1. Introduction

Many new possibilities appear in optical design when modern optical devices and techniques are used in an imaging system. An example of this is the use of adaptive optics (or its slower counterpart called active optics) to enhance image quality as it is done in astronomical telescopes. Furthermore, active optics has appeared during the last few years in more conventional systems. It is expected that with newer, cheaper and better active technologies yet to appear, the use of active optics will increase in the development of original and flexible imaging techniques. The locally magnifying imager proposed here is based on such use of active optics elements.

In many applications, like video surveillance, a specific zone of interest must be observed with higher sampling within the entire field of view. Increasing the object sampling in the full field of view (FFOV) by using a detector with a high number of pixels is not a solution since it increases the bandwidth required to transmit and record the images. A solution quite often used in surveillance is to combine two lenses, a wide-angle lens to image the FFOV and a Pan/Tilt/Zoom (PTZ) lens to obtain higher magnification in a zone of interest. This however requires at least two cameras and is limited to a single zone of interest per PTZ lens. Furthermore, a software algorithm is needed to present a recombined image to the operator.

In most lenses, the magnification between the object and the image is constant, or very slightly changing, creating systems with almost no distortion. Distortion can be described as a variable magnification across the FFOV of a lens. Because of that, it is often considered an unwanted aberration that optical designers want to reduce. However, it can sometimes be a useful parameter in the design of some particular system where variable magnification is required. An example of this is a fish-eye lens [1] in which the barrel distortion is necessarily present to allow imaging a full hemispheric object onto the image plane.

Distortion does not directly affect image quality. However, since distortion requires different magnifications, meaning different optical power and varying local focal length (LFL) across the FFOV, it often indirectly affects image quality by producing other aberrations as well. As shown in earlier work [2], distortion can be modified by optical surfaces which are far from the aperture stop, near the front optical element or close to the image plane. This is because there is a small beam diameter to lens diameter ratio in these regions. It is thus possible to change the magnification of rays from a part of the FFOV without affecting the other rays. Also shown in earlier work, a small undesired local deformation on the front lens will have larger impact in some parts of the FFOV, depending on some parameters as the LFL [3]. These findings led to the present paper where controlled distortion is now desired. The idea behind the locally magnifying imager is to use an active optics element to change in real-time the direction of some rays coming into the system, thereby changing the local magnification of these rays in a zone of interest. Contrary to ordinary zoom systems which change the magnification in the whole field by a given constant, thus changing the FFOV, the FFOV can remain constant with a locally magnifying imager. To achieve a constant FFOV, a zone of lowered magnification is used around the zone of increased magnification to compensate.

Our previous paper [4] described the general idea of this system, without the mathematical analysis presented here. It also showed simulations of such a system and explained the differences in other similar systems. The locally magnifying imager is different from several systems using active optics. For example, active zoom lenses [5] change the magnification everywhere and consequently the FFOV. Foveated imaging systems [6,7] use an active optic in the pupil plane to correct aberrations which increase the MTF locally but a detector with a high number of pixels is required. Manually deformed mirrors have been tried but are hard to control [8]. Finally, unlike Panomorph lenses [9], the locally magnifying imager has zones of higher magnification that can be dynamically adjusted instead of being static.

This paper first presents a mathematical description of the system in section 2. Section 3 follows with results obtained from an experimental prototype and finally a discussion is presented in section 4 before concluding the paper at section 5.

2. Spatio-temporally adjustable magnification lens

2.1 Concept of the locally magnifying imager

This novel optical system is based on the idea of locally changing the magnification or the LFL in a small region of the FFOV using an active optical component which introduces a field dependent optical power. This active optical surface can be anything that can change in real-time the direction of some rays of light, whether it is reflective, refractive, diffractive, etc. To create a small localized zone of modified magnification in the image plane, only a small fraction of rays require a change of direction by a zone of modified power on the active surface. This change of direction is illustrated in Fig. 1 , also showing two other fields unaffected by the active surface.

 

Fig. 1 Schematic representation of the concept behind a locally magnifying imager. Only a small fraction of the rays are affected by the active optical surface, the rays from the blue field in this case. By changing the object angle of these rays, a different magnification is produced at the image plane. For clarity, all the other optical surfaces except the stop are hidden.

