Study of asymmetric or decentered multi-view designs for uncooled infrared imaging applications

: Multi-view architectures using lens arrays can bring interesting features like 3D or multispectral imagery over single aperture cameras. Combined with super-resolution algorithms, multi-view designs are a way to miniaturize cameras while maintaining their resolution. These optical designs can be adapted for thermal infrared imagery and can thus answer the size, weight and power (SWAP) challenge with advanced imagery functions. However, in this spectral range, the choice of an uncooled microbolometer detector imposes a high numerical aperture for the system which increases the size of the optics and makes diﬃcult a multi-channel arrangement combined with a single focal plane array (FPA). In this paper, we theoretically investigate several asymmetric or decentered multi-view designs that allow both a high aperture for the optical channels and the use of a single FPA for the sub-images. Ray-traced designs will illustrate this study and their image quality will be checked with modulation transfer functions (MTF) for diﬀerent ﬁeld points.


Introduction
Multi-view architectures can bring interesting features like 3D or multispectral imagery over single aperture cameras [1]. Another interesting application is multiscale imagery [2][3][4][5][6][7]. Multiscale imagery combines a single aperture lens with multiple lens modules, each associated with its own focal plane array (FPA) to obtain a wide field of view gigapixel camera. Bio-inspired multi-view systems can provide lightweight and compact wide field of view systems as well [8][9][10][11][12][13][14][15]. Indeed, they are made of multiple simple channels looking at different parts of the field of view like insect eyes. Moreover, some of them can be compatible with wafer technologies [16][17]. Finally, with super-resolution algorithms, multi-view designs are a way to miniaturize cameras while maintaining their resolution [18] and bio-inspired architectures can therefore provide compact cameras with both a wide field of view and a good angular resolution [19][20].
A straightforward way to make a multi-view camera is the use of lens arrays. This arrangement, called TOMBO by a research team [18], has the ability to produce the multiple images on a single FPA and with a minimum of optical components to align. Indeed, the optical element is a monolithic lenslet array. Numerous studies about TOMBO-based designs have shown the various possible applications of multi-view systems. Indeed, 3D imagery [21][22], multispectral imagery [22][23][24][25] and polarization imagery have been investigated. This design has been applied for a wide range of spectral bandwidths, from the visible to the thermal infrared waveband. In the infrared, an ultra-compact TOMBO-based cryogenic camera has been made [26][27][28]. Another infrared TOMBO-based camera has been designed for uncooled detectors like microbolometers [29][30][31]. Both architectures use super-resolution algorithms to reduce the length of the camera. Indeed, replacing a conventional lens of focal length F by an array of NxN sub-lenses of focal length F/N reduces the length of the optical system by N. However, miniaturizing the optical system while maintaining a constant field of view (FOV), a constant F-Number (F#) and the same focal plane array with a pixel pitch t pix results in a decrease of the angular resolution (IFOV) and of the number of resolved points N b for each channel [32]. However, if each channel provides nonredundant information, the combination of the sub-images can contribute to a final image with an enhanced resolution thanks to the theorem of Papoulis [33]. To obtain this non-redundancy, subpixel shifts between the sub-images can be introduced for example. Finally, TOMBO designs rely on a relationship between the lowest possible F# and the FOV for an optimal use of the pixels of the FPA [34]: Equation (1) means that the detector area dedicated to a channel is equal the size of the lens. Moreover, verifying Eq. (1) guaranties for a given FOV and the lowest possible F# neither no field overlap nor unused pixel area. Equation (1) can be problematic for multi-view systems with a high aperture, such as systems using uncooled infrared detectors. Indeed, we notice that in reference [31], the size of the aperture of a channel is greater than the image size associated to the desired FOV. So the FPA is not optimally used. To overcome this constraint, a two-stage freeform lens arrays has been proposed to decenter the apertures of the multi-view system so that no overlap between the channels remains and the FPA is optimally used [35]. In this case, decentering the apertures leads to tilt the optical axis which is not perpendicular to the image plane anymore. Then, aberrations of decenter have to be corrected with freeform optics that are optics with no axis of symmetry. Freeform surfaces are indeed seriously considered for improving the image quality or for the miniaturization of decentered catoptric configurations [36][37][38] and could be an original use for multi-view systems. By decentering a multi-view system, Carles' team was able to design a small FOV multi-view system with a high aperture.
In this paper, we will explore different strategies of asymmetric or decentered multi-view systems to break the rule of Eq. (1). The aim is to design high-aperture multi-view systems allowing an optimal use of a single FPA with smaller field of views than the ones allowed with conventional TOMBO designs. In Section 2, the aperture issue for fast conventional TOMBO designs will be highlighted. Then, in Section 3, non-circular pupils will be investigated. In Section 4, some decentered freeform lens array systems, based on reference [35], will be studied and the issue of the lateral chromatism will be explained. Finally, folded lens channels or folded monolithic catadioptric channels will be proposed to solve the chromatism issue. Ray-traced designs will illustrate this study and their image quality will be checked with Modulation Transfer Functions (MTF) for different field points. These designs will be compatible for peripheral arrangements of the optical channels, that is arrangements like 2 × 2 or 2 × 3 channels. In particular, we will study 6-channel optical architectures in a 2 × 3 arrangement, for a horizontal field of view of 30°and for an F# of 1.2. The various designs use a 1024 × 768 pixels microbolometer detector with a pixel pitch of 17 µm.

