Previous experience with other space telescopes, and in particular the Hubble Space Telescope (HST), has shown that the ability to obtain groundbreaking discoveries relies heavily on the quality and understanding of the telescope’s point spread function (PSF - the normalized image of an unresolved source). The critical elements are: (a) that the PSF is of the highest possible quality; (b) that the PSF is as stable as possible; and (c) that the PSF can be accurately modeled and understood during the data analysis stage. HST has a stiff monolithic temperature- controlled primary mirror. Changes in the HST PSF therefore arise almost exclusively due to variations in the distance of the secondary mirror from the primary mirror. These changes occur at a level of microns both on an orbital timescale ("breathing") due to thermal variations associated with day-night transitions, and on a timescale of years due to desorption.

The situation will be quite different for JWST. The 6.5 m primary mirror consists of 18 semi-rigid segments. Each segment has 7 controllable degrees of freedom (tip, tilt, clocking, piston, two translations, and radius of curvature) and the secondary mirror has an additional 6 degrees of freedom (its radius of curvature cannot be varied). The telescope is passively cooled, but due to the changing attitude of the telescope with time (as it observes targets at different positions on the sky) the telescope will never be fully in thermal equilibrium. Thermal variations combined with the existence of (at least) 132 degrees of freedom will result in a parameter space of JWST PSFs that is much higher-dimensional than that for HST.

# Impact of the Pupil Shape on the PSF

To lowest order the PSF of an imaging system is determined by two things: the shape and size of the effective entrance aperture (pupil) and the deviations from perfect optics across the pupil. In diffraction-limited optics such as HST and JWST, these “wavefront” errors can be expressed as slight changes in optical path lengths for parallel rays entering the aperture. Often, the shape of the pupil is well known and relatively simple, as in the case of a circular or annular aperture. However, more complex pupils are increasingly common for large segmented systems (the JWST and Keck pupils are prime examples). Errors in the wavefront can arise from a variety of sources, including imperfections in the system’s optics (which tend to be stable) or atmospheric variations (as in the case for ground-based observations), and small displacements or thermal variations of the optics that change over hours to weeks. The figure below demonstrates how the pupil shape affects the PSF, building from an open circular pupil of diameter ~6.5 m (left), a hexagonal pupil of diameter ~ 6.5 m (center), and a realistic representation of the JWST pupil (right).

In this figure, the PSF corresponding to each aperture is displayed on a logarithmic grayscale from 1.0e-7 to 1.0e-3 counts. The total number of counts in each of these simulated PSFs is 1.0.

# PSF Dependence on Wavefront Errors

In order to quantify the impact of expected variations of the PSF over time due to wavefront errors, we consider a number of different realizations of the JWST PSF provided to us by the JWST Project. These PSFs differ in the specific distribution of the wavefront error across the pupil; all satisfy, a recent version (T) of the wavefront error budget, both in total rms wavefront error and in its distribution as a function of wavenumber (cycles across the pupil). As long as the error budget is satisfied the telescope optics meet their high level requirements. Comparing these PSFs provides an upper limit to the PSF differences that could be encountered in the normal course of observations. Some wavefront errors, primarily higher frequencies and those arising inside NIRCam, are likely to be highly correlated between different times, and therefore using completely independent realizations can overestimate the temporal variation in the properties of the PSF.

The above figure shows an example of the JWST pupil support (left) and an example of a JWST Rev V OTE error budget Optical Path Difference (OPD) (right). The OPD is displayed on a linear scale from -600nm to 600 nm. The OPD includes wavefront error contributions from the JWST Optical Telescope Element (OTE), the Integrated Science Instrument Module (ISIM), and the Near-Infrared Camera (NIRCam) with all reserves, and stability and image motion. The RMS WFE for this OPD is ~150 nm.

In the above figure the PSF corresponding to the same OPD is displayed on a linear scale between 0 and 0.2 in a small field of view (0.6" x 0.6"). On the right the same PSF is show in logarithmic grayscale from 1.0e-5 to 1, over a larger field of view (4" x 4"). The Logarithmic scale is useful to identify very faint structures in the PSF that extend far from the center, in particular the effects of the hexagon shaped mirrors.

The above figure shows the radial average of the PSF (left) and the encircled energy (right) for the corresponding NIRCam PSF with the F200W filter. The Strehl ratio is 80% for the F200W PSF, with an encircled energy of 88.6% at 0.5". These results are consistent with the JWST requirement that the image quality be diffraction-limited at wavelengths of 2 microns and above.

# Overall Image Quality

We expect that the overall image quality of JWST will be very good, even at wavelengths below 2 micron. In the adjacent figure, we show a comparison between simulated JWST + NIRCam images and actual HST Advanced Camera for Surveys (ACS) images of the Hubble Ultra-Deep Field. These images, kindly made available by Massimo Stiavelli, show that JWST can achieve depth and resolution in the far visible and near-IR that will be better than achievable with HST for the wavelengths they have in common.

The figure shows an ACS image of the HUDF field (top left) in V-, I-, and Z-band, compared with a simulated JWST/NIRCam image in F0707W, F090W, and F115W (top right). A 500 x 500 pixel detail of the galaxy group in the upper-right of the field is shown beneath each image. Even though JWST is not optimized for optical observations, its large primary mirror produces a PSF that is small enough to compare favorably with HST/ACS imaging.

# Point Spread Modeling Software

There are numerous tools available to model the point spread function from an optical system. Many of the simulated PSFs shown here made use of the JWPSF software tool (Cox and Hodge 2006), while the newer **WebbPSF** adds support for a larger fraction of JWST's observational capabilities (most notably, adding support for NIRISS, FGS, and the coronagraphic modes of NIRCam and MIRI). Both programs are available for download by interested users.