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Two important features are seen in Fig. 1. First, the active surface has to be placed in a region where the beam footprint diameter is small compared to the lens diameter to allow a separation of the beams. This happens in regions far from the stop. In the schematic of Fig. 1, the active surface is the first one of the system, but this does not need to always be the case. Still, this choice simplifies calculations and the active surface will be considered at the front of the system for most of this paper. The second important feature visible from this figure is that the beam diameter has to be small and close to the chief-ray to make sure that the active surface has a similar optical impact on each ray for a given field, ensuring that the geometrical aberrations added to the rays by the active surface remain under a certain limit. To do so, a high F/# can be forced with an iris at the stop or a wide-angle lens with its natural small beam diameter could be used. A mathematical description of the system follows at subsection 2.2 and then a description of the aberrations and limits will be presented in subsection 2.3.

2.2 Mathematical description of the system

The first step in describing this system mathematically is to obtain an equation of the ratio of magnification (RoM), the ratio of the new magnification to the original one (without the active surface), as a function of the active optics surface shape. The system with an active reflective surface as the first surface in front of an imager is schematized in Fig. 2 . For simplicity, the imager is reduced to only an entrance pupil (EP). For this simple example, it is considered that the original EP position of the optical system, that is before considering the displacement of the EP caused by the active surface at some fields, is constant in the FFOV. This simplification is often too harsh in wide-angle lenses since the EP position is affected by the original lens distortion profile [2]. More accurate results can be obtained in these panoramic lenses by considering the effect of the LFL as described in earlier work [3]. Also, all the rays from a given object point are considered to be close around the chief-ray so that when a chief-ray is traced from a point in the image plane toward the center of the stop and then to the object, only the angular deviation ψ produced by the active surface will affect the magnification. Finally, only the additional aberrations introduced by the active surface in the imager are considered.

 

Fig. 2 Schematic of the problem with two chief-rays separated by an infinitesimal distance on the active surface. For this example, the active surface is chosen to be a mirror. The angle on the mirror with the horizontal at the two positions r and r + δr where the rays hit are different and given respectively by ψ_r and ψ_{r + δr}. L₀ is the distance between the EP and the mirror.

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First, if the active surface is a deformable mirror, so that the reflected angle is given by${\phi }_r={\theta }_r-2{\psi }_r$, the LFL can be written as Eq. (1) [3].

Finally, multiplying by $\frac{\delta r/L_0}{\delta r/L_0}$and using the definition of the derivative and the derivative of arctan(r), Eq. (5) is obtained, which simplifies to Eq. (6) if the angles θ are small, meaning that r<<L₀, an approximation valid with systems having FFOV under 30 degrees and no original distortion. For systems with larger FFOV, having initial distortion, more complex equations have to be used.

This equation fits well with previous ZEMAX simulations that were done with Grid Sag defined surfaces and that were calculated with a more complex method [4,10]. It is also the equation that would be obtained with the paraxial imaging equation, replacing the curvature by an element of focal length f_curv. Similarly, if the reflecting mirror is replaced by a refracting surface, RoM(r) is given by Eq. (7), where the + sign means that a positive slope on the active surface bends the rays away from the optical axis contrary to the reflecting case.

2.3 Fundamentals limits of this system

2.3.1 Amplitude of the active surface

The first important limit of a locally magnifying imager is the required amplitude on the active surface. For a continuous deformable mirror, to produce useful angular deviations, large amplitudes are required and it can quickly become a limit to the performances of such imagers. For a desired spatial distribution of RoM(r), Eq. (6) allows to calculate the shape of the mirror and thus the required amplitude on it. Two special cases are analyzed here, the case of a constant RoM in the whole zone of interest and the case of a large RoM at the center of the zone of interest quickly decreasing around it.

As a first example, consider the case of a zone of increased magnification with a constant RoM. Equation (6) shows that for a constant RoM(r), Z”(r) must be null (the trivial case RoM = 1) or constant. For a value of RoM > 1 (increased magnification), the solution is quickly found to be a positive parabola in the zone of interest. The example is schematized at Fig. 3 for an imager with no original distortion. Since

 

Fig. 3 Schematic showing the chief-rays for three fields. With a flat mirror, the original red rays are reflected toward the EP and reach the image plane. With the mirror in the correct shape, the green rays are now the rays reflected toward the EP. This creates 3 zones, one of higher magnification from r = 0 to r = a, one of lower magnification around it from r = a to r = b and one where the mirror stays flat, keeping the original magnification, from r = b to r = D/2. The ray hitting the extreme part of the mirror, at r = D/2, comes from an angle θ_max and hits the image at a distance H from the center. Consequently, the FFOV is constant since this angle is the same before and after changing the shape of the mirror.

