Abstract

In our development of multiscale, gigapixel camera architectures, there is a need for an accurate three-dimensional position alignment of large monocentric lenses relative to hemispherical dome structures. In this work we describe a method for estimating the position of the objective lens in our AWARE-10 four-gigapixel camera using the retro-reflected signal of a custom-designed auto-stigmatic microscope. We show that although the physical constraints of the system limit the numerical aperture of the microscope probe beam to around 0.016, which results in poor sensitivity in the axial direction, the lateral sensitivity is more than sufficient to verify that the position of the objective is within optical tolerances.

© 2013 OSA

1. Introduction

The AWARE project consists of a family of gigapixel scale cameras based on a multiscale optical design [1, 2]. Multiscale design utilizes optics of different size scales to obtain both high resolution and a large field-of-view (FOV), while keeping the optical design simple and scalable. This concept is illustrated in Fig. 1. A large objective lens collects the incoming field points to a curved intermediate image surface, which is further corrected and relayed onto the imaging sensor by micro-camera optics [Fig. 1(a)]. If the objective lens is monocentric [3], the number of resolvable pixels can be increased by adding more micro-cameras and scaling the objective [Fig. 1(b)]. This is the basis of the designs for the AWARE cameras, including the first prototype demonstrated by Brady et al. [4].

 

Fig. 1 Illustration of multiscale optical design. (a) One channel of a multiscale camera with objective lens and one micro-camera. (b) Array of identical micro-cameras covering a larger field.

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The use of multiscale design imposes a unique optical alignment challenge, where the objective lens must be accurately aligned along several optical axes corresponding to each micro-camera in order to maintain consistent performance across the field. AWARE cameras utilize an aluminum dome fabricated with a five-axis milling process as a support structure with counter-bore holes to hold and align each micro-camera [5]. The features in the dome can be machined with high precision; however, as with any machining operation, fabrication tolerances result in finite discrepancies between actual and nominal dimensions. In the case of AWARE-10, a camera with a resolution of four gigapixels currently under construction, the holes were machined with a pointing tolerance of 0.05° and decenter of 0.025mm. Figure 2 illustrates the design of the AWARE-10 dome and its assembly to the objective lens. The maximum cumulative error arising from the machining tolerances of the mechanical components comprising the dome, cover plate, and objective lens flange places the objective lens to within 100µm of the dome’s nominal center. We need an adequate measurement method to verify the location of the objective lens with respect to the dome structure within this limit. Ideally if all the axes of the holes intersect at the center of the dome, only two measurements would be necessary to triangulate the position of the objective. However, given the uncertainties in the location and pointing direction of each hole due to machining tolerances, a much larger number of measurements must be taken in an attempt to average out these deviations.

 

Fig. 2 Solid model of basic mechanical components of AWARE-10. (a) Assembly of dome and objective lens. (b) Cross-sectional view showing relevant dimensions. Note inset showing the lip where the micro-camera sits.

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A number of effective tools for measuring the position of the center of curvature (COC) of an optical element in Cartesian space exist, such as the Point Source Microscope (PSM) [6], measuring deviations in total indicated runout obtained through rotation of the optical element [7], and alignment telescopes. However, the physical constraints imposed by the design of AWARE-10 prohibit the direct application of most commercially available metrology tools. Figure 2(b) shows the cross sectional view of the dome assembly, where the relatively long distance from the outside of the dome to the COC of the objective and the small diameter of the counter-bore micro-camera holes prevent the use of a higher numerical aperture (NA) tool like the PSM. Furthermore, the lack of a single optical axis prohibits the use of most lens centering tools. We describe the design and construction of a low NA auto-stigmatic microscope (ASM) [8] used to triangulate the position of the objective lens.

