1. Introduction

In their 1970 patent [1], Luis Alvarez and William Humphrey invented a pair of conjugate, rotationally asymmetrical, aspheric surfaces such that the lateral movement of these surfaces across the optical axis resulted in an element of variable optical power. This paper presents two designs, which use these basic concepts to extend the capabilities of a miniature, multi-modal microscope (4M) device by providing it with an additional object depth scanning feature. The 4M device is a miniaturized microscope first developed for the detection of pre-cancer [2]. The pre-cancer tissues have a finite depth and experiments have suggested that in order to study the various morphologic and biochemical changes in tissue, object depths up to 200 microns need to be sampled. Thus in order to fully develop the various modalities envisioned by Descour et al. [2] such as optical sectioning, it is necessary to have a 4M device which can image object planes at different depths on to a stationary image plane. The two designs discussed in the paper present different ways of solving the same problem. The first design is a variable power 4M device, which directly applies the Alvarez-Humphrey principle to vary the optical power of the system, thus enabling it to image different object depths on a fixed image plane. The second design is a novel design combining the work of Alvarez and Humphrey [1] with the theoretical work done by Iwona Palusinski et al. [4] on induced aberrations. This design has two pairs of Alvarez-Humphrey surfaces. Once again the lateral displacement of the central element changes optical power. However, this design is a great improvement on the previous design as it has far superior imaging performance and allows more room for the actuator due to the separation of the Alvarez-Humphrey surfaces.

In a traditional zoom lens design, there are at least two moving elements. In a miniaturized optical system with moving elements, the actuators required to move the elements take up a lot of space and hence it is critical to minimize their number. In addition, the exact synchronizing of the movements of the actuators that would be needed for a traditional zoom lens design would also be very difficult to achieve. This created the need for a zoom lens design having only one moving element. This need led to the idea of replacing the last lens of an existing 4M device design [2,3] (as shown in Fig. 1) with a modified Alvarez-Humphrey device, where only the first plate is moved laterally across the optical axis. Until recently manufacturing Alvarez plates was prohibitively expensive, due to the costs associated with manufacturing rotationally asymmetrical aspheric surfaces in glass. However, the very nature of injection molded processes [8] as well as the photolithographic process [5] allow for a cost effective and easy way of making such surfaces. Thus to the best of our knowledge, this paper presents the first design solutions that use the Alvarez-Humphrey concept to solve a current optical design problem.

The first design is a direct application of the concepts laid out by Alvarez and Humphrey. However, it requires the plates to be placed very close to each other and also requires a relatively large (±250µm) lateral displacement of the moving Alvarez plate. The actuators designed by Jeremy Rogers, Tod Christiensen et al. [6] have already been used to move the scanning grating in previous 4M devices and the aim was to modify them and use them to move the Alvarez plate as well. However, realizing this small size and large lateral displacement required by the AAP design would mean a complete re-design of the actuator. In addition, the imaging performance of the AAP design is diffraction limited only on axis and for the central object conjugate. For off-axis field locations and for object conjugates at the far end of the range, the performance is simply not diffraction limited and therefore unacceptable for a 4M device. The second design solves all these problems. It is a unique design involving four Alvarez-Humphrey surfaces. However, as mentioned before, the symmetrical nature of the design and the incorporation of higher order terms to correct induced aberrations, ensure that this design has diffraction limited performance over all the fields and over the entire object range. In addition, the 1.5 mm space on either side of the moving element and the lower lateral displacement range of +/- 135µm allows for the utilization of the actuators designed by Jeremy Rogers, Tod Christiensen et al. [6] with just a few minor changes. Both designs have similar prescription data (i.e., Field Of View, Numerical Aperture, Back Focal Distance, Magnification) as other 4M devices so that they can be interchangeable members of the 4M device family. The paper discusses the optical aspects, advantages and disadvantages of both designs.

