Investigation of the uncertainty contributions of the alignment of PTB's double-ended interferometer by virtual experiments

An initial alignment procedure for an earlier prototype [15] was revised during the development of the current interferometer [16]. Further improvements led to the following strategy being devised:

The entire alignment process is carried out with only the 532 nm laser, since the 633 nm laser is mainly used for finding the integral order of interference via the method of exact fractions [23].

First, the fiber must be positioned in such a way that it is within the focal point of the collimation lens (L1). To this end, the lens is placed at a distance of approximately its focal length in front of the fiber tip. The lens is then tilted and the fiber is moved laterally until the back reflections of the lens surfaces overlap in the center of the end of the fiber. This brings the fiber tip approximately into the optical axis of the lens. The lens mount has a removable mirror attached behind the lens perpendicular to its optical axis. The light reflected from this mirror is coupled back into the fiber. To further improve the fiber position, the fiber tip is mounted on an x–y translation stage that automatically scans over a lateral shift of 100 μm in both directions. During this scanning stage, the recoupled intensity is measured at the reverse end of the fiber. The fiber tip is then adjusted in such a way that it is in the center of the measured intensity distribution. The final displacement is less than 10 μm from the optical axis. To align the fiber axially, a shearing interferometer is used in order to create a collimated beam. The axial alignment is experimentally determined to be better than 2 μm.

In the next step, the reference mirrors are aligned to be in autocollimation to the incoming beam. This step is similar to [24] but the reference mirror is tilted instead of the fiber tip being moved. The tilt of each mirror over two different axes is scanned by means of the three piezo pushers of the mirror mount while the light recoupled into the fiber is measured. The optimum tilt coincides with the center of the intensity distribution measured. For the alignment of RM_A, an automated shutter between BS_C and BS_B is closed and only the back reflection of RM_A couples into the fiber. Similarly, the tilt of RM_B is aligned with the shutter between BS_C and BS_A being closed. The standard deviation of repeated tilt adjustments is less than 0.7 μrad.

For further alignment of the interferometer, it is necessary to obtain images of the interference pattern on the two cameras. Each imaging system consists of two lenses (L2, L3) with focal lengths f₁ = 500 mm and f₂ = 40 mm, positioned at a distance of f₁ + f₂ from each other. A small aperture with a 2 mm diameter is placed at the focal point between the two lenses in order to block unwanted beams that result from multiple reflections in the wedged optical plates. Each camera is adjusted along the optical axis to achieve a sharp image of the corresponding gauge-block surface on its CCD. The sharp image allows the center position of the gauge block in the recorded image to be calculated and the central length to be evaluated. Each imaging system is aligned independently of the rest of the system along an optical bench that can then be aligned as one unit with regard to the interferometer. Since the beam exiting the vacuum-tight chamber is collimated, it is sufficient to align the imaging units via a simple lateral movement and tilting until the focal spot is in the center of the aperture. Via lateral movement, the field of view of the camera can be adjusted until most of the beam, and thus a reasonable image area, has been captured.

To prevent a cosine error (described in section 4.1), the beams traveling in opposite directions between BS_A and BS_B must be parallel. At the same time, high interference contrast must be achieved. Due to the small spatial coherence length of the light emitted from the multimode fiber, this can be achieved only by precisely overlapping these two beams. A removable crosshair between the collimation lens (L1) and the entrance window (W_C) is used to produce a reference point. The lateral shifts between the crosshair images in the passing beams (P) and the reference beams (Ref) can be observed by the two cameras. The beam splitters BS_A and BS_B are each tilted over two perpendicular axes in such a way that the images of the crosshairs overlap on both cameras. If necessary, the imaging systems can be realigned simultaneously to maintain the desired image area on the cameras. Since transmission through the beam splitters is also altered by tilting them, the autocollimation of the two reference mirrors must be realigned. As the reflections of these mirrors produce the reference crosshairs for aligning the beam splitters, this alignment procedure of beam splitters and reference mirrors must be carried out iteratively until the image of the crosshair is overlapped on both cameras while the reference mirrors are in autocollimation. After this step, the two opposite beams nearly overlap and are nearly parallel. To improve the parallelism, one of the beam splitters can then be very finely aligned until no fringe is visible in the interference image on the cameras. The tilt necessary is so small that the autocollimation is not significantly affected. This results in an angle alignment of the beam splitters that is better than 1.7 μrad, which would correspond to one visible interference fringe on the camera.

Finally, the tilt of the gauge block is aligned until no interference fringe is visible on camera A. Here, if the parallelism of the gauge block is good, no fringe will be visible on camera B, either.