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To better understand Eq. (9) and to get a scale of magnitude, Fig. 4 shows the required amplitude on the deformable mirror for the case of a mirror having D = 100 mm and a zone of interest with a diameter of 20 mm (α = 0.2), as a function of both the RoM and the ratio H/f.

 

Fig. 4 Graphical representation of Eq. (9), with RoM and H/f variable, and with values of D = 100 mm and α = 0.2. For realistic use, amplitudes of hundreds of µm are required.

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For increasing values of the ratio of H/f, or equivalently larger values of tan(θ_max), as in wide-angle lenses, the required amplitude can become very high. For lenses with longer focal length, the amplitude is small, but the distance L₀ between the mirror and the imager is large. For increasing values of RoM, the increase in amplitude is significant from 1 to 2, but gets slower with ever increasing RoM, an effect of the (RoM-1)/RoM term in Eq. (9). This figure is for given values of α and D, but looking at Eq. (9) shows that if α is kept constant and D doubled, the amplitude required is also doubled. Also, if D is constant and α is doubled, the amplitude is 4 times higher.

The second example is with a zone of interest having a large RoM at the center and quickly decreasing around it. A useful example of this is having a Gaussian shaped bump on the deformable mirror. Using a Gaussian deformation of amplitude A on the mirror and having a FWHM of 2(2ln2)^1/2σ as Z = Aexp(-r²/2σ²), taking the second derivative, inserting it in Eq. (6) and rearranging the terms gives Eq. (10) for the amplitude as a function of the RoM at the center of the mirror (r = 0), again using from Eq. (8) that L_o = Df/2H.

2.3.2 Entrance pupil diameter

Another important limit in a locally magnifying imager is the requirement on the entrance pupil diameter. Since the major part of the image is unaffected by re-magnification, the detector is normally located at the best-focus image plane before adding the active surface. Adding an element with an optical power (a curvature) changes the best focus position in the zone of interest and also in the zone where the original magnification is restored. These additional aberrations produced by the active surface on these selected beams affect the image quality in these zones. To keep them at a tolerable level (within the depth of focus), the entrance pupil diameter has to be small enough since the depth of focus is proportional to the square of the F/#.

To modify distortion, the deformed surface is generally limited to a relatively low spatial frequency. This means that the surface is relatively smooth and when the surface is expanded in Taylor series around the position where the chief-ray hits the surface, for the very small beam diameter from a given field point as explained in subsection 2.1, the surface can be well represented by neglecting terms higher than the second order term, meaning an off-axis parabola, with possibly a different curvature in the sagittal and tangential direction. In that case, the most important aberrations added by the surface are defocus and astigmatism [11]. Depending on the shape of the mirror and on the incident angle, astigmatism could become significant, but for usual applications of a locally magnifying imager, it was found by ZEMAX simulations that defocus is the primary limiting aberration. Also, even if distortion requires a low spatial frequency, having an active surface with a high number of actuators [12] can be useful to create several zones of interest.

As a way to quantify the limiting F/# of the imager to keep a tolerable focus, Rayleigh quarter-wave criterion on peak-to-valley optical-path-difference (PV OPD) in the exit pupil is used [13]. The first thing to consider is the effect of the active surface on the entrance pupil diameter for the whole system. An active surface producing a magnification of RoM in front of an imager originally having an entrance pupil diameter of EP for a given field object point will resize the local entrance pupil of the total system for the rays from this field point to RoM×EP and the exit pupil will stay the same. This change of size for the EP is a consequence of the local magnification in the object-side image of the stop and no additional vignetting is produced because the beam keeps its original diameter in or after the stop. For RoM > 1, this increase of size of the entrance pupil is desirable since it ensures that the relative illumination in the image plane stays constant by capturing RoM times more light in the beam (RoM_xRoM_y if both dimensions are considered), but also spreading it on RoM times more pixels, a consequence of conservation of radiance. For an initial plane wave, the resulting PV OPD in the Z direction after reflection on the mirror, an approximation of the PV OPD in the direction of the beam for small angles, is then given by Eq. (11). This is for a parabola of curvature Z” and it includes a factor 2 for the reflection on the mirror. If the angle is not small, a term 1/cos(θ) = ((r² + L₀ ²)^1/2)/L₀ must be added.