2. The 4M device

The optical design for the 4M device has been adequately described in the papers by Descour et al. [2], and Lee et al. [3]. However, since the main purpose of the designs described in this paper is to add an object depth sampling capability to the 4M microscope objective, it would be appropriate, at this stage, to give a brief system level view of the 4M Device. The optical imaging part of the 4M device images the light being reflected off the tissue on to the CCD image plane at a paraxial magnification of -4, as shown in Fig. 1. The imaging lens layout consists of a commercial off the shelf lens, followed by three hybrid lenses. These hybrid lenses are simply plano-convex lenses, whose sag is less than 60µm. The space in between the last lens and the CCD image plane is used for the rest of the non-imaging optics such as the grating and illumination set-ups. The silicon substrate has a length of 13mm and a width of 10mm [2]. The aim of this research was to replace the third sol-gel lens with the Alvarez plates system and actuator. Thus the maximum space available to accommodate the entire Alvarez plate assembly is about 4 to 5 mm along the optical axis and 7–8 mm in the lateral direction (given the 10 mm substrate width). Table 1 contains a list of the basic parameters of the existing 4M device. In the subsequent designs, an effort has been made to keep the basic parameters constant.

 

Fig. 1. Optical Layout of 4M device. The Alvarez setup is expected to replace the third hybrid lens. i.e., the lens shown furthest away from the folding mirror.

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Table 1. Basic optical parameters of the 4 M devices

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3. The adjacent Alvarez plate design (AAP)

Since the new designs were meant to be merely an addition to the already existing family of 4M devices, the basic parameters like the silicon substrate size, the size and location of the off-the-shelf lens and the first two hybrid lenses were left unchanged. This left a small area about 4–5 mm along the optical axis and 7–8 mm across the optical axis in which to place the entire object depth scanning apparatus complete with the actuator for moving the elements. The size constraints along with the previously mentioned problems regarding controlled synchronous movement meant that the zoom-lens setup could have at most one moving element. Furthermore, since there was less space along the optical axis and more space in the lateral direction, lateral movement across the optical axis rather than longitudinal movement along the optical axis would be preferred. This led to the investigation of the Alvarez-Humphrey surfaces [1]. In the 4M device, the optics can be made using either the photolithographic [5] or injection molding [8] technologies. In both these technologies, the main cost is just making the aspheric photo-mask (in the case of the photolithographic technology) or the aspheric mold (in the case of the injection molding technology). This cost when averaged over a large volume of production is negligible. Thus no significant additional costs are incurred in going from a spherical to an aspheric surface. Thus in the AAP design, the third hybrid lens was replaced by the Adjacent Alvarez plate setup.

 

Fig. 2. Schematic of the Alvarez Humphrey Surfaces

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According to the equations formulated by Alvarez and Humphrey¹:

The ‘sag’ or z co-ordinate of the first surface is defined by the equation:

Hence the equation of the conjugate surface will be:

Now if the first surface is displaced by a distance ‘d’ in the ‘x’ direction, the equation of that surface will be:

The composite lens optical thickness will be:

where A,B,C, and D are the coefficients of the ‘(${\mathrm{xy}}^2+\frac{1}{3}{x}^3$)’, ‘x²’, ‘xy’, and ‘x’ terms respectively, while ‘E’ is a term for constant thickness. Z₁ and Z₂ are the sags of the first and second Alvarez plates respectively. Z is the total sag of the conjugate pair, i.e. the total optical depth. ‘x’ is the lateral direction, perpendicular to the optical axis. ‘y’ is the direction perpendicular to the optical axis and to the direction of motion of the Alvarez plates, while ‘d’ is the distance moved by the Alvarez plate in the ‘x’ direction.

In the Alvarez surfaces discussed in the patent [1], the symmetrical movement of the plates causes all terms containing ‘x’ and ‘y’ to cancel out leaving only the (x²+y²) term and some constants. Thus with symmetrical movement of the plates, the two plates together form a variable power spherical lens in which the power is a function of the coefficient ‘A’ and the displacement ‘d’. However, this is not exactly the case with the single moving plate. Here along with the variable power term ‘Ad(x²+y²)’, the terms ‘(Ad²+2Bd)x’ and ‘Cdy’, which are basically variable prisms, are also present. In the AAP design, the coefficients ‘B’ and ‘C’ were initially set to zero. This leaves only ‘Ad²’ as the prism term, and this term is negligible for small values of ‘d’ due to its dependence on the square of the value of ‘d’. However, after obtaining the initial designs, a lens design program was used to optimize the variable prisms to get the design with the best imaging performance across all the fields and over all the object planes.