The absolute positions of the optical components are not known with enough accuracy to create a perfect virtual copy of the interferometer. However, it is sufficient to study the tilts of the components, since the influence of the absolute positions of flat surfaces is negligible. Therefore, the positions of the components are defined approximately and their tilts are aligned according to the proposed alignment method of the real interferometer. The crucial relative positions between the lenses and their focal points can be calculated via ray tracing. In order to reproduce the real interferometer as accurately as possible, a virtual alignment procedure is implemented that is following the same steps as in the real system.

First, the distance between the point source and the collimating lens (L1) is set to the focal length; the collimated beam travels in a straight line and is aimed at RM_A.

Since it is possible to calculate the position and direction of all rays at any point in the interferometer, there is no need to iteratively align the beam splitters and the reference mirrors. To align the tilts of the beam splitters BS_A and BS_B, a virtual plane is introduced between the collimation lens (L1) and the window (W_C) perpendicular to the beam direction. Via ray tracing to this plane, the position as well as the corresponding directions of a number of rays are defined. Each of these rays is then separately traced along the two interferometer arms to a second virtual plane at the position of the gauge block. In order to overlap both beams, each pair of rays must strike this plane at the same position. Additionally, the two opposite rays must be parallel.

The reference mirrors are optimized by fulfilling the criterion that the incidence of all rays to the corresponding mirror must be perpendicular.

The orientation of the lenses in the imaging systems is defined by their optical axes and focal lengths. The tilt of the whole imaging system is optimized in such a way that the incidence of all rays is perpendicular to the camera plane. The system is then laterally moved in such a way that the camera plane is centered on the bundle of incoming rays. This is followed by the axial movement of the virtual camera (relative to L3) toward the sharp image plane of the gauge-block surface. The position of this plane is calculated by tracing rays from a virtual point source in the center of the corresponding gauge-block surface to their focal spot behind the imaging lenses.

Due to the wedge of the windows and beam splitters, the circular beams change their shapes slightly. Hence, the tilts of the previously aligned beam splitters still produce an angle between the beams from the opposite arm and the beam from the reference mirror onto the camera. This is equal to visible fringes in the interference image on the camera. Following the alignment in the real system, BS_B is again tilted until the angle between the two beams on camera A is minimal (i.e. corresponds to vanishing interference fringes). To this end, rays from the virtual camera pixels are aimed at the light source along the pathway of the gauge block, passing beam (P_A) as well as the pathway via RM_A (Ref_A). The tilt of BS_B is varied in such a way that the optical path differences between each pair of rays which start at the same position on the camera plane along the two beam paths are equal. Then, the tilt of RM_B and the axial position of camera B are again checked and realigned.

In the last step, the tilt of the gauge block is aligned in the following way. The optical path difference between the rays reflected from the corresponding gauge block surface (R_X) and those reflected from reference mirror (Ref_X) is calculated for a bundle of rays. The gauge block tilt is then varied in such a way that the optical path difference for every pair of rays is equal, which is similar to a camera image without fringes in the real experiment.

Virtual length measurement is performed by 'ray aiming' the defined beams onto the virtual camera pixels. To closely reproduce the real experiment, one can calculate the optical path lengths of all six beams for a whole camera array of 1024 × 1024 pixels. As mentioned earlier, the pathways via the reference mirrors cancel each other out and vanish in the equation for calculating the length (see equation (2)). Therefore, in order to calculate the measured length, it is sufficient to calculate the passing (P_X) and reflected (R_X) beams from the gauge block. In order to further increase the performance, only the four rays that hit the corresponding gauge-block surface in the center are traced. The beams are aimed at these points to obtain the starting angle at the light source and then traced to the camera. From the four resulting optical pathways, the measured length is calculated in accordance with equation (2).

Due to the increased number of beams in the DEI, the cosine error does not depend exclusively on the misalignment of the gauge block. The parallelism of the two beams from the two interferometer arms that pass the gauge block in opposite directions are also of crucial importance. The measured length in an SEI and a DEI is

$$\begin{equation}{L}_{\text{SEI}}=L\cdot \mathrm{cos}\enspace \alpha \end{equation} \tag{ 3 }$$

$$\begin{equation}{L}_{\text{DEI}}=\frac{1}{2}L\cdot \left(\mathrm{cos}\enspace \alpha +\mathrm{cos}\enspace \beta \right),\end{equation} \tag{ 4 }$$

where α and β are the angles of the incoming beam to the surface normal of the gauge block on sides A and B respectively.

The angle between the two opposite beams can be substituted by γ = α + β. It is notable that the equation for the DEI is reduced to that for the SEI, not only if the beam splitters are perfectly aligned and γ = 0, but also if α = β. Furthermore, the equation for SEI can be extended if both sides of a gauge block are measured and the length is given as the average of both measurements. This means that the gauge block is measured once with each side wrung to a reference plate, while in each measurement, there is a misalignment of α or β.