To get the corresponding PV OPD in the exit pupil, the PV OPD in the original entrance pupil of the imager is first calculated. From the mirror to the original entrance pupil, the transverse radius r of the wavefront is scaled by a factor 1/RoM as explained earlier and the radius of curvature R is scaled by a factor 1/RoM, a consequence of propagating a wavefront with a radius of curvature R = (2Z”)⁻¹ on a distance L_o and Eq. (6). The PV of a spherical wavefront being given by PV = (r²/R)/(1 + (1-r²/R²)^1/2), the PV OPD is also scaled by 1/RoM. Since the imager adds few aberrations at a high F/# like the one used here, it is then considered that the additional PV OPD is equal in the entrance pupil and in the exit pupil, finally giving Eq. (12).

3. Experimental verification of the system

3.1 Experimental prototype

To test experimentally the concept of a locally magnifying imager, a commercial camera and lens are used in combination with a ferrofluidic continuous deformable mirror [14]. The mirror has a full diameter of 100 mm with the 91 actuators located in a central 40-mm diameter region. The reason for using such a mirror is the high amplitudes it allows. However, this liquid mirror must remain horizontal and the adaptive optics setup in which it is used is limited to measuring amplitudes of about 40 µm on the mirror by the Shack-Hartmann sensor while the low-current driving electronic limits the mirror to amplitudes of 150 µm. This soft limit depends only on the over-heating electronic and actuators since the ferrofluid can achieve amplitudes of over 1 mm with the right magnetic field. For the slopes, the maximum amplitude difference from two neighbor actuators is about 50 µm in a separation of 2.8 mm, producing a maximal slope of 1 degree. The limit on the sensor is not a problem in our case since a precise shape can be set on the mirror using the adaptive optics loop and then, by scaling the voltage (and thus the current) in the actuators by a constant, it is possible to get higher amplitudes scaled approximately by this constant. This explains why scale factor is the independent variable instead of amplitude in the experiments. The driving electronic capability is however limiting and other tests with a better amplifying electronic could allow better amplitudes. Another obstacle with this mirror is the low natural reflectivity of about 4%. For imaging uses, a metal liquid-like film layer could be deposited on it to achieve a better reflectivity, but in our case, for simplicity, a brightly illuminated object was used. The schematic of the setup is shown at Fig. 5 , with the camera making a small angle (<15 degrees) with the vertical which has the same negligible effect as the small angles did in section 2.2. The distance L₀ between the entrance pupil and the center of the mirror is set to 217 mm according to Eq. (8). The lens is a f = 12.5 mm Fujinon HF12.5SA with an entrance pupil measured to be 25 ± 1 mm behind the metallic frame on the object side. It has a manually adjustable focus ranging from 0.1 m to infinity and a manually adjustable F/# ranging from 1.4 to 22. For all the following images, a F/# of 16 is used. According to Eq. (13), this is the limit for a RoM of 1.4 and some defocus is expected when higher RoM are produced. As for the camera, a 2592x1944 pixels (5 megapixels) AVT Guppy F-503 having pixels of 2.2 µm is used. Figure 6 shows the reference image obtained with the deformable mirror in a flat position. The test pattern is a ceiling light diffuser, the plastic cover under a neon light, because of its brightness and regularly spaced pattern. Each small square in the regular pattern, as seen in Fig. 6, is used as a target for measurements. All the images are cropped in the same way to show the zone of interest and the region around it.

 

Fig. 5 Schematic of the experimental setup used. The camera + lens are pointed at the ferrofluidic deformable mirror to look by reflection at the object, a light diffuser on the ceiling. Not shown in the figure is an adaptive optics closed-loop used to set and measure the surface.

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Fig. 6 Reference image obtained with a flat deformable mirror. For the images to follow, only the cropped region is presented. In the central region, all the small targets are equally spaced. At the edge of the container, liquid meniscus effects produce some unimportant distortion.

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The ferrofluidic deformable mirror used for the experiments is able to produce almost any shape, up to a limiting amplitude and spatial frequency. However, to effectively show the use of the proposed systems, only the two family of shapes presented in subsection 2.3.1 are presented here, the parabolic shape producing a zone of constant RoM and the Gaussian shape producing a high localized magnification in the center and quickly decreasing around it.

3.2 Parabola producing a zone of constant magnification

To efficiently show the effect of having a large zone of constant positive or negative magnification, a zone of interest having a diameter 40 mm compared to the full mirror diameter of 100 mm is chosen. The zone of settling back is not very important here and a quick recovery back to a flat mirror is used outside the 91-actuator region. As predicted by Eq. (9), this α of 0.4 requires large amplitudes on the continuous deformable mirror to produce the desired slopes. Experimentally, a parabola was produced on the deformable mirror, using the Shack-Hartmann wavefront sensor in the adaptive optics closed-loop bench to measure the surface. Once the shape is measured, the closed-loop is removed from the setup. The parabola was first obtained with a scale factor of + 1 on the voltages and then this scale factor was varied from −5 to + 5 to produce a wide range of amplitude of the parabola. Seven images were taken, but only the four most interesting are shown at Fig. 7 .