2. Instrument design

The basic schematic for the ASM is shown in Fig. 3(a). A laser diode operating at 658nm is used as a point source and focused by a lens with focal length f to the COC of the objective lens. This probe beam reflects off the surface of the objective, passes a second time through the lens, and is imaged onto an electronic sensor. Lateral translations of the objective lens (Δy) result in an angular displacement (α) of the retro-reflected signal as shown in Fig. 3(b).

 

Fig. 3 Diagram of ASM operation. (a) Propagation of beam at nominal objective lens position. (b) Shift of return signal when objective is laterally misaligned.

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When tracing the chief ray of the reflected ASM probe beam, small lateral shifts of the objective lens result in angular displacements that are essentially linearly related to the lateral displacement, according to

α2ΔyR.
By evaluating the retro-reflected signal using a paraxial matrix analysis [9], the displacement of the focused signal on the sensor (ys) as a function of Δy can be expressed as
ys=2ΔyR[d0(1d1Rf)+d1R],
where d0 is a function of d1 and f as defined by the Gaussian Lens Formula. Applying the physical constraints of the AWARE-10 mechanical design results in values of d1 = 206mm and R = 64.8mm. Measurement sensitivity (S) is defined to be the ratio between the displacement of the objective lens from the probe beam axis and the shift in the signal beam at the sensor surface and is given by

S=|ysΔy|.

The AWARE-10 optical system was designed to provide high quality images when the spherical objective and the micro-cameras are aligned to within 100µm of design specifications. Acceptable images are obtained at up to 200µm alignment errors. Since the pixel size of the ASM sensor that dictates the position accuracy of the displaced beam is much smaller than that (see below), a lens with focal length of f ≈70mm will provide an S value of ~1 in the optical system to achieve sufficient accuracy for adequate alignment of the objective lens.

To reduce cost and fabrication time, the use of off-the-shelf (OTS) components to build the ASM is highly desirable. Figure 4 shows a CAD model with all system components and a photograph of the as built system. The 50/50 beam splitter, iris, 1-inch diameter (SM1) lens tubes, XY translation mount, and Z translation mounts are OTS parts available from ThorLabs (Newton, NJ). The sensor is a kit camera from Lumenera (Ottawa, Ontario) with a 5.2µm pixel pitch. The dome adapter is the only part that requires custom machining. A 70mm achromatic doublet was not a standard part, so a 75mm lens was chosen instead. This has the impact of slightly increasing d0 to d0 = 118mm in the matrix analysis, which increases the sensitivity to S = 1.14.

 

Fig. 4 Components of the ASM. (a) CAD model of ASM. Note the lens tube holding the lens has been cross-sectioned for viewing convenience. (b) Photograph of as built ASM.

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Zemax (Radiant Zemax, Redmond, WA), a commercial ray tracing software package, was used to simulate the complete system and confirm the validity of the paraxial assumption before final assembly. The beam was stopped down to 6.5mm at the lens to account for the aperture introduced by the dome adapter. This results in an NA of 0.016 for the probe beam. Figure 5(a) shows the results of this simulation at Δy = 0. As shown in Fig. 5(a), when there is no misalignment of the objective lens, the incident probe beam wavefront matches that of the curvature of the objective lens so that the rays reflect and follow the same path back to the ASM. Figure 5(b) shows the spot diagrams at various objective displacements. As shown by the spot diagrams, the aberrations are controlled to well within the diffraction limited spot diameter of 28µm. Figure 5(c) shows a plot of ys versus Δy for this Zemax simulation. The results show that in the regime of small objective displacements, ys does indeed behave as a linear function of Δy with a sensitivity value of S = 1.13. This agrees well with the paraxial chief ray model derived above since all the rays in this system are well within the paraxial approximation.

 

Fig. 5 ASM Zemax simulation results. (a) Ray trace at nominal objective position. (b) Spot diagrams of return signal at various lateral shifts of objective. (c) Linear fit of signal shift (ys) versus objective shift (Δy).

