1. Introduction

Extending the depth of focus is an interesting field of research. Various approaches have been proposed to solve this topic. Some by using aperture coding which later on require digital decoding (post-processing) [1-4], some by using aperture apodization [5], some by placing complicated diffractive optical elements [6-8] and others by all-optical means where phase mask is added to the entrance pupil of the imaging lens [9]. The polarization of light can add additional degree of freedom that may be used for the compromise done in the optimization process of the imaging system [10-14]. Recently, a new approach in which a birefringent lens was fabricated [15] and produced two focal lengths (for the ordinary and the extraordinary polarization states). By proper design of the lens the two focal lengths can be chosen such that the focusing range is extended [16]. However the fabrication of such a lens is complicated and expensive.

A birefringent lens provides focusing action due to its shape and high depth of focus due to judicious use of the inherent property of the birefringent material. In the present system, the focusing action is provided by a properly designed ordinary lens system and the high depth of focus is obtained by inserting a birefringent plate [17] in between the imaging lens and the detector- as if a birefringent lens has been conceptually divided into two parts (focusing and enhancement of depth of focus due to birefringence). This direction simplifies the fabrication of the system. The motivation is simple. The object in infinity and the object located near the imaging system are focused at slightly different planes and the detector needs to be shifted along the axis of the system in order to visualize a focused image of both. The near and distant objects are never in focus simultaneously. The parallel plate made of birefringent material that we design is placed between the lens and the detector. In effect, the system will have two focal planes, one for the ordinary beam and the other for the extraordinary beam. If the thickness of the plate is such that the focal plane of the ordinary beam for the distance object overlaps with the focal plane of the extraordinary beam for the near object (or vice versa) and the detector is placed in that particular plane of overlap, both the near as well as the distance object will be in focus simultaneously. Although due to slightly defocused beams of orthogonal polarizations present in the same plane, the overall image quality may not be of very high quality. Nevertheless, the depth of field of the imaging system will increase.

Since in most of the imaging modules U >> V ≈ F where U is the distance between the object and the imaging lens, V the distance between the lens and the detector and F the focal length of the lens. Thus, since the imaging condition is:

when large change in U occurs, very small change in V is required in order to re-focus the image. Due to this assumption the difference in the position of the detector required in order to overlap both ranges is very small and can be only few tens of microns. Therefore, a thin birefringent plate can be used. This plate whose width is properly designed can be positioned anywhere between the detector and the imaging lens such that the refractive index of one of the polarization states (e.g. the ordinary) will create an effective optical path as if the detector is positioned in the position required for the far field (FF) imaging and the other polarization state (e.g. extraordinary) will pass through effective optical path as if the detector is positioned as required for the near field (NF) imaging. Since we overlap the NF and the FF ranges both images (each arriving in different polarization state) will be superimposed one on top of the other.

Note that the NF image is focused for near ranges and well defocused for far ranged and the far field image vice versa, the superimposed result will have extended depth of focus by combining both ranges while the contrast may be reasonably high (above 35% with the overall point spread function taking into account the overall effect of both images) even when no digital processing is applied. Thus, the suggested approach is not only very simple, versatile and cheap but it is also an all-optical concept. Although it includes the generation of imaging artifacts that should be treated digitally. An example for birefringent material that can be used for this purpose is a Calcite or YVO₄ which are simple for fabrication and also have very strong birefringence i.e. large difference in the refractive index between the two principal axes (the two states of polarization).

In section 2 we present some design considerations. Experimental investigation of the proposed approach in real images is performed in section 3. The paper is concluded in section 4.

2. Design considerations

As previously mentioned the insertion of the thin birefringent plate between the imaging lens and the detector can significantly increase the obtained depth of focus. Since the plate is thin it does not change the volume of the imaging system. Let us now compute the required width for such a birefringent plate.

Assuming that the difference in the optical paths in free space is Δ and the birefringent material has ordinary and extraordinary refractive index of n_o and n_e respectively, then the width of the birefringent plate should be:

Note that Eq. 2 is valid for the normal incidence of the incoming beam.