Aperture issue for fast conventional TOMBO designs
Advanced optical features such as 3D imagery, multispectral imagery or super-resolution require combining several images observing the same scene, i.e. the same FOV. In the case of a TOMBO architecture with 2 × 3 optical channels making the best use of a 1024 × 768 pixel FPA, 2 optical channels can be arrange according to the lateral size of the FPA and 3 optical channels can be arranged according to the longitudinal size of the FPA . The sub-images are ordered along the horizontal size of the FPA as illustrated in Fig. 1.
In this figure, we can see that some field points are not shared by all channels. Indeed, the extreme channels 1 and 3 have their FOV vignetted by the edge of the detector. Thus, the field points represented in purple and orange are not common to all three channels and are therefore unusable when recombining the sub-images. As a result, there are areas of unused pixels between the different shared sub-images of the optical channels. In Fig. 1, the shared horizontal FOV is 30°for a system opened at F/1.2 and then F#<1/FOV. Therefore, the sub-image is smaller than its optical channel and unused pixel areas appear. It can be seen in this figure that the first stage of lenses 1 having the largest diameter are joined together along the longitudinal part of the FPA, so it is not possible to bring closer the sub-images to reduce the number of unused pixels. If we want to validate Eq. (1) with 30°of FOV, which means that the optical channel has the same size as its associated sub-image (then no unused pixel area), the F-number should be 2 that it is not compatible when using a microbolometer.
The theoretical maximum detector occupancy for such a FOV and aperture can be calculated. For this, we will use the following dimensions: L FPA the length of the FPA, W FPA the width of the FPA, S the size of a square sub-image and D the diameter of the lens 1 (drawn in blue); which are represented in Fig. 2. The percent occupancy Occ is the ratio of the area of sub-images (the red squares) to the area of the detector: The size of the sub-image S can be expressed according to the focal length f and FOV as: The length of the FPA can be expressed according to D and S. D being linked to the F# by F# = f /D, we obtain the following relation: By combining Eqs. (2), (3). and (4), we can express the percent occupancy according to the FOV: For a given F# of 1.2 and a 1024 × 768 pixels microbolometer, we can plot the occupancy of the detector according to the FOV, as illustrated in Fig. 3. For a FOV greater than 47°, the sub-image is larger than the optical channel and therefore the occupation of the FPA is maximum with 89% of the pixels used. For a FOV of 30°, the occupancy of the FPA is only 47%. It is an inherent limitation of fast TOMBO architecture that we propose to overcome through different strategies.