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Change in sensitivity of the system due to axial shifts can be calculated by adding a slight perturbation to either d0 or d1 in Eq. (2). An axial misalignment of 400µm to either value results in a change in sensitivity of less than one percent and can effectively be ignored. This result was confirmed with Zemax simulations. As predicted, this system is well suited for lateral displacement measurements but is relatively insensitive to axial misalignments.

3. Measurement method

As previously mentioned, the low NA of the ASM prevents accurate measurement of the COC along the axis of the ASM. This means when the ASM is inserted into a hole in the dome [Fig. 6(a)] and a measurement is taken, the COC of the objective is determined to lie somewhere along a line parallel to the axis of the ASM but offset by the amount measured. Ideally, the axis of the ASM and the axis of the hole into which the ASM is inserted should be coincident and each line should represent the pointing direction of the corresponding hole.

 

Fig. 6 COC measurement procedure. (a) Illustration of insertion of ASM into dome for measurement. (b) Diagram of vector description of a pair of skew lines.

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Since in reality each machined hole has some deviation from nominal, these skew lines are unlikely to intersect and a simple triangulation method is insufficient. Instead, several measurements were taken from various angles and an estimation method utilizing a simple statistical analysis of the proximal points of the resulting skew lines was used to determine the COC position.

In this approach, each skew line is described by two vectors, a position vector (a and c) and a direction vector (b and d), as shown in Fig. 6(b). The closest, proximal points between two skew lines are connected by a vector n perpendicular to both lines. The relationship between these vectors are given by

a+sb+pn=c+td.
The positions of these proximal points can be determined by solving for s, t, and p in Eq. (4). To estimate the COC position, the proximal points for every possible pair of skew lines were calculated and the mean value of all these points can be taken as an estimate for the COC.

4. Calibration

Three calibration steps are required to properly setup the ASM. The first step is to align the focus of the probe beam to lie on the axis of the dome adapter. This was achieved by spinning the ASM on a precision v-block and adjusting the position of the laser diode until the focused spot on the sensor placed at the nominal COC position did not move with rotation [Fig. 7(a)]. A plastic clamp was used to press the ASM barrel against the v-block so that the ASM could spin with minimal resistance. The sensor arm was removed to make this rotation easier.

 

Fig. 7 Photographs of first two calibration steps. (a) Procedure for aligning ASM probe beam to the axis of the dome adapter. (b) Procedure for finding the nominal zero position of return signal.

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The second calibration step is illustrated in Fig. 7(b), where the return spot is calibrated by attaching the sensor arm and placing a diffuse surface at the COC location. The centroid of the returning image on the sensor is measured by taking the center of mass of the intensity and set as the nominal zero position of the objective lens.

The third calibration step is illustrated in Fig. 8(a). The ASM is placed on a three-axis translation stage and positioned relative to the objective lens such that the return spot is at zero. Then, the ASM was translated in 10µm steps and the centroid position of the return spot was measured for up to 400µm. A sample measurement at ys = 40µm is shown in Fig. 8(b). The results of this calibration step are shown in Fig. 8(c). The measured sensitivity S is around 1.12, which is in good agreement with the Zemax model. The slight difference in sensitivity from the Zemax model can easily be explained by differences in d0 and d1 due to mechanical mounting. Exactly matching the predicted sensitivity is not important as long as the system is calibrated and the actual S is known. The residuals between the data and the linear fit are 3µm or less, meaning this simple centroid locating algorithm can detect sub-pixel shifts in the return spot.

 

Fig. 8 Final calibration step. (a) Procedure for measuring ASM sensitivity. (b) Sample image of returned spot signal at Δy = 40µm. The green circle indicates the calibrated center position from step 2 and the red ‘X’ indicates the calculated centroid of the spot. (c) Linear fit of measured signal shift (ys) versus objective shift (Δy).