In the following we have examined the proposed approach. The testing was done with Zemax which is industrial software that is most commonly used for lens design and optical analysis. Note that the suggested approach can be combined together with other all-optical approaches for extending the depth of focus as described in Ref. [9] and that way to result with even further extended depth of focus. For the simulation we have used an imaging triplet with effective focal length of F=5mm and F number of 2.8. Using Eq. (2) the required width of the birefringent plate should be as follows: we use YVO₄ plate having approximately n_o=1.99 and n_e=2.22 for the visible range, and thus the optical path difference of 130 microns results with:

As we are about to see an optical path difference of 130 micron is enough to obtain continuous and longitudinally extended range of focus. Thus, a birefringent plate of 1.25mm was inserted into the Zemax simulator. When the birefringent plate is added, the through focus MTF (modulation transfer function) chart for the NF is obtained in one of the polarization states and the FF is obtained in the other state. Both superimposed in the detector plane. This super imposing was seen in the numerical simulations in Zemax.

In Fig. 1(a) we present the Zemax simulations displaying the through focus MTF chart. This chart presents the MTF for spatial frequency of 60 cycles per mm (relatively high frequency but not a cut off of the system) for different axial positions of the object. The horizontal units are in mm (measured in the detector plane). When the object is shifted from position of 25cm away from the camera towards the FF distance, it is equivalent to moving the detector plane. The red MTF corresponds to P-polarization and the pink one to the S-polarization state. Both curves are superimposed in the MTF chart. The movement of the object from 25cm to FF is equivalent to movement of 0.13mm in the detector plane maintaining high imaging contrast of above 0.7. If only one curve (rather than both) would have been used (i.e. without applying the suggested approach) then the movement of the object from 25cm to infinity would have reduced the contrast of the MTF to less than 0.1 (i.e. the spatial frequency of 60 cycles per mm would have become irresolvable). As mentioned before the simulation of Fig. 1(a) was performed using Zemax while the birefringent plate was defined as part of the optical configuration of the system.

Note that since the chart of Fig. 1(a) presents the MTF (absolute value of the spectral transmission) of the ordinary and the extraordinary polarization states passing through the birefringent material it is hard to see the real average response taking into account both polarizations. Therefore, in order to improve the clarity we present the overall point spread function (for both polarizations) at infinity (Fig. 1(b) left) and for 25cm (Fig. 1(b) right). From Fig. 1(b) one may see that the quality of imaging at infinity as well as at 25cm is very good since the overall point spread function has width of approximately one pixel of the detector (about 6 microns). In Fig. 1(c) we present the ordinary polarization when the extraordinary is in focus and vice versa. From the figure it is seen that the defocused polarization can always be neglected since it contributes only very low frequency background.

 

Fig. 1. (a). The through focus MTF chart for ordinary and extraordinary polarization states at 60 cycles per mm for object at infinity and at 25cm with birefringent plate added between the lens and the detector. (b). The overall point spread function of both polarizations obtained at infinity (left) and at 25cm (right). (c). The point spread function of the ordinary polarization when the extraordinary is in focus and vice versa.

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In order to extend the obtained overall extended depth of focus (EDOF) even further, one may add the binary and low spatial frequency phase element described by Ref. [9]. This element allows extending the depth of focus in an all-optical manner as well and it is to be positioned at the entrance pupil of the imaging lens. In addition to that we have added our birefringent plate between the lens and the detector. In this case as seen in Fig. 2 although the contrast is reduced (to 0.4) much larger depth of focus can be obtained and the object even at distance of 15cm may well be resolved (with contrast of above 0.4 at spatial frequency of 60 cycles per mm).

 

Fig. 2. The through focus MTF chart at 60 cycles per mm for object at infinity and at 15cm with birefringent plate added between the lens and the detector.

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3. Experimental investigation

In this section we test the proposed approach on real images. The results of Fig. 3 were obtained numerically by combined usage of Zemax and Matlab softwares. Figures 3(a) and 3(b) present images obtained when the detector is positioned such that the FF and then the NF objects are in focus, respectively. Figure 3(c) presets the all-optical result obtained when a birefringent plate with width of 1.25mm is inserted into the imaging module including a detector of 2M pixels and a lens having focal length of 5mm and F number of 2.8. The close range business card is positioned at 10cm from the camera. One may see that both the close as well as the far field object is well resolved.

Note that the images of Figs. 3(a) and 3(b) were obtained by experimentally capturing two images while shifting the detector. The shift of the detector between the two images was computed following the shift obtained in the through focus MTF charts of the Zemax when a birefringent plate of 1.25mm is added to the system. The image of Fig. 3(c) is obtained by superimposing the two images of 3(a) and 3(b) as anticipated from the numerical Zemax simulation for the through focus MTF charts.

 

Fig. 3. Numerical testing applied on real images. (a). Image captured when far filed objects are in focus. (b). Image captured when a near field object which was a business card positioned 10cm away from the camera was in focus. (c). Image captured when the birefringent plate approach was applied.