Multi-channel architecture with non-circular pupils
The relationship between aperture and field of view has been identified as an inherent limitation for fast TOMBO multi-channel architectures. This relationship has however been obtained for circular pupils. In nature, non-circular pupils are quite common [39], so in this section, we investigate non-circular apertures to obtain a high aperture for each optical channel while increasing the number of resolved points of the sub-images looking at the same field of view. This strategy works for multi-view configurations for which the channels are not adjacent on one side, for example in the case of arrangements like 2 × 2 or 2 × 3 channels. Figure 4 illustrates the way to define the shapes of the non-circular pupils for a 2 × 3 channels configuration. Starting with the expected resolved points for the sub-images sharing the same field of view that defines the diameter of adjacent circular pupils, increasing their diameter will lead to an overlap of the pupils on the adjacent channels. Therefore, we propose non-circular pupils made by a half-disc whose aperture validates Eq. (1) and by a second half-disc having a greater aperture. The orientations of the non-circular pupils are chosen to avoid overlapping areas. Non-circular pupils allow expanding the aperture of the channels in the peripheral area of the multichannel configuration while constraining the apertures of the channels in the neighbouring area. To estimate the equivalent aperture of these non-circular pupils, we can first estimate the area of the pupils with the following equation: Where, Φ eq is the equivalent diameter of a circular pupil having the same area as the noncircular pupil, Φ 1 is the diameter of the half-disk in the neighbouring area and Φ 2 is the diameter of the half-disk in the peripheral area. With the definition of the F# (F#=Φ/f, f the focal length of the system), Eq. (6) becomes: with F# eq the equivalent radiometric F-number of a circular pupil having the same area as the non-circular pupil, F# 1 the F-number of a system with a Φ 1 diameter pupil and F# 2 the F-number of a system with a Φ 2 diameter pupil. In the example of Fig. 4, the non-circular pupils are a combination of a circular pupil with an F# 1 of 1 in the peripheral area with a circular pupil with an F# 2 of 1.4 in the neighbouring area. The designed non-circular pupils lead therefore to an equivalent radiometric F# eq of 1.15. Then by simply replicating this optical channel for different orientations, a 2 × 3 channels system shown in Fig. 4 is obtained. For a 17.4 mm (L FPA ) by 13.1 mm (W FPA ) FPA, the 6 channels fit into a 20.8 by 15.5 mm rectangle. The multi-view design has a horizontal field of view of 30. The architecture of an optical channel is illustrated in Fig. 5(A), showing the asymmetrical shape of the first lens oriented vertically. Each channel produces sub-images of 282 × 282 pixels looking at the same field of view, as represented in Fig. 5(B), (to be compared with the sub-images of 249 × 249 pixels produced by a classic TOMBO configuration described in Section 2). The occupancy of the FPA rises to 61%. In order to obtain a good image quality with a contrast at Nyquist frequency higher than 20%, 4 aspherical surfaces up to the 8th aspherical order are needed. The polychromatic MTF are shown in Fig. 6(A) and spot diagrams for different field points are shown in Fig. 6(B). In these figures, we show 4 points of the field of view, including the opposite fields of view 15°and −15°, because the non-circular pupil breaks the rotational symmetry. Non-circular pupils make possible to go beyond the limits of a classical fast TOMBO architecture by exploiting the free spaces in the peripheral area of a multi-channel configuration. However, the gain remains low compared to the growing complexity of the optical surfaces.