One major difference between length measurements in a DEI compared to measurements in an SEI are the reference points on the gauge block or reference plate. In SEI length measurements, ISO 3650 defines the central length of gauge blocks as the perpendicular distance from the reference plate to the center of the measuring face. For interferometric measurements in practice, this means the center of the free face must be determined to allow the optical path length of its reflected light to be directly measured. The optical path length reflected from the reference plate must be interpolated to the corresponding point which is hidden by the gauge block.

Since, in DEI measurements, the two gauge block end faces are imaged independently, there is no such direct reference between the two end faces. Therefore, it is usually assumed that the gauge block is a perfect cuboid where all faces meet at a right angle and the opposite measuring surfaces are parallel. However, according to ISO 3650, the tolerance of the perpendicularity of a side face with a measuring face of a 100 mm grade K gauge block, for instance, is 100 μm. Therefore, real gauge blocks might have a shape like that in the inset of figure 2.

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For a perfectly aligned system there is no difference in the measured length for gauge blocks with different perpendicularities. However, for a slight gauge block tilt the influence of the cosine error increases with decreasing perpendicularity. In figure 2, the measured length deviation in dependence on the perpendicularity deviation of a 100 mm gauge block is shown. The dots represent results of simulated length measurements. As long as the interferometer is optimally aligned and no cosine error occurs the measured length deviation is negligible (solid line). However, if the gauge block is slightly tilted, it is to be seen that the influence of the cosine error increases with a higher deviation from perpendicularity. For a gauge block tilt of ω = 30 μrad, the cosine error can be up to 3 nm, whereas for a perpendicular gauge block it would be less than 0.07 nm. For a wavelength of λ = 532 nm, this tilt corresponds to one visible interference fringe on the cameras along the gauge block's short axis of 9 mm, which is the same magnitude as the alignment accuracy.

Thus, the influence of the cosine error in connection with non-perfect gauge blocks differs from that of perfect gauge blocks. A perpendicularity deviation of Δp is equivalent to an angle deviation ɛ = arctan(Δp/L) from 90°. The distance between the two central points of the end faces is therefore L' = L/cos(ɛ). This can be described as an effective length for the cosine error in equation (4), while the gauge block is tilted by ɛ. The measured length can then be described by

$$\begin{equation}{L}_{\text{DEI}}={L}^{\prime }\enspace \mathrm{cos}\left(\varepsilon +\omega \right)\end{equation} \tag{ 5 }$$

$$\begin{equation}=L\frac{\mathrm{cos}\left(\varepsilon +\omega \right)}{\mathrm{cos}\enspace \varepsilon },\end{equation} \tag{ 6 }$$

where ω is an additional gauge block tilt due to misalignment. With α = ω and β = −ω, this equation is consistent with equation (4). The result is a shift in the tilt of the gauge block at which the measured length is at its maximum (see figure 3).

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For small tilts of the gauge block, this error is mainly independent of the absolute gauge-block length L. This can be seen in the derivative with respect to the gauge block tilt

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}\alpha }{L}_{\text{DEI}}=-y\enspace \mathrm{cos}\enspace \alpha -L\enspace \mathrm{sin}\enspace \alpha ,\end{equation} \tag{ 7 }$$

which is −y for α = 0.

Besides the alignment error, the deviation from perpendicularity has to be considered while correcting for wavefront aberrations in DEIs. Nonetheless, in the following simulations, all gauge blocks are assumed to be perfectly perpendicular between their measuring faces and their side faces to explicitly distinguish between the error sources. Only in the Monte Carlo runs with non-perfect gauge blocks in section 6 we considered this shape deviation.

Another simplification of the gauge block's shape is the parallelism of the two end faces of the gauge block. Although the tolerance is 0.1 μm for grade K gauge blocks that are shorter than 300 mm, the deviation can be dramatic for other measurement standards. Additionally the parallelism of the measuring faces depends on the choice of supporting points while mounting the gauge block. We will assume in this study, that exactly the Airy points are chosen, although in the real experiment this is only an approximation. In SEI length measurements, the measuring method does not change in case of deviations from parallelism of the measuring faces. The reference plate must still be perpendicular to the incoming beam, resulting in a tilted front face of the gauge block. In DEIs, on the other hand, both end faces are observed at the same time and only one face can be tilted in an ideal fashion.

To be close to the definition, the gauge block should be aligned in a completely perpendicular orientation on one side, neglecting the opposite side. However, the uncertainty due to the camera misalignment would be smaller if both sides of the gauge block were misaligned equally (see section 4.4). Similarly to measurements in an SEI it should be stated, which side of the gauge block has been used for the alignment or if the length has been measured in both alignment states. Due to the parallelism of the two opposing beams, a measurement in the DEI with one measuring face aligned perpendicular to the incident beam corresponds to a measurement in an SEI where the same side is wrung to the reference plate.