3.1 Basic calculations for the adjacent Alvarez plate design (AAP)

At this point some of the trade-offs inherent to the Alvarez plate system should be mentioned. The value of the coefficient ‘A’ affects the sag, curvature and range of lateral displacement for the moving Alvarez plate for a given range in object conjugates. Increasing the value of ‘A’ increases the curvature, which in turn increases the sag and causes larger amounts of aberrations. On the other hand, the change in optical power is proportional to the product of ‘A’ and the displacement ‘d’. Thus the main advantage in increasing the value of ‘A’ is that it decreases the amount of displacement needed to achieve the same change in optical power. Meanwhile, the coefficient ‘D’ affects the slope of the prism along the ‘x’ direction of the Alvarez plate and is often used as a compensating term to reduce the sag caused by the curvatures due to ‘A’. Thus choosing the actual Alvarez plate design depends on balancing the maximum displacement available, with the maximum sag allowed and the image quality, which is affected by the higher aberrations. For instance, grayscale photolithographic hybrid glass processing methods such as the sol-gel technique used in previous 4M devices currently allow for lenses to have sags up to 100µm.[2] Meanwhile recent advances in miniature actuator technology also indicate that displacements up to +/-100 µm are currently achievable. [6] In fact, in their work on long throw linear magnetic actuators Fischer and Guckel have demonstrated that even displacements up to +/-250µm were possible [7]. Thus in the AAP design, an attempt has been made to keep within these parameters.

The object plane distance varies by 200µm, starting from 250µm and increasing to 450µm. Thus the mid-point occurs at 350µm. Assuming the first three lenses of the setup to be identical to the previous 4M devices the optical powers required from the Alvarez plate setup for the various object planes was as follows: ϕ₂₅₀=0.153mm^-1, ϕ₃₅₀=0.114mm^-1 and ϕ₄₅₀=0.06mm^-1. Thus the required change in optical power Δϕ=ϕ₂₅₀ - ϕ₄₅₀=0.093mm^-1. Next, various values for the coefficient ‘A’, starting with the value that corresponded to a +/-100µm lateral displacement, were examined. As previously mentioned, when the value of ‘A’ was increased there was a reduction in image quality due to higher amounts of aberrations. After extensive trials, diffraction limited performance was found possible only for values of ‘A’ lower than 0.186 mm^-4. This value for ‘A’ corresponds to a required lateral displacement range of +/- 250µm. Thus a total lateral displacement range of 500µm is required for the Alvarez element.

The two Alvarez plates have convex surfaces on the outside and conjugate Alvarez surfaces on the inside. Initially the radius of curvature for the outside surfaces was set at 10mm each in order to get the basic 0.114mm^-1 power required for the mid-range object. The design was then simulated using the commercially available lens design software, ZEMAX with the Alvarez plates being simulated using the ‘Extra Data Editor’. Once the basic design was completed, the front and back surface curvatures of the Alvarez plates, the displacement ‘d’ and the coefficients ‘A’ and ‘D’ were set as variables and the system was optimized for maximum performance.

At this stage, a word of caution is needed. While ZEMAX is successful in simulating the Alvarez-Humphrey surfaces and tracing rays through them, the author has found, to his surprise, that ZEMAX does not correctly calculate any of the paraxial data such as Paraxial Magnification, Effective Focal Length, F-number, Exit Pupil Location and Diameter and location of principal planes. Thus only ‘real-ray trace’ data such as Spot diagrams (showing location and nature of spots), MTF curves and Working F-number can be trusted. In addition, since the Alvarez-Humphrey surfaces are behind the aperture stop the paraxial quantities such as Object Space NA and Entrance pupil diameter and location can also be trusted. Hence while calculating parameters such as the magnification, the author has used a set of separated point objects and calculated the magnification by seeing the separation in the corresponding point images.

Table 2 lists the basic optical parameters of the AAP design, while Table 3 gives a detailed list of the exact values of all the required dimensions. The specifications for the Alvarez-Humphrey surfaces have been included in Table 4. Currently for ease of manufacture and to illustrate the basic concept, no other coefficients were used. However, further improvement can be achieved by setting other coefficients, which correspond to prism surfaces and astigmatic surfaces, as variables and optimizing for best performance over the entire object.

Table 2. Basic optical parameters for adjacent Alvarez plate design.

