We describe the use of wavefront coding for the mitigation of optical aberrations in a thermal imaging system. Diffraction-limited imaging is demonstrated with a simple singlet which enables an approximate halving in length and mass of the optical system compared to an equivalent two-element lens.
©2009 Optical Society of America
As the cost of uncooled thermal imaging detectors decreases year-on-year in accordance with Moore’s law, the total cost of infrared systems is increasingly dominated by the manufacturing costs of the lenses . With conventional optical design, multi-element aspheric lenses are required to provide wide-field, near-diffraction-limited imaging with the fast optics required for good radiometric sensitivity. We report here the design and manufacture of a thermal imaging lens that uses wavefront coding  to mitigate off-axis aberrations and enable a field-of-view (FoV) that is approximately double that of a conventional singlet lens and functionality comparable to a more complex compound lens.
In recent years, wavefront coding has been shown to significantly increase tolerance to manufacturing inaccuracies and various aberrations, in particular those related to defocus such as thermal and chromatic defocus, field curvature and astigmatism [2–5]. Wavefront coding involves a spatial phase-modulation at the exit pupil so as to produce a specific point-spread function (PSF) and a distinctively blurred, or one might say, encoded image. The important characteristics are that the modulation-transfer-function (MTF) then exhibits no nulls for the design frequency range; nominally those frequencies falling below the Nyquist frequency of the detector array , and the PSF is approximately invariant with respect to an increased range of variations in optical aberrations. The absence of nulls in the MTF  enables a high-fidelity image to be recovered by digital inversion whilst the invariance of the PSF simplifies inversion by enabling a single kernel to be used. The phase function may be rotationally symmetric [8–11] or anti-symmetric [2, 12–14], however antisymmetry offers, in general, a better trade of aberration mitigation against reduction in signal-to-noise ratio (SNR).
The performance enhancement of a wavefront-coded imaging system is subject to some limitations: the signal-to-noise ratio in the recovered image is necessarily lower than in the recorded image; accurate manufacture of an antisymmetric phase-function with a peak-to-valley height of just a few microns is difficult to achieve using conventional manufacturing techniques; and less widely reported, modest variations in the PSF introduce image artefacts. These are important factors in the trade-off design of a wavefront-coded imaging system.
We have previously reported the possibility of using wavefront coding to enable elimination of the Petzval field-flattening element from a two-element infrared lens to yield a singlet with software image recovery for correction of aberrations [15, 16]. Without wavefront coding, the performance of the optimised singlet lens was limited by high levels of off-axis aberrations. We showed that introduction of an antisymmetric phase-function into the front surface of the singlet combined with digital image-recovery, would enable high-quality imaging across an extended field-of-view. We report here the optimisation and manufacture of this imaging system and results from improved image restoration algorithms. To provide flexibility for this demonstration, the phase-function was implemented as a discrete phase-mask as a prelude to future incorporation of the phase-function into the front surface of the singlet. The singlet lens and phase-plate are both machined from germanium and used with a long-wave, thermal infrared (8–12µm) uncooled focal plane array (FPA).
In the next section we describe the design, manufacture and assessment of the singlet imaging system; image recovery and experimental results are described in section three and in section four we present conclusions.
2. Development of a wavefront-coded thermal infrared singlet
2.1 Conventional thermal IR imaging system: from two lenses to singlet
We appraise here the merit of wavefront coding for simplification and size-reduction of an exemplar high-performance fast-lens, as shown in Fig. 1(a). The original F/1, 75mm focal-length lens employed a meniscus front element with an aspherical back surface, to minimise coma and spherical aberration. A Petzval rear element was included to reduce field curvature and astigmatism and yield virtually diffraction-limited performance across a field-of-view (FoV) of 9×7 degrees. When integrated with an uncooled, long-wave infrared FPA of 320×240 pixels on a 38µm pitch (micro-bolometer from FLIR Systems) optical aberrations are insignificant.