 

Fig. 7 Images of zones of increased and decreased magnification by using a parabola in the zone of interest. In all four images, the magnification is constant in the central zone and different from the original magnification of Fig. 6. The voltage in the actuators was scaled with the following factor: (a) −5 (b) −1 (c) + 1 (d) + 5

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The position of each small target on a horizontal line passing through the center is measured for each of the 7 captured images, including the reference, and Fig. 8 shows the results graphically. From this Fig. 8, to get a measurement of the magnification in the central zone of interest, a linear fit is done for each line and the resulting values of the central magnification, in pixels/target, are plotted as a function of the scale factor at Fig. 9 . It is important to see that the RoM in the zone of interest scales according to Eq. (6) with the amplitude of the parabola. The RoMs produced are rather small, being 1.17 with the scale factor of + 5. These small RoMs are a consequence of the large amplitudes required for a parameter α of 0.4. Larger RoMs have been achieved in a constant zone by placing the camera farther (increasing L₀) and/or using a smaller zone of interest [4]. Also to be noticed is the zone of transition where the liquid goes back to a flat horizontal mirror after the parabola. As expected, the RoMs in that zone increase when the zone of interest has a RoM <1 and decrease when the zone has a RoM >1, a consequence that the FFOV is conserved and that the number of pixels is constant.

 

Fig. 8 From the 7 images with a parabola, graph of: (a) the positions in pixels as a function of the target number for targets on a horizontal line and (b) the relative position showing the displacement in pixels of each target with respect to the reference of Fig. 6. In both graphs, the scale factors are −5, −3, −1, 0, 1, 3 and 5. The slopes in the central zone are linearly fitted and results are at Fig. 9.

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Fig. 9 From Fig. 8, a linear fit is done in the central region and from the slope of these fits, by dividing by the original magnification, RoMs are plotted as a function of the scale factor, along with their 95% confidence bound. As expected, it can be seen that the magnification in the zone of interest compliantly follows a fit having the shape of Eq. (6).

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3.3 Gaussian bump producing a localized high magnification

As explained in subsection 2.3.1, a Gaussian shaped deformation on the mirror will produce a region of variable magnification, with a maximum in the center. A Gaussian shaped deformation was produced and then it was scaled to obtain different magnifications. Figure 10 shows 4 of the 8 images taken. Figure 11 shows the position of each target on a horizontal line passing by the center of the zone of interest.

 

Fig. 10 Images produced by Gaussian shaped deformations, creating a zone of increased magnification in the center and quickly dropping around it. In all four images, the magnification drops below the original magnification from Fig. 6 in an annular zone around the center. The voltage in the actuators was scaled with the following factor: (a) 0.5 (b) 1.0 (c) 1.3 (d) 1.5. With a scale factor of 1.5, the central region is a bit out of focus.

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Fig. 11 Graph of: (a) the position in pixels of each target on a horizontal line from the 8 images taken and (b) the relative position showing the displacement in pixels of each target with respect to the reference of Fig. 6. In both graphs the scale factors are 0, 0.5, 1.0, 1.1, 1.2, 1.3, 1.4 and 1.5. With increasing scale factor, the magnification in the center increases too and it is more visible by looking at the derivative in the center region as plotted in Fig. 12.

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To clearly see the magnification at the center, the derivative of this graph is taken and only the central zone is shown in Fig. 12 . For numerical differentiation, a sampling of 2 measurements per target is used and the magnification is obtained by multiplying the differences by a factor of 2. The results are then divided by the original magnification to get the RoM. By smoothing this figure, it shows that a RoM up to about 3.4 is achieved in the center when the scale factor is 1.5. However, it must be noted that the apparently higher magnification achieved with a Gaussian bump rather than with a parabola is only a consequence of having a smaller region needing a high curvature and thus easier to produce.

 

Fig. 12 Resulting spatial RoM as a function of the target # for the 8 images taken with Gaussian shaped bumps on the deformable mirror. With increasing scale factor, the magnification in the center can become significant. The 8 curve scale factors, from the bottom to the top around their maximum are: 0, 0.5, 1.0, 1.1, 1.2, 1.3, 1.4 and 1.5.