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Table 2 lists the basic optical parameters of the AAP design, while Table 3 gives a detailed list of the exact values of all the required dimensions. The specifications for the Alvarez-Humphrey surfaces have been included in Table 4. Currently for ease of manufacture and to illustrate the basic concept, no other coefficients were used. However, further improvement can be achieved by setting other coefficients, which correspond to variable prisms and variable astigmatic surfaces, as variables and optimizing for best performance over the entire object.

Table 3. Specifications for the AAP design.

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Table 4. Specifications for the Alvarez Humphrey surfaces in the AAP design. Note: The signs of the coefficients do not have to be changed since the conventions in Zemax does that automatically

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3.2. Performance of the AAP

All the subsequent graphs and calculations are at the image plane. The design is at its best in the middle of the object range, i.e. near 350µm. The layout of the system, as shown in Fig. 3, corresponds to this object distance. At these conjugates the system spot size on the image plane is smaller than the Airy disk diameter on axis and increases slightly for the outer field positions as shown in Fig. 4. Figure 5 shows the MTF performance of the system at the image plane for this mid-range object conjugate. In fact, the imaging performance of the system remains remarkably unchanged over most of the object range between 250µm and 400µm. Beyond this range, i.e. from 400µm to 450µm, the performance begins to degrade very rapidly, especially at the outer field positions. This is clearly shown in Fig.6., which plots the RMS spot size versus field position for object planes at 350µm and 450µm. Finally, the lateral displacement of the moving Alvarez plate is linearly proportional to the change in object depth for the central region from about 300µm to 400µm. At object depths less than 300µm and greater than 400µm, there is a slight non linearity in the curve.

 

Fig. 3. Optical Layout of the Adjacent Alvarez Plate Design

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Fig. 4. Spot sizes for various fields. Obj. Dist=350µm

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Fig. 5. MTF curves for various fields. Obj. dist.=350µm

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Fig. 6. Rms spot radius v/s field position for different object locations.

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3.3 Object resolution

For the AAP design, the diffraction limited resolution, according to the Rayleigh Criterion (i.e., resolution=0.61λ/N.A.), varies between 1.15µm and 1.37µm as the object space NA varies between 0.379 to 0.444 for the different object planes. The MTF curves shown in Fig. 5 indicate that the performance of this system is close to being diffraction limited for the fields closer to the optical axis. For the field further away from the axis, the MTF curves are much lower than the diffraction limit. This is one of the drawbacks of the AAP design.

Thus from an imaging and spatial frequency point of view, the 4M device with the Alvarez plates works to practically diffraction limited specifications on axis over object depths ranging from 250µm to 400µm. However, for outer field positions and for object plane locations beyond 400µm, the imaging performance is not diffraction limited.

4. The separated Alvarez plate design (SAP)

The major drawback of the previous system is the stringent demands it places on the design of the actuator in terms of size and lateral range. Jeremy Rogers, Tod Christiensen et al. [6] had developed an actuator for the lateral displacement of a scanning grating used in previous 4M devices and the attempt was to use a similar actuator design to move the Alvarez-Humphrey element. However, these actuators had a lateral range of ± 100 µm. In addition, they held the moving element symmetrically in the center and extended to about 1.5mm on either side of it. In the AAP design, the lateral range of the actuator would have to be ± 250µm. In addition, the actuator would have to hold the moving element to the side since the distance between the conjugate Alvarez plates is only 220 µm. Thus the AAP design would require a complete re-designing of the actuator. Apart from the difficulties in actuator design, another concern regarding the AAP design was that the imaging performance was just not as good as previous 4M devices. Hence the need of the hour was to separate the moving element from the stationary Alvarez plate, to reduce the lateral displacement to more manageable proportions and to improve the imaging performance of the system. However, as will be explained in the subsequent section, separating the Alvarez-Humphrey surfaces was not a trivial matter and required a complete re-designing of the entire system from scratch. This new approach led to the Separated Alvarez Plate (SAP) design, which is a completely novel way of applying the Alvarez-Humphrey method.