Multi-channel architecture with decentered pupils and freeform surfaces
Another strategy to increase the aperture of the optical channels is to decentre the pupils in relation to their corresponding sub-images. The optical axis of each channel has to be tilted, and this can be obtained by using prism shape lenses [20,[40][41]. G. Carles' team adapted this approach for an infrared multi-aperture with high angular resolution and compatible with microbolometers [35]. They designed a two-element freeform architecture in germanium for an 8°field of view, for an F# of 1.6 and for a wavelength of 10 µm. However high aperture and high resolution induce an important tilt of the optical axis. Therefore, asymmetric and off-axis aberrations appear, which can be removed with freeform optics as shown in reference [35]. Unused areas between the sub-images of common field of view are drastically reduced, so this design approach can get around the coupling constraint between F# and FOV of classical TOMBO architectures. So we reproduced a decentered and tilted optical channel following the features of the multi-view design of reference [35] to evaluate its image quality in a broadband infrared spectrum (8 µm to 12 µm). In G. Carles' team work, the system is in a 3 × 3 optical channel configuration, with a central conventional channel and 8 decentered pupil channels. Fig. 7 illustrates the central channels and two lateral channels.
Then, for a FOV of 8°, an F# of 1.6 and side-by-side images, the minimum tilt is around 26°. The angle α of the tilt of the optical axis is equivalent to the deviation angle of the prisms introduced in the lateral channels as illustrated in Fig. 7. In the particular case of Fig. 7 where the chief ray of the central field point is perpendicular to the first surface of the prism, a simple relation links the vertex angle of the prism A with the axis tilt angle α and the refractive index n: Then, for a FOV of 8°and an F# of 1.6, the prisms introduce in the lateral channel have a vertex angle of about 7.2°. However, using prisms generates chromatic dispersion that can be problematic for broadband imagery. With important axis tilt α, even with high Abbe-number materials like germanium, important lateral chromatic shift can be observed. Using Eq. (9), the chromatic shift D chrom on the FPA between an image point at 8 µm and the one at 12 µm can be calculated as: D chrom = f (arcsin(n Ge@12µm sin(A)) − arcsin(n Ge@8µm sin(A)), with α 12µm and α 8µm the tilt of the optical axis for the wavelengths 12 µm and 8 µm. The refractive index of germanium for each wavelength (n Ge@12µm and n Ge@8µm ) is given by Zemax software. In our example, the focal length can be calculated by the following equation: with a L FPA of 1024 pixels with a pixel pitch of 17 µm, and a FOV of 8°, we obtain a focal length of 41.3 mm. Then a chromatic shift D chrom of 22 µm is calculated. A lateral decentered and tilted optical channel of this 3 × 3 channel configuration is illustrated in Fig. 8. The prism is fused with the front lens to form a prism shape lens and all surfaces are freeform and described by XY polynomial coefficients. Since the axis tilt α is important (26°), we observe an important lateral chromatic shift for all the field points, as shown in the spot diagram of Fig. 9(A). On the spot diagram, we estimate a chromatic shift on the FPA of around 20 µm which is consistent with the distance D chrom calculated with Eq. (10). We notice as well that the orientation of the prism-shape lens is different than the orientation of the prisms in Fig. 7. Indeed the orientation of the prisms in Fig. 7 may not be the best to limit the amount of lateral chromatism and this could explain the small difference between the estimated chromatic shifts. The chromatic shift measured in the spot diagrams is larger than the size of 17 µm of a pixel and degrades significantly the image quality, as can be seen in Fig. 9(B) in which the tangential MTFs (in solid line) are below the 20% contrast threshold at Nyquist frequency.
In the case of a 30°field of view multichannel with decentered pupils system opened at F/1.5, we placed the stop diaphragm on the second lens, as illustrated in Fig. 10(A). This architecture is asymmetric along the vertical axis, and symmetric along the horizontal axis. In this system, the decreases of both the focal length and the axis tilt angle, in comparison with the previous example, limit the amplitude of the lateral chromatism induced by the prisms. Therefore, a good image quality over a broadband infrared spectrum is observed in Fig. 10(B) and in Fig. 10(C) (as this architecture contains a horizontal symmetry plane and a vertical asymmetry plane, we show 6 points of the field, 4 in the vertical axis, 2 in the horizontal symmetry axis). Squared sub-images of 320 × 320 pixels were obtained as illustrated in Fig. 11(A). The occupancy percentage is 78%. A top view of ray tracing of extreme field points (±15°) of the 2 × 3 optical channel system is illustrated in Fig. 11(B). The 6 channels fit into a 46 mm by 44.2 mm rectangle. For clarity, the shapes of the lenses are not drawn. In this figure, we see that the rays of extreme field points for each optical channel (one colour by channel) do not overlap each other, so it is possible to make optics and assemble them with no mechanical overlaps. We can conclude that tilting the optical axis by decentering the pupil positions of each channel greatly increases the number of resolved points in the sub-images of the different optical channels. However, the prism introduced to tilt the axis can generate important lateral chromatism and can limit the use of such system for broadband imagery. Fig. 11. A) Prints on the FPA of the 2 × 3 square sub-images of a multichannel architecture with decentered pupils and freeform surfaces for a 30°horizontal FOV and a F/1.5 aperture. B) Top view of a ray tracing of extreme field points (±15°) of a 2 × 3 optical channel system. Each color corresponds to one optical channel. For clarity, the shapes of the optics have not been drawn.

Multi-channel architecture with folded lenses
Decentering the pupil position is an effective way to greatly increase the number of resolved points in the sub-images of the different optical channels. Instead of using prisms to tilt the optical axis, plane folded mirrors can rather be used to prevent lateral chromatism. To limit the number of optical elements to be aligned, we first integrate the plane mirrors inside the optics, so that the optical channel becomes a combination of two folded optics. There are some freedoms for the size of the front lens but the rear lens must remain compact and must have a smaller size than the size of the sub-image in order to be tightly assembled with the rear optics of the other optical channels. The stop diaphragm should thus be on the rear optics. Figure 12 shows a starting point for a folded optical channel with a field of view of 30°and with an aperture of F/1.2. The stop diaphragm is on the first surface of the rear optic. Thick optics are designed to be folded by a plane mirror tilted by 45°(the thickness should be longer than the diameter of the optic). The thickness of the optics can be more efficiently increased by the use of materials with a high index of refraction like germanium.   Since plane mirrors do not introduce any off-axis or chromatic aberrations, a good polychromatic image quality is obtained with 4 aspherical surfaces as shown in Fig. 13(B) and Fig. 13(C). No freeform surfaces are thus required. Folded optical channels by plane mirrors allow having sub-images of 325 × 325 pixels looking at the same field of view, as illustrated in Fig. 14(A), and a FPA occupancy of 81%. Thus, it represents an important gain compared to a classic TOMBO configuration with the same features. The bulky front lenses are placed far enough from their corresponding sub-images so that the folded channels can be replicated with various orientations without overlapping as shown in Fig. 14(B). The 6 channels fit into a 44.8 mm by 42.6 mm rectangle.