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Another thing to consider is that the shape on the mirror can be more complex, having multiple zones of variable magnification. Figure 13 shows a quick example of that, having 2 zones of increased magnification separated by a zone of decreased magnification.

 

Fig. 13 Image with 2 zones of interest separated by a zone of lower magnification. Unlike all the previous examples having a single zone of interest, a real application of this kind of lens could have more than one zone of interest.

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4. Discussion of the limits of locally magnifying imagers

As described in subsection 2.3, this imager has some fundamental limits, mainly on the amplitude required on the active element and on the diameter of the entrance pupil to limit aberrations. A closer look at Eq. (9) and Eq. (10) shows that the amplitude required is proportional to the square of the length of the zone of interest, that is α in Eq. (9) or σ in Eq. (10). This means that it would be easier to use this technology when only a small fraction of the total mirror surface is in the zone of interest. Also from these equations, the inverse proportionality in L₀ means that for smaller amplitudes on the active surface, it is better to use the deformable surface far from the imager lenses, which means a large active surface diameter D or a small ratio H/f = tan(θ_max) according to Eq. (8). Since using a large active surface diameter is often costly, this seems to limit this kind of imager to a long focal length, or in other words, small FFOV.

However, there are two possible solutions to this problem. First, other technologies than continuous deformable mirrors are able to change the direction of rays like diffraction-based LC SLM [15] and could provide lower requirement on the active surface at the expense of other challenges. Also, since the large amplitude is a consequence of the large angular deflections needed, discontinuous micro-mirror systems [16] could be advantageously used in such systems since they can easily allow deflections of over 3 degrees. However, when using this kind of optic, all the additional challenges related to diffraction have to be taken into account. Finally, new, better and cheaper deformable optics tend to appear with time and the concept explained here could be used with a yet to be discovered type of active surface. The second possible solution to this problem of high amplitudes is to use the active surface at another position than at the first surface of the system. By using a static optical element on the object side which produces a large change of direction to rays, a small angular deviation on the active surface could be amplified to a larger angular deviation in the object space. This would then require smaller amplitudes for a given RoM. This solution also includes placing the active surface far from the stop near the image plane instead of near the front lens. The drawbacks of this method are increased aberrations and more complex calculations for the active surface shape.

Because of the non-linear relation between RoM and Z” in Eq. (6) and 7, another limit of the deformable surface is on the achievable precision of the desired shape. A given error on the surface curvature produces a non-linear error on RoM. From Eq. (6), given a relative error Δ(L₀Z”) on the product of Z” and L_o, the relative error ΔRoM is given by Eq. (14). With a numerical example, a relative error of 1% (Δ(L₀Z”) = 0.01) will produce relative RoM errors of respectively 0.10%, 1.01% and 4.17% with RoM of 1.1, 2 and 5, meaning that the larger the RoM is, the more impact on the desired magnification is caused by a given relative error on the surface curvature or the distance L_o. It was not possible to measure it in our experiment because of the limited amplitude that can be measured by the Shack-Hartmann.

As for the limit on the entrance pupil diameter, which controls the beam diameter or the F/# of the imager, it is restrictive too. A closer look at Eq. (13) shows that larger values of D and/or lower values of H and f decrease the requirement on the minimum F/#, meaning that large FFOV lenses could be less limited by aberrations produced by the deformable surface. This is a consequence of the tendency of wide-angle lenses to have very small beam diameters compared to the front lens diameters because of the short focal length. Another solution is working with higher wavelengths so that the λ/4 criterion is less restrictive. However, when working with high F/# and increasing the wavelength as in the infrared, it must be remembered that the diffraction limited spot size scales linearly with λ and the F/# and that at high F/# as in this paper, diffraction already has a high impact on PSF and usable pixel size.

5. Conclusion

The locally magnifying imager presented here opens a new possibility in optical design, the use of real-time distortion to vary as needed the magnification of an image. This paper first presented the concept behind it and then followed with a mathematical description of the imager. This allowed obtaining Eq. (6), describing the achieved magnification change (RoM) in terms of surface amplitude and distance to entrance pupil. From this, it was found that the two major limits of this imager are the required amplitude for the active surface and the required F/# to keep additional aberrations created by the active surface under λ/4. With that in mind, an experimental prototype was assembled and results were presented for two cases, one with a constant magnification and one with magnification varying with field. In both cases, the results presented agreed with the theory.

Future work includes a deeper analysis of cases when the active surface is not the first of the system, better ways to counter the aberration problem and a closer look at some applications in distortion correction [17] and microscopy.