4.1 Problems associated with separating the Alvarez plates

The mathematical derivation of the working of the Alvarez plates in Alvarez and Humphrey’s patent [1] assumes that the plates are very close to each other. Hence it assumes that the coordinates of the point at which the ray of light exits the first Alvarez surface and the coordinates of the point at which it intersects the second Alvarez surface are the same. However, if the plates are separated then this is not the case and there can be a significant difference between the co-ordinates. This difference causes induced aberrations, which are dependent on the displacement of the Alvarez plate. The theory governing the induced aberrations was first developed by Iwona Palunsinski et al. in their paper on lateral shift variable aberration generators [4]. In their publication, Palusinski et al. outline the method of calculating the induced aberrations and illustrate it by calculating the induced coma due to the prism terms in the Alvarez plates. For the SAP design, the induced aberrations due to the ‘A(${\mathrm{xy}}^2+\frac{1}{3}{x}^3$)’ term, which is the crucial term causing the variable optical power, were calculated.

The basic formulae mentioned below are based on the work done by Palusinski et al. [4]. The induced aberrations due to the ‘A${\mathrm{xy}}^2+\frac{1}{3}{x}^3$)’ term are calculated in the following manner:

The sag of the two surfaces due to this term is given by :

Assuming that the first plate has been displaced by a distance ‘d’, the net wavefront ‘W(x,y)’ of the wave after it has passed through both Alvarez plates with refractive index ‘n’ is given by:

Since the plates are not close to each other ‘x’ and ‘y’ are not equal to ‘χ’ and ‘γ’. If the distance between the Alvarez plates is assumed to be ‘S’, then

After carrying out the above calculations, all the terms having the displacement term ‘d’ in the second or higher order were set to zero as they were negligible due to the small value of ‘d’. In spite of this there were still a large number of terms left. These terms contained values of ‘x’ and ‘y’ up to the eighth order.

Thus it was clear that the resulting induced aberrations were extremely complex involving powers up to the eighth order. According to Palunsinski et al. if the Alvarez plates have terms corresponding to the powers of the induced aberrations, then they can be optimized to cancel out these induced aberrations. Hence the basic forms of these terms were isolated and then set up as variables and optimized. However, a large number of these terms were linearly dependent on the lateral displacement of the Alvarez plate. Thus, while ZEMAX was able to optimize and cancel the induced aberrations to achieve excellent performance for a given object location, it was unable to do so over the entire range. Every time the Alvarez Plate was moved, the amount of the induced aberration would change and hence the aberration correction scheme would fail. Hence due to the dependence of the induced aberrations on the displacement of the Alvarez plate, it proved impossible to exactly apply, as is, the theoretical work done by Palunsinski et al to this particular design. Thus separating the Alvarez plates would require an entirely new approach.

4.2 The separated Alvarez plate design

As mentioned in the above paragraph, while calculating the induced aberrations, all terms which had higher orders of the displacement term ‘d’ were considered negligible. Hence the terms left were those that did not depend on the displacement and those that had a linear dependence on the displacement, i.e. terms that had d¹ in them. Now for two sets of Alvarez plates, which are similar to each other, the terms linearly dependent on displacement would cancel out if the plates moved equally in opposite directions. This led to the idea of having four Alvarez surfaces. Thus the four surfaces would basically form two conjugate Alvarez- Humphrey pairs. When the central element is moved down, from the point of view of the first Alvarez pair the rear surface is moving down, while from the point of view of the second Alvarez pair the front surface is moving down. Moving the front surface down has the same effect as moving the rear surface up. Thus by moving the central element, the displacement ‘d’ is equal and opposite for the two pairs of Alvarez plates.

So when the middle Alvarez element is laterally displaced the two Alvarez pairs behave in exactly opposite fashion resulting in a complete canceling out of the displacement dependent induced aberrations. Now having two completely identical pairs of conjugate Alvarez-Humphrey surfaces would not serve any purpose, as along with the canceling out of the induced aberrations, the optical power would cancel out as well. Thus the value of the coefficient ‘A’ for the two pairs is slightly different so that while the induced aberrations are reduced (if not completely cancelled out), there is still a net change in optical power, to allow the different object planes to be imaged on to the same image plane. However, the displacement independent induced aberrations are still present. But, since these induced aberrations do not change with displacement they can be easily corrected by using the higher order terms mentioned by Palusinski et al [4]. The resulting layout is a very well corrected system in which the optical power changes with the lateral displacement of the Alvarez surface, with minimum incidence of induced aberrations. In effect, the SAP design could be thought of as a very well corrected variable lens. The SAP design is implemented by replacing the rear surface of the second hybrid lens by the first Alvarez surface. This is followed by the central element that has an Alvarez-Humphrey surface on both sides. This central element is followed by the last element, which has an Alvarez surface in the front and a convex surface at the back.