A singlet lens of equivalent f-number and focal length, as shown in Fig. 1(b), is obtained by removal of the Petzval element followed by re-optimisation of both surfaces of the front element to minimise aberrations although high levels of field curvature and astigmatism are unavoidable: up to 10 waves at the primary wavelength of 10µm. It is noteworthy that the removal of the Petzval element has enabled a 45% reduction in the optical track; from 142mm to 78mm and a similar fractional reduction in mass. The variation of the MTF with FoV is illustrated in Fig. 2(a) for this singlet for frequencies up to the Nyquist frequency of the detector. Note the significant disparity between the sagittal and tangential MTFs arising from the astigmatic wavefronts and the presence of nulls in the MTF with increasing FoV, which result in irrecoverable loss of information. The pixelated PSFs  shown in Fig. 2(b), illustrate the large spatial variation that results in the blurring of the detected image.
The re-optimized singlet was manufactured by single-point diamond machining. An example image of the World Trade Centre area in Stockholm acquired with the singlet is displayed in Fig. 3(a). As expected, the image is sharp only in the central area and is significantly degraded towards the edges of the image.
Optimal implementation of wavefront coding involves a trade of noise amplification against FoV: mitigation of higher off-axis aberrations requires increased amplitude of phase modulation and the resultant MTF suppression causes higher levels of noise amplification during image recovery. For optimal image quality, we have chosen to mitigate aberrations only within a FoV of 7°, which involves up to six waves of aberration.
2.2 Design and fabrication of the germanium phase mask
Antisymmetric phase-functions of the general form 
where |x|<1, |y|<1 are normalised pupil-plane co-ordinates and α and β are real variables, have been shown to reduce the sensitivity of the PSF to defocus-related aberrations. α and β determine the amplitude of phase-modulation and hence the magnitude of aberration that may be mitigated. Optimisation of image quality, as defined by mean-square-error in the recovered image, yields two optimal combinations of α and β: one for which β~0 (a pure cubic) and one for which α≈-3β  (a generalised cubic). The optimal values of α or of (α, β) for pure- and generalised-cubic phase functions were determined by numerical optimisation based on the method employed by Sherif et al. . The merit function contains components that are related to (1) minimisation of the variation of the MTF across the FoV and (2) maximisation of the on-axis value of the MTF at half of the Nyquist frequency. This corresponds to a weighted measure of the highest MTF that is invariant within the selected field of view. The generalised cubic phase-function was found to offer superior MTF and PSF characteristics than the cubic function due to its superior off-axis performance. Unless very high values of α were used for the pure-cubic function (with commensurately high values of noise amplification), the MTFs, as shown in Fig. 4(a), exhibited nulls at diagonal off-axis field angles. For the generalised-cubic function, optimisation yielded α=9.779µm and β=-30.530µm, representing a peak-to-valley surface sag for the germanium phase plate of approximately 29µm (peak-to-valley OPD of 8.8λ, λ=10µm). As described below, the manufactured mask had a slightly suboptimal peak-to-valley OPD of 24µm and the calculated MTFs and PSFs for this mask are shown in Fig. 4(b). Although there are modest variations across the full FoV, there are no zeros in the MTFs. The pixelated PSFs in Fig. 4(b) also show how the constancy of the PSF is superior for the generalised-cubic phase-function for field angles away from the horizontal and vertical axes than for the pure-cubic. We conclude therefore that the generalised-cubic phase-function generally promises higher image quality than the pure-cubic in this type of design where off-axis aberration dominate and hence this is the phase-function we have chosen to implement.
2.3 Manufacture of the phase mask
For minimum complexity and weight the phase modulation should be incorporated into a surface of the singlet, but for experimental convenience in this proof-of-concept investigation, a discrete phase-mask has been implemented, although this has a negligible effect on the recorded images. Manufacture of the mask was by single-point diamond machining. Measured and calculated interferograms for the phase mask are shown in Fig. 5 and indicate a modest error in the form of the manufactured mask and also a peak-to-valley deviation of only 24µm compared to a design value of 29µm. The main consequence is that the resultant MTF, as shown in Fig. 4(b), is on average about 30% higher than the optimised value and there is a small reduction in the FoV for which good aberration mitigation is possible. Better control of the machining process will negate this issue.