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Note that the superimposing assumption of the MTF coming from both polarizations (P and S polarization states) was verified numerically using the Zemax simulations. In addition since the polarizations are orthogonal they are summed as intensities, rather than as fields, in the camera (after the time averaging operation during the detection) and therefore this implied that the OTF (optical transfer function) of each polarization (rather than the coherent transfer function for instance) may be summed. From Fig. 1(c) one may see that the digital superimposing of both images as was done in Fig. 3 is relatively good assumption since the point spread function of the defocused polarization always contributes only very low frequency background level.

In the next step we have fabricated the designed element and placed it into a real imaging module. We have used a Videology camera having VGA resolution of 640 by 480 pixels with a lens of focal length of 8mm and an F number of 2.5. Note that during the fabrication of the birefringent plate that we eventually used for our experiment, it was important to take care that the direction of the crystal’s optical axes in the plate will be parallel to the faces of the plate and that the plate itself will be positioned in a parallel way to the detector plane.

The experimental results can be seen in Fig. 4. In both figs. of 4(a) and 4(b) imaging of close as well as far objects were captured while the suggested birefringent element with width of 1.25mm was inserted in the imaging module for the left side of the images of 4(a) and 4(b) and no element was used, for comparison purposes, in the right side of the images of 4(a) and 4(b). The close object of Fig. 4(a) (business card) was at distance of 15cm while the FF resolution chart was at distance of 120cm. In the experiment of Fig. 4(b) the close object was at distance of 25cm. Figure 4(a) was done in close doors while Fig. 4(b) was captured outside.

Although the obtained results are preliminary, one may see the improvement in the focus quality of the images in the left side (with the birefringent element) provided better focusing for the close range while maintaining similar imaging quality for far field objects (e.g. this is expressed in the capability to read the letters in the business card when the birefringent element is added in).

Note that in order to have fair comparison, two cameras were used to capture the images of Fig. 4. The distance between the lens and the detector was identical in both of them and it was fixed. In one of the systems the birefringent plate was added between the lens and the detector.

 

Fig. 4. Experimental testing of the proposed approach. In both figures of (a) and (b) imaging of close as well as far objects was performed while the suggested birefringent element was inserted in the imaging module for the left side of the images of (a) and (b) and no element was used, for comparison purposes, in the right side of the images of (a) and (b).

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One of the main problems with the suggested approach is that for wide fields of view of the imaging system the rays coming at large angles pass in different direction through the birefringent element and therefore obtain smaller extending of the depth of focus. In addition, a double image effect may occur depending on the direction of arrival of the optical rays. In order to correct those effects we have applied simple Wiener deblurring filter. The digital processing was preliminary and intended only to show that the distortion can be reduced. The obtained results are seen in Fig. 5. In the left part of Fig. 5 we have applied the deblurring processing while for comparison purposes the right image is without the processing. One may see that the double image effects seen around bars (seen as white shadow) were significantly reduced in the left part image.

Note that although the Wiener post processing resembles other digital approaches for EDOF, since the corrections are relatively minor (those are second rather than first order corrections), the filter may be realized with small spatial deblurring kernel which implies lower computational complexity. This is of course in contrast to other deconvolution EDOF approaches requiring higher complexity of post processing.

 

Fig. 5. Numerical deblurring algorithm to reduce the double image effects. In the left part of the image the algorithm was applied. For comparison reasons we present the right image obtained without the numerical processing.

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The images of Figs. 4 and 5 have some chromatic aberrations caused due to the dispersion in the birefringent plate. This problem can be reduced by either using less dispersive birefringent material, or by correcting it with various techniques as those used in lenses (e.g. by addition of a diffractive element). More over digital post processing can reduce the severity of this problem as well.

4. Conclusions

In this paper we have showed theoretically and later on verified numerically and experimentally that a simple and a low cost solution of thin birefringent plate made out of calcite or YVO₄ when inserted between the imaging lens and the detector can allow extensive increase in the depth of focus. The approach was demonstrated for axially continuously-extended depth of focus that significantly reduces the minimal focusing distance of a given imaging system. Such a configuration allows simultaneous imaging of close range business cards as well as having high quality imaging for the far field objects as required in various imaging applications (as in the cameras of the cellular phones).

The proposed solution is basically an all-optical one and does not require high computational load for the processing unit of the camera module although numerical deblurring algorithms may assist in reducing various artifacts generated in the captured images.