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5. Results

ASM measurements were taken from 13 holes as indicated in Fig. 9(a). Also defined in Fig. 9(a) are the coordinate axes used in this analysis. A pair of symmetric return spots appeared at oblique angles (i.e. non-zero angles relative to the Z-axis). The larger the angle, the greater the separation distance between the two spots became. An example of this separation is shown in Fig. 9(b). This separation can be explained by non-concentricity between the surfaces of the objective lens. Nominally, the reflected signals from the front and back surfaces of the objective lens should be coincident on the sensor. Factors such as epoxy between glass layers and grinding tolerances can account for this non-concentricity. This very slight non-concentricity was accounted for by taking the position halfway between the two spots to be the effective COC position.

 

Fig. 9 Measurement procedure. (a) Points on dome where measurements were taken highlighted in red. (b) Splitting of signal at oblique angles indicating non-concentric objective lens.

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The results of the proximal point analysis are shown in Fig. 10. Two data sets were taken: one where the ASM was pressed flat against the seat inside each hole as stably as possible (instantaneous measurement) and one where the ASM was intentionally randomly shifted around laterally over several exposures to get an effective center (average measurement). The average measurement method was used to account for and minimize the effects of the inherent play that exists between the hole and the ASM barrel. By taking several images while moving the ASM the extent of the play between the two components can be measured. These images are then combined using a maximum comparison method where the image intensities are compared and only the highest value is retained per pixel. This is more accurate than using an average (or a simple sum) of the image frames because it does not bias the centroid algorithm toward a side where more measurements happened to be taken. For the results in Fig. 10(b) twenty images were taken per measurement location and combined using the maximum comparison method. The centroid of this resultant image was calculated and used as the effective position for the skew line analysis.

 

Fig. 10 Results of measurements. (a) Results from instantaneous measurement method. Color of data points indicate distance from the COC estimate and line lengths in the principal directions indicate standard deviation. (b) Results from average measurement method. Note positive Z direction is towards the dome [see Fig. 9(a) for definition of coordinates.]

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As shown, the COC estimates from the two data sets are in good agreement. The objective lens position is within the 100µm range set by the machining tolerance budget. The standard deviation of the data is a measure of the amount of random deviations in the position and pointing of the ASM due to machining tolerances and improper seating. As expected, the standard deviation of the average measurement is lower than that of the instantaneous measurement since seating errors were averaged out. The spread of the proximal points is larger in the Z direction. This too is expected since measurements were only taken on one side of the Z-axis, similar to how axial resolution is typically lower than lateral resolution in traditional stigmatic imaging systems.

6. Conclusion

We have demonstrated a method for using an ASM built from OTS components to estimate the position of a monocentric ball lens with respect to a machined dome. Our method takes into account randomly distributed machining errors and gives an indication of their range. Although this system was specifically designed for use in the AWARE-10 camera, the design is flexible enough for use in other AWARE systems or applications where a sphere must be aligned to an inconveniently shaped component.

Acknowledgments

We would like to thank Dr. Kai Hudek for his valuable input on the estimation problem and the Defense Advanced Research Projects Agency (DARPA) for financial support via award HR-0011-10-C-0073.

References and links

1. D. J. Brady and N. Hagen, “Multiscale lens design,” Opt. Express 17(13), 10659–10674 (2009). [CrossRef]   [PubMed]  

2. E. J. Tremblay, D. L. Marks, D. J. Brady, and J. E. Ford, “Design and scaling of monocentric multiscale imagers,” Appl. Opt. 51(20), 4691–4702 (2012). [CrossRef]   [PubMed]  

3. D. L. Marks and D. J. Brady, “Gigagon: a monocentric lens design imaging 40 gigapixels,” in Imaging Systems (IS), (Optical Society of America, 2010), paper ITuC2.