Care has been taken to allow for 1.5 mm space on either side of the moving Alvarez element to allow for the actuator. In case of the previous AAP design, the value of the coefficient ‘A’ could not increased beyond a certain limit due to the higher order aberrations that resulted. This led to the limiting of the value for ‘A’, which is turn led to an increase in the lateral displacement range of the actuator. In the case of the SAP design, some of these higher order aberrations cancel out and hence we can afford to have higher values for ‘A’ and thereby reduce the lateral displacement range needed to just ±135 µm.

However, the SAP design differs from the previous design in that it changes the entire existing 4M device design. The basic size of the silicon substrate, the non-imaging optics and the initial Edmund Scientific Lens are the same. However, all the subsequent plano-convex sol-gel lenses have been modified. This design also makes use of the recent advances in sol-gel techniques that allow for lenses to have curvatures on both sides of the glass substrate. Thus all four hybrid elements in the SAP design have curvatures on both sides. Table 5 lists the basic optical parameters of the SAP design. Table 6 and Table 7 list the exact values of all the specifications of the SAP design. Table 8 lists the higher order coefficients needed to correct the induced aberrations. Once again the author would like to caution the reader that ZEMAX is, for some reason, unable to calculate the paraxial quantities involving the Alvarez-Humphrey surfaces properly and hence only ‘real ray’ trace data has been used and paraxial quantities states such as paraxial magnification have been inferred from the real ray trace data.

 

Fig. 7. Optical layout of the Separated Alvarez Plate Design

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Table 5. Basic Optical Parameters for the SAP design

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Table 6. Specifications for the SAP design

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Table 7. pecifications for the Alvarez-Humphrey surfaces in the SAP design

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Note: The signs of the coefficients do not have to be changed since the conventions in Zemax do that automatically. The above coefficients are the commonly used ones. The performance of a system with these surfaces is almost as good as the AAP design. However, so far no attempt has been made to correct the induced aberrations. Correction of these aberrations involves the inclusion of higher order terms for the third surface. These higher order terms allow for that further increase in performance that leads to diffraction limited performance across all the fields and for all the object plane locations. As mentioned before, induced aberrations containing terms up to the eighth order are present. However, only the terms up to the fourth order were required to achieve diffraction limited performance. (*)→ Surface 3 is only partially defined here and has higher order terms which have been listed in table 8.

However, the SAP design differs from the previous design in that it changes the entire existing 4M device design, as shown in Fig. 7. The basic size of the silicon substrate, the non-imaging optics and the initial Edmund Scientific Lens are the same. However, all the subsequent plano-convex sol-gel lenses have been modified. This design also makes use of the recent advances in sol-gel techniques that allow for lenses to have curvatures on both sides of the glass substrate. Thus all four hybrid elements in the SAP design have curvatures on both sides.

Table 8. Higher order coefficients for the third A-H surface required for correction of Induced Aberrations

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4.3 Performance of the separated Alvarez plate design (SAP)

Once again all the performance parameters stated are at the image plane. The SAP design has much better aberration correction and far superior imaging performance than the AAP design at the same field points. The spot diagram shown in Fig. 8 corresponds to an object depth of 350µm. The spots shown are clearly smaller than those in Fig. 4, which correspond to the AAP design at the same object depth. Fig. 10 plots the RMS spot size versus the field positions for object depths of 350µm and 450µm. A quick comparison with Fig. 6 clearly shows that the SAP design has much better performance across different field positions than the AAP design. Figure 10 also indicates that the performance of the SAP design remains very good across the entire object range and does not degrade for objects distances greater than 400µm like the AAP design. The MTF characteristics for this system are also diffraction limited for the entire range, as shown in Fig. 9. Like in the previous case, the relationship between the object distance and the lateral displacement of the Alvarez plate is linear for the central part of the range, but develops slight non-linearities for object depths below 300µm and above 400µm

 

Fig. 8. Spot diagrams for all field positions(obj,dist=350µm)

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Fig. 9. MTF curves for object distance=350µm

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Fig. 10. Rms spot size v/s field position for different object distances.