Multi-channel architecture with folded monolithic catadioptric elements
Folded channels seem to be an interesting strategy to make a fast multi-view system with a high angular resolution. Indeed, folded plane mirrors do not add any off-axis or chromatic aberrations and no freeform surfaces are required to have a good image quality. However, two folded optics for each optical channel have to be made and aligned along a folded optical axis. This remains quite tricky. In this section, we investigate the possibility of designing a folded optical channel in a single catadioptric component.
Catadioptric components have been explored to make flat cameras [42][43] or panoramic cameras [44][45]. More recently an infrared monolithic catadioptric element has been studied to replace the multiple optics of a common optical design [46][47] but without the constraint of making several images on the same detector. Since optical powers were implemented on the mirrors, freeform surfaces had to be used. Compared to the example of references [46][47], multi-view systems require sub-images with less resolved points so we were able to design a folded monolithic catadioptric channel without adding optical powers on the mirrors. So, only aspherical surfaces were used. The folded monolithic catadioptric channel illustrated in Fig. 15. A) Germanium monolithic channel for 30°horizontal field of view opened at F/1.2. The monolithic channel consists of two plane mirrors and two aspherical surfaces. The surfaces marked with a red * are aspherical. B) Polychromatic MTF for the folded catadioptric channel and for different field points. Contrast is greater than 20% for all the field points at the Nyquist frequency. C) Example of spot diagrams for different field points for wavelengths of 8 µm and 12 µm. Fig. 15(A) has a horizontal field of view of 30°and an aperture of F/1.2. The optical component is made of germanium.
The stop diaphragm was placed on the second folding mirror to efficiently correct the off-axis aberrations and to limit the size of the second mirror and the size of the rear refractive surface. Indeed, their sizes have to be carefully controlled during the optimization process to avoid an overlap of the channels. Moreover, the inclination of the mirrors, as well as their respective positions, have to be chosen to place the front refractive surface and the first folding mirror far enough from the position of the sub-image to avoid overlap between the optical channels while maintaining a good compactness of the monolithic catadioptric component. With only 2 refractive surfaces to form the image, this architecture has few degrees of freedom to fulfill all these conditions and astigmatism is difficult to correct. Nevertheless, good image quality has been obtained, as shown in Fig. 15(B) and Fig. 15(C). The sub-images size is 330 × 330 pixels (83% of FPA occupancy), as presented in Fig. 16(A) and Fig. 16(B) shows that there is no overlap between the channels in a 2 × 3 configuration. The 6 channels fit into a 49.4 mm by 47.8 mm rectangle.

Summary
Multi-view TOMBO architectures and their ability to easily integrate different optical elements in each channel like filters or polarizers can answer a wide range of applications requiring advanced imagery functions. Moreover, the compactness achieved by these architectures can answer SWAP constraints and make them suitable for UAV applications for example. However, in the infrared spectral band, using SWAP detectors like microbolometers requires fast optics that limits the angular resolution of a standard TOMBO configuration, that is, a combination based on lenslet arrays. Indeed, due to important size of pupils, the size of the sub-images sharing a common field of view is drastically reduced. In this paper, we showed that for a 2 × 3 channels configuration, asymmetric or decentered multi-view designs can be used to improve the angular resolution of these systems. Multichannel designs with non-circular pupils, with decentered pupils and freeform optics, with folded optical channels, or with folded monolithic catadioptric components were studied. The strategies developed lead to increasing of the size of the front lens arrays which allows an increase of the focal lengths of the channels at constant F-number, and thus an increase of their number of resolved points. Consequently, the studied optical systems are no longer subject to Eq. (1). They all show an improvement of the size of the sub-images over a classical configuration, most of them reaching even the maximum possible size for sub-images produced on a single rectangular focal plane array. Nevertheless, they represent a challenge for manufacturing. Indeed, even if we managed to use only aspherical surfaces for most of the studied designs, technologies developed for freeform optics should be considered especially to precisely define references and coordinate systems in order to be able to measure all surface angles, decenter and surface form deviation correctly. So, fiducials are needed for manufacturing as well as for referencing the surfaces. Another challenge in multichannel architectures would be to prevent crosstalk between the sub-images. A conventional TOMBO system uses a separation layer between the lens arrays and the detector. However, this solution would not be obvious, because the window of the microbolometer is in between the detector and the last lens array. Other solutions could be studied, such as baffling upstream the optical system.

Disclosures
The authors declare no conflicts of interest