3. Image acquisition and restoration of coded images
The phase-mask, singlet and uncooled FPA were mounted in an aluminium housing with means to acquire images with and without the phase-mask. An image recorded with the phase-mask in place is shown in Fig. 3(b). In the central part of the image plane, where the PSF is practically constant, deconvolution can yield an acceptable restoration, but due to the spatial variation of the optical PSF, good image quality across the full FoV requires a spatially variant kernel to be used in image recovery. To assess the importance of spatially variant PSFs in image recovery we employed the Van Cittert algorithm  for spatially-variant restoration and Richardson-Lucy  for spatially invariant restoration. The spatially-variant pixelated PSFs used in the Van-Cittert restoration were obtained from ray-traced modelling based on the experimental measurements of the phase-function depicted in Fig. 5(b). For computational convenience, PSFs were calculated for a 49×37 array of field intervals and linear interpolation was used to calculate PSFs at intermediate locations. For spatially invariant image recovery using the Lucy-Richardson algorithm the on-axis PSF was used as the kernel.
Examples of images restored using the Richardson-Lucy and Van Cittert algorithms are shown in Fig. 6. From a comparison between these images and that shown in Fig. 3(a) it can be seen that the region of good image sharpness has been increased from less than ±2° for the singlet alone to about ±3.5° for the wavefront-coded images. The quid pro quo however is that some reduction in signal-to-noise-ratio is also evident, particularly for larger field angles where the MTF is more suppressed. In the central region this reduction is a factor of six. The spatially variant restoration is seen to yield better image sharpness at larger field angles than does the spatially invariant restoration, but this is at the cost of increased computational expense.
The reduced image quality in the corners of the images, for field half-angles of about 5°, corresponds to areas outside the zone for which the lens was optimised. In these areas the PSF is more extended and rapidly varying leading to greater noise amplification and, for spatially-variant recovery, artefacts are introduced by a poor interpolated estimate of the PSF.
We have described the optimisation and experimental demonstration of wavefront coding for the mitigation of aberrations in an infrared imaging system. To the best of our knowledge this is the first such system that has been reported in the open literature. We have shown that this technique clearly increases the FoV for which acceptable image sharpness is obtained. The associated reduction in signal-to-noise ratio is not insignificant, but from our experimental demonstration well acceptable for many applications. This supports the conclusion that provided the recorded signal-to-noise ratio is sufficiently high, wavefront coding provides good scope for significant improvement in the trade of cost, volume and weight against performance for low-cost thermal imaging systems. Future work will include incorporation of the phase-function directly into the singlet surfaces and improvements to wide-field image recovery. It is noteworthy that the enhanced depth-of-focus of wavefront coding also provides a degree of athermalisation and when combined with diffractive surfaces for achromatisation, enables the manufacture of high performance singlet lenses moulded at low-cost from chalcogenide glass.
This research was funded by the UK MoD. Images were recorded at FLIR, Sweden.
References and links
4. G. Muyo and A. R. Harvey, “Wavefront coding for athermalization of infrared imaging systems,” Proc. SPIE Vol. 5612, p. 227–235 ( 2004). [CrossRef]
6. G. Muyo and A. R. Harvey, “The effect of detector sampling in wavefront-coded imaging systems,” J. Opt. A: Pure Appl. Opt. 11, 054002 ( 2009). [CrossRef]
8. D. Zalvidea and E. E. Sicre, “Phase Pupil Functions for Focal-Depth Enhancement Derived from a Wigner Distribution Function,” Appl. Opt. 37, 3623–3627 ( 1998). [CrossRef]
9. W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875–877 ( 2001). [CrossRef]
11. S. Mezouari, G. Muyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A 23, 1058–1062 ( 2006). [CrossRef]
12. S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE Vol. 5108, 1–12 ( 2003). [CrossRef]
15. G. Muyo, A. R. Harvey, and A. Singh, “High-performance thermal imaging with a singlet and pupil plane encoding,” Proc. SPIE 5987, 162–169 ( 2005).
16. G. Muyo, A. Singh, M. Andersson, D. Huckridge, and A. Harvey, “Optimized thermal imaging with a singlet and pupil plane encoding: experimental realization,” Proc. SPIE 6395, U211–U219 ( 2006).
17. G. D.B oreman, Modulation transfer function in optical and electro-optical systems (SPIE Press, Bellingham, WA, 2001). [CrossRef]
18. T. Vettenburg, A. Wood, N. Bustin, and A. R. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc SPIE 7429, 742903 ( 2009). doi: 10.1117/12.825119. [CrossRef]
19. W. C. Karl, “Regularization in image restoration and reconstruction,” in Handbook of Image and Video Processing, A. Bovik, ed., (Academic Press2000).