus of the nominal optical design. The focus position is varied from z = - 2.0 mm to z = + 2.0 mm around the Zemax focus by step of 0.2 mm (to be optimized with the result of the present simulation) with Zemax. For each of the PSF we will compute the EE50 and FWHM and search for the smallest one.

In the next section, we just consider for the study the center of the focal plane and we simulate the PSFs at 1 µm.

2.

Computation of the reference PSF at focus

To estimate the reference position, we will compute the EE50 and the FWHM at z = 0. We project on X and Y axis and fit a gaussian:

The results of the fit of the projection give us the FWHM according the X-axis and the Y-axis:

For the radius of the EE50, we find

We recover well the knowledged relation which validate the computation:

3.

Determination of the best focus position

To find the focus, we use a minimization of the previous estimators FWHM or EE50 by changing the z position: first we use the over sampled PSF at different positions of the focus and fit the minimum of the position. We should find the z = 0 from Zemax.

To minimize the estimators, we follow this procedure:

We apply the different steps for the EE50:

We find:

For FWHM, we obtain the Figure 10:

Figure 7:

Process of focus/defocus

Figure 8:

Example of fit of 2D Gaussian without pixelisation and no noise, without dithering

Figure 9:

EE50 variation in function of z, from a 2D Gaussian fit, over sampled PSF

Figure 10:

FWHM variation in function of z, from a 2D Gaussian fit, over sampled PSF

We find the focus at z_minFWHM = 1.3 µm.

We observe that the EE50 minimum is too large and give a large incertitude in the z minimum determination. We have a better result with the FWHM as expected. Anyway, we don′t find z = 0: it is probably due to the approximation in the method as PSF is not a perfect gaussian.

Anyway, we will take it as the reference for the next section with the real PSF.

4.

Application for realistic PSF

We will apply the method on pixelised PSF with noise.

a) Application for pixelised PSF with noise

In this part, we pixelize the PSF image and add a detector noise and a white noise representative of the photon noise as seen on Figure 4.

For each PSF we then introduce the realistic properties of the detector. We consider 1 electron/second/pixel with long exposure of 560 seconds.

We have an image as shown on Figure 11

Figure 11:

Example of fit of 2D Gaussian with pixelisation and noise

After X and Y-axis projection we extract the FWHM and by doing 100 realisations, we compute also the error and find:

The value is closed to the perfect case, but the error is larger due to the noise and to the pixelisation.

We then have produced with this FWHM, 100 focus-defocus realisations as shown in Figure 12.

Figure 12:

Variation of the FWHM according to the position z of the focus with pixelisation and noise, without dithering (100 iterations)

We find the minimum, representative of the position of the focus:

Compared to the focus determined with an over sampled PSF, we have now an error of 16.1µm approximately. The margin is very short compared to the budget of 20µm. So we propose to improve the method by adding dithering to reduce the error on the focus evaluation.

b) Improving of the focus measurement by using dithering

To improve the FWHM value, we use dithering which means that we move the FWHM around the center of the PSF with an offset of of a pixel and compute a mean FWHM by position. Then we reapply the minimization procedure in function of z. We dither with 4 (2*2), 9 (3*3), 25 (5*5) positions.

We show as an example the case of a dither with 9 positions on Figure 13.

Figure 13:

Example of Gaussian RMS dithered versus focus (9 dithers)

We can observe the distribution and determine the error position:

We find now:

Compared to the z_min of the over sampled PSF, we have an error of ∼1µm compared to the reference′s focus. That’s better than the last 16.1 µm.

The final evaluation in function of the dithering is shown on figure 14:

Figure 14:

Evolution of the focus according the number of dither

We see that with ∼10 dither, we recover well the focus with a small error.

We have significantly improved the position error inside the requirement of 20 µm and this demonstrate that the dithering is needed and will help to recover the minimum position with a good accuracy.

We propose to have at least 25 dither (5*5) to ensure a good precision.