An incorrectly measured length can be the result not only of a cosine error, but also of an incorrect camera position. First, the image of the gauge block is used to detect the center of the gauge block. To this end, sharp edges in the image are necessary. Second, all beams must have the same optical path distance from the gauge block's measuring surface to the camera, regardless of their direction. The latter concept will be explained in the following.

In figure 4, the path of the beam reflected from the gauge block R_B is shown for a tilted gauge block. The two imaging lenses focus a point of the gauge block's surface onto the sharp image plane s. The beam passing the gauge block P_B is also imaged onto the plane s. Due to the nature of the imaging system, both beams have the same optical pathway between the gauge block's surface and the plane s. However, if the camera is axially misaligned, the optical pathways of the two rays change differently. Thus, if the camera distance to the imaging lenses is too long, for the change in the optical path length ΔL_P,B < ΔL_R,B applies, and hence, the measured length is shortened (see equation (2)).

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Figure 5 shows how a misalignment of one camera can influence the measured length.

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The error can be calculated by

$$\begin{equation}{\Delta}{L}_{\text{cam}}=\frac{d}{2}\cdot \left[1-\frac{1}{\mathrm{cos}\left(2\alpha \cdot {f}_{2}/{f}_{3}\right)}\right],\end{equation} \tag{ 8 }$$

where d is the displacement of the camera from the sharp image plane, α is the tilt of the gauge block and f₂ and f₃ are the focal lengths of the lenses in the imaging system. Since this effect is combined with the cosine error, the measurement of the gauge block can be not only too short (as usually observed), but also too long (see ▽ in figure 6).

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It is especially important that this defocusing effect be taken into account for DEIs in which both end faces of the gauge block are imaged to a single camera. Several approaches to set up a DEI propose, for example, simply adding a corner cube or a roof mirror to an SEI [7, 10, 12]. In these setups, it is not possible to optimally adjust the focus on both end faces. In SEIs this effect may also occur for non-parallel gauge blocks.

A lateral displacement Δx of the fiber tip tilts the collimated beam entering the interferometer by the angle $\omega =\mathrm{arctan}\left({\Delta}x/{f}_{1}\right)$. Thus, the two opposite beams at the gauge block are tilted by the same amount ω, but in opposite directions. With β = −α, equation (4) reduces to

$$\begin{equation}{L}_{\text{DEI}}=\frac{1}{2}L\cdot \left(\mathrm{cos}\enspace \alpha +\mathrm{cos}\enspace \beta \right)\end{equation} \tag{ 9 }$$

$$\begin{equation}=L\cdot \mathrm{cos}\enspace \omega .\end{equation} \tag{ 10 }$$

In this case, the influence of the camera position is negligible, since the reflected beam at the gauge block (R_X) has the same direction of propagation as the beam coming from the other side that passes the gauge block (P_X). There is also no relative influence of a rhombic gauge block shape, since the gauge block is not tilted (only the incoming beam is tilted). In figure 7, experimental measured length differences are plotted against the lateral shift of the fiber tip together with simulated data and a fit of equation (10).

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The tilt of the beams introduced by a lateral fiber displacement can be largely compensated by aligning the beam splitters BS_A and BS_B. The remaining astigmatism, as well as a spherical wavefront deformation introduced by axial translation, can be compensated by wavefront corrections. Since there are many other contributions to wavefront distortion, the axial position is set to its optimal position and is not investigated further here. For the same reason, the astigmatism introduced is not investigated in this paper further. However, the fiber tip translation allows a comparison between the experimental measurements and the simple theoretical equation, since most other alignment effects are negligible.

Generally, it is not possible to separate the different effects in the real interferometer. To observe alignment errors in the real experiment we used a 200 mm silicon single crystal gauge block which provides high stability over time and is well known from laboratory-internal comparisons carried out in the past. In figure 8 the results for the length measurement of this gauge block in dependence on the gauge block tilt is shown. The measured gauge block has a clear rhombic shape with a distinct deviation from right angles. Therefore, the maximum value of the measured length is not centered on a perpendicular incidence on the measuring surface. The orientation of the gauge block is depicted along the graph. Furthermore, the deviation is not symmetric to the gauge block tilt, indicating the influence of a defocused camera.

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The effective length for the cosine error (see equation (6)) can dramatically increase the uncertainty due to a tilted gauge block, since the slope of the cosine becomes larger for perfectly aligned gauge blocks as the angle deviation from 90° increases.