4. D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012). [CrossRef]   [PubMed]  

5. H. S. Son, A. Johnson, R. A. Stack, J. M. Shaw, P. McLaughlin, D. L. Marks, D. J. Brady, and J. Kim, “Optomechanical design of multiscale gigapixel digital camera,” Appl. Opt. 52(8), 1541–1549 (2013). [CrossRef]   [PubMed]  

6. R. E. Parks and W. P. Kuhn, “Optical alignment using the Point Source Microscope,” Proc. SPIE 5877, 58770B, 58770B-15 (2005). [CrossRef]  

7. P. R. Yoder, Mounting Optics in Optical Instruments, 2nd ed. (SPIE, 2008), Chap. 12.

8. W. H. Steel, “The autostigmatic microscope,” Opt. Lasers Eng. 4(4), 217–227 (1983). [CrossRef]  

9. E. Hecht, Optics, 4th ed. (Addison-Wesley, Reading, Mass., 2002), Chap. 6.2.1.

References

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  1. D. J. Brady and N. Hagen, “Multiscale lens design,” Opt. Express 17(13), 10659–10674 (2009).
    [Crossref] [PubMed]
  2. E. J. Tremblay, D. L. Marks, D. J. Brady, and J. E. Ford, “Design and scaling of monocentric multiscale imagers,” Appl. Opt. 51(20), 4691–4702 (2012).
    [Crossref] [PubMed]
  3. D. L. Marks and D. J. Brady, “Gigagon: a monocentric lens design imaging 40 gigapixels,” in Imaging Systems (IS), (Optical Society of America, 2010), paper ITuC2.
  4. D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
    [Crossref] [PubMed]
  5. H. S. Son, A. Johnson, R. A. Stack, J. M. Shaw, P. McLaughlin, D. L. Marks, D. J. Brady, and J. Kim, “Optomechanical design of multiscale gigapixel digital camera,” Appl. Opt. 52(8), 1541–1549 (2013).
    [Crossref] [PubMed]
  6. R. E. Parks and W. P. Kuhn, “Optical alignment using the Point Source Microscope,” Proc. SPIE 5877, 58770B, 58770B-15 (2005).
    [Crossref]
  7. P. R. Yoder, Mounting Optics in Optical Instruments, 2nd ed. (SPIE, 2008), Chap. 12.
  8. W. H. Steel, “The autostigmatic microscope,” Opt. Lasers Eng. 4(4), 217–227 (1983).
    [Crossref]
  9. E. Hecht, Optics, 4th ed. (Addison-Wesley, Reading, Mass., 2002), Chap. 6.2.1.

2013 (1)

2012 (2)

E. J. Tremblay, D. L. Marks, D. J. Brady, and J. E. Ford, “Design and scaling of monocentric multiscale imagers,” Appl. Opt. 51(20), 4691–4702 (2012).
[Crossref] [PubMed]

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref] [PubMed]

2009 (1)

2005 (1)

R. E. Parks and W. P. Kuhn, “Optical alignment using the Point Source Microscope,” Proc. SPIE 5877, 58770B, 58770B-15 (2005).
[Crossref]

1983 (1)

W. H. Steel, “The autostigmatic microscope,” Opt. Lasers Eng. 4(4), 217–227 (1983).
[Crossref]

Brady, D. J.

Feller, S. D.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref] [PubMed]

Ford, J. E.

Gehm, M. E.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref] [PubMed]

Golish, D. R.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref] [PubMed]

Hagen, N.

Johnson, A.

Kim, J.

Kittle, D. S.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref] [PubMed]

Kuhn, W. P.

R. E. Parks and W. P. Kuhn, “Optical alignment using the Point Source Microscope,” Proc. SPIE 5877, 58770B, 58770B-15 (2005).
[Crossref]

Marks, D. L.

McLaughlin, P.

Parks, R. E.

R. E. Parks and W. P. Kuhn, “Optical alignment using the Point Source Microscope,” Proc. SPIE 5877, 58770B, 58770B-15 (2005).
[Crossref]

Shaw, J. M.

Son, H. S.

Stack, R. A.