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5. Comparison of the advantages and drawbacks of both designs

The chief differences between the two designs are as follows:

(i) In the SAP design, the moving element is separated by 1.5 mm from the elements on either side of it. Hence there is plenty of space for the actuator. However, in the AAP design, the separation between the glass substrates of the Alvarez plates is only 220µm. The most suitable micro-actuators found during research (those discussed by Jeremy Rogers et al in their paper [6]) held the moving element symmetrically in the middle and extended up to 1.5mm on either side. Thus a custom made actuator for the SAP design can be easily implemented with present actuator technology, while making an actuator for the AAP design would present a greater degree of difficulty.

(ii) The lateral displacement range required for the moving element in the SAP design is only ±135µm, while that required for the AAP design is ±250µm. Once again, the smaller range means that the requirements on the actuator for the SAP design are far less stringent.

(iii) The imaging performance for the SAP design is as good as the previous 4M devices [2,3]. It is diffraction limited across the entire field and maintains this high performance over the entire object range. On the other hand, the AAP design is diffraction limited only on axis and not for the outer fields. Secondly, the performance of the AAP design also drops off very quickly for objects depths ranging from 400 µm to 450 µm. Thus the imaging performance of the SAP design is clearly superior is all respects to the AAP design, when compared at identical field points.

(iv) However, there are some areas where the AAP design scores over the SAP design. For instance, in the SAP design the image is laterally displaced across the CCD image plane by around 280µm. In the AAP design, the lateral displacement is only 184µm. Thus in the case of the SAP design there will be a greater movement or ‘jitter’ of the image.

(v) Another drawback for the SAP design is that the Field of View of this system is reduced to 200µm from the 250µm FOV for the rest of the 4M device family, including the AAP design. Secondly the vignetting is also higher, especially for the outer fields. In fact, for some fields the vignetting can be as high as 5%. However, the reduction in the Field of View and the increase in vignetting have nothing to do with nature or working of the Alvarez plates in the SAP design. It is simply due to the increase in the overall length of the system due to the space left on each side of the moving element. In addition, a vigentting of 5% is a very minor drawback and the resulting drop in MTF performance and illumination is not discernible.

(vi) The final and really only serious drawback in the SAP design is the increased sag required for these systems. This increased sag is not really due to increased curvatures but instead due to increased prism terms. Hence it should not really increase the difficulty associated with the manufacturing process. Currently the plates in the SAP design require sol-gel depths of up to 170µm as against the 100µm depth required by the AAP design.

6. Conclusions

The primary aim of this thesis was to present a design that applied the concept of the Alvarez-Humphrey surfaces to a miniature modern optical device. In that respect, the Adjacent Alvarez plate design is a more direct application of the Alvarez-Humphrey concept. It is able to achieve the required change in optical power through the movement of only one element and is therefore much more feasible than the traditional zoom lens designs that require the synchronized movements of two or more elements. However, while this design does work in principle it would be difficult to fabricate since it requires an actuator with a higher range and smaller size than currently available. The imaging performance of the AAP design is also not up to the diffraction limited standard of the other 4M devices.

The separated Alvarez plate design corrects all these problems. This design consists of four separated Alvarez surfaces. The lateral displacement of the central element changes the optical power of the system. However, due to the symmetry of the system and the incorporation of the theoretical work done by Palusinski et al. [4] concerning the use of higher order terms to correct for the induced aberrations, this system is able to do a much better job of aberration correction. Thus this design gives superior imaging performance than the AAP design. In fact, this design is diffraction limited over the entire object conjugates and over the entire field. In addition, there is enough space for the actuator and the requirement on the lateral range of the actuator is also almost 50% less than for the AAP design. However, with the advantages there are also some inevitable drawbacks for the SAP design. These drawbacks, as mentioned in the previous section, include a slightly larger ‘jitter’ of the image, a 20% reduced FOV, 5% vignetting and most importantly an increased requirement on the sag from 100µm to about 170µm

However, before attempting to fabricate the above designs, it would be necessary to tolerance the aspheric surfaces and develop a method to test them. The tolerancing procedure will depend on the method of fabrication. For the photolithographic process using sol-gel, a tolerancing procedure relating surface height departure of the aspheric to imaging performance will have to be developed. The testing of these surfaces will also present significant challenges. The Shack-Hartman sensor has been used before for the measurement of such surfaces [9]. However, once tested and toleranced the Separated Alvarez Plate system will be a unique method of adding an object depth scanning ability to a miniature microscope.