H. S. Son, A. Johnson, R. A. Stack, J. M. Shaw, P. McLaughlin, D. L. Marks, D. J. Brady, and J. Kim, “Optomechanical design of multiscale gigapixel digital camera,” Appl. Opt. 52(8), 1541–1549 (2013).
[Crossref] [PubMed]

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref] [PubMed]

Steel, W. H.

W. H. Steel, “The autostigmatic microscope,” Opt. Lasers Eng. 4(4), 217–227 (1983).
[Crossref]

Tremblay, E. J.

Vera, E. M.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref] [PubMed]

Appl. Opt. (2)

Nature (1)

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lasers Eng. (1)

W. H. Steel, “The autostigmatic microscope,” Opt. Lasers Eng. 4(4), 217–227 (1983).
[Crossref]

Proc. SPIE (1)

R. E. Parks and W. P. Kuhn, “Optical alignment using the Point Source Microscope,” Proc. SPIE 5877, 58770B, 58770B-15 (2005).
[Crossref]

Other (3)

P. R. Yoder, Mounting Optics in Optical Instruments, 2nd ed. (SPIE, 2008), Chap. 12.

E. Hecht, Optics, 4th ed. (Addison-Wesley, Reading, Mass., 2002), Chap. 6.2.1.

D. L. Marks and D. J. Brady, “Gigagon: a monocentric lens design imaging 40 gigapixels,” in Imaging Systems (IS), (Optical Society of America, 2010), paper ITuC2.

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Figures (10)

Fig. 1
Fig. 1

Illustration of multiscale optical design. (a) One channel of a multiscale camera with objective lens and one micro-camera. (b) Array of identical micro-cameras covering a larger field.

Fig. 2
Fig. 2

Solid model of basic mechanical components of AWARE-10. (a) Assembly of dome and objective lens. (b) Cross-sectional view showing relevant dimensions. Note inset showing the lip where the micro-camera sits.

Fig. 3
Fig. 3

Diagram of ASM operation. (a) Propagation of beam at nominal objective lens position. (b) Shift of return signal when objective is laterally misaligned.

Fig. 4
Fig. 4

Components of the ASM. (a) CAD model of ASM. Note the lens tube holding the lens has been cross-sectioned for viewing convenience. (b) Photograph of as built ASM.

Fig. 5
Fig. 5

ASM Zemax simulation results. (a) Ray trace at nominal objective position. (b) Spot diagrams of return signal at various lateral shifts of objective. (c) Linear fit of signal shift (ys) versus objective shift (Δy).

Fig. 6
Fig. 6

COC measurement procedure. (a) Illustration of insertion of ASM into dome for measurement. (b) Diagram of vector description of a pair of skew lines.

Fig. 7
Fig. 7

Photographs of first two calibration steps. (a) Procedure for aligning ASM probe beam to the axis of the dome adapter. (b) Procedure for finding the nominal zero position of return signal.

Fig. 8
Fig. 8

Final calibration step. (a) Procedure for measuring ASM sensitivity. (b) Sample image of returned spot signal at Δy = 40µm. The green circle indicates the calibrated center position from step 2 and the red ‘X’ indicates the calculated centroid of the spot. (c) Linear fit of measured signal shift (ys) versus objective shift (Δy).

Fig. 9
Fig. 9

Measurement procedure. (a) Points on dome where measurements were taken highlighted in red. (b) Splitting of signal at oblique angles indicating non-concentric objective lens.

Fig. 10
Fig. 10

Results of measurements. (a) Results from instantaneous measurement method. Color of data points indicate distance from the COC estimate and line lengths in the principal directions indicate standard deviation. (b) Results from average measurement method. Note positive Z direction is towards the dome [see Fig. 9(a) for definition of coordinates.]

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

α 2Δy R .
y s = 2Δy R [ d 0 ( 1 d 1 R f )+ d 1 R ],
S=| y s Δy |.
a +s b +p n = c +t d .

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