2.1.

Corner Cube Interferometers

In its simplest form, an interferometer is made up of a beamsplitter (BS) and two corner cube reflectors (CCRs) as illustrated in Fig. 1(a).

Fig. 1

(a) Corner cube interferometer, (b) modified from (a) for differential or angular measurements.

Since the heterodyne beams have orthogonal polarization, the optics of the interferometer has to be polarization-dependent. A polarizing beamsplitter (PBS) is, therefore, used to separate the two laser beams with frequencies f1 and f2, building the start point of the measurement paths. Quarter waveplates in the measurement paths are used to rotate the beam polarization in order to switch their reflectivity behavior on their return to the PBS. As an example, the beam propagating in the upward direction in Fig. 1(a) is first reflected by the PBS, meaning it is s-polarized. After passing the quarter waveplate once, the polarization in the measurement path is circular. On the way back, the polarization is rotated linear again by the quarter waveplate, but it is now p-polarized such that the beam is not reflected, but transmitted by the PBS.

For the detection of the beating signal of the recombined beams, a polarizer at 45 deg and a photodiode can be used. This simple version of an interferometer provides a linear displacement measurement of the CCRs with respect to the BS.

By rotating one of the measurement beams by 90 deg, for example by using a prism (P), as illustrated in Fig. 1(b), two parallel measurement beams are created, resulting in a differential measurement between two individually moving objects or in an angular measurement in case both CCRs are mounted on a common support. Such types of differential interferometers are insensitive to movements parallel to the measurement direction. Translations of the interferometer (PBS and P) or the CCRs perpendicular to the measurement direction result in a shift of the output beam, and hence a decrease in measurement signal or even signal loss when the detector no longer intercepts the beam.

2.2.

Plane Mirror Targets

The disadvantage of interferometers based on corner cubes is the limited transverse movement capability of the measurement mirrors with respect to the BS optics. Even simple applications such as the measurement of $xyz$-stage systems require such a degree of freedom. When using plane mirrors instead of corner cubes, as illustrated in Fig. 2(a), such transverse movements are tolerated. In order to observe the signal-generating temporal interference (i.e., beat frequency), rather than a spatial fringe pattern smearing out such a signal on the detector, the plane mirror must be aligned perpendicular to the beam with wavelength precision. When assuming a beam size of 3 mm and accepting a shift of a quarter wavelength, the angular alignment of the plane mirror has to be perpendicular to the beam within $50\mu \mathrm{rad}$. This strict alignment requirement can be relaxed by modifying the interferometer design to include a corner cube in the plane mirror interferometer as depicted in Fig. 2(b). The resulting double-pass interferometer combines the simple alignment of the CCR-based interferometers illustrated in Fig. 1 and the transverse movement capability offered by the plane mirror. The resolution is also doubled because the beams propagate through the interferometer twice. A tilt of a plane mirror results in a pure shift of the output beam, while the wavefront remains untilted. The parallelism of the output beams is maintained making this interferometer tolerant against angular alignment errors of the plane mirrors which have to be perpendicular to the beam propagation within mrads instead of microrads.

Fig. 2

(a) Plane mirror interferometer, (b) double-pass plane mirror interferometer.

2.3.

Rotating Targets

The interferometers presented thus far enable the measurement of translating targets, but also small-range angular measurements are possible. However, for the interferometric measurement of spindles, the target mirror will be rotating and hence requires further modifications to the interferometer design. This is especially true for measurements perpendicular to the axis of rotation where a mirror with rotational symmetry, such as a cylindrical or spherical mirror, is required. Figure 3(a) shows a simple, single-pass interferometer including a rotating target. A lens or lens system focuses the interferometer beam onto the rotating mirror, with the focus being at the rotation center. This allows obtaining an interferometer signal at all rotation angles. Thereby, the focus of the beam may be placed at the surface, resulting in a small measurement spot, or at the center of the spherical target mirror, resulting in a larger spot on the surface and making the system more tolerant against mirror surface imperfections. Though the depicted interferometer is single-pass, the focused geometry makes its alignment tolerant against alignment errors,³ such that alignment requirements are comparable to the previously presented double-pass interferometer.

Fig. 3

(a) Interferometer for including a rotational measurement, (b) beam deflection off of a spherical mirror.

The measurement range along the beam propagation direction is limited by the focused geometry. Restricting the target mirror is to move within the depth of focus of the beam.

A prerequisite for measuring a rotating target is that the rotating mirror must be precisely centered on the axis of rotation. Techniques for measuring under this condition are well-known.³ If the mirror is not well-centered, then the resulting wobble motion causes the wavefront to become tilted (reducing the interferometer signal) and in extreme cases can even cause the reflected beam to not propagate back into the interferometer, as illustrated in Fig. 3(b).

3.1.

Measurement of the Sphere Center

To allow the measurement of a rotating spherical target mirror that is not centered on the axis of rotation or that is moving in space, the interferometer beam has to maintain its pointing at the center of the sphere, which in the first step requires the center of the sphere to be measured in the directions perpendicular to the beam propagation direction $z$. The measurement resolution should be better than $1\mu \mathrm{m}$ for this $x/y$ position measurement in order to keep the geometrical error of the interferometric measurement small for a 36-mm diameter spherical mirror (Fig. 4).

Figure 5 illustrates how such a measurement can be realized in an interferometer and further details can also be found in Ref. 4. The output beams of the interferometer, i.e., after the PBS, have orthogonal polarization, which allows them to be separated again by a second PBS. Before doing so, the interferometer depicted in Fig. 5 splits the output beam using a nonpolarizing BS. In one of the generated output beams, the interference signal is normally generated, using a polarizer at 45 deg and detected on a photodiode. In the other output beam, the two polarizations are separated by a PBS and the position of the beam reflected off the sphere is detected in a two-dimensional (2-D) position-sensitive detector (PSD). A transverse movement of the sphere with respect to the interferometer results in a change of beam position on the PSD. The resolution that can be obtained depends on the detector and optical geometry, and typically reaches the submicron range. Thus, the interferometer in Fig. 5 measures the position of the sphere with nanometric resolution in the $z$ direction and micron resolution in the $x/y$ directions.

Fig. 5

Measurement of the transverse position of the spherical target mirror using a position sensitive detector (PSD) that can be used to track the center position of the sphere in a closed loop.

Having the position of the sphere measured in the $x/y$ plane, the interferometer itself can be installed on a translational $x/y$ stage and realigned in a closed loop using the PSD signal to keep pointing at the center of the sphere. The coupling of the laser into the tracking interferometer must be realized in a way to not affect the pointing of the beam entering the interferometer. Closed-loop tracking of the measurement beam gives the sphere full freedom of movement in the $x/y$ plane. The movement in the $z$-direction remains limited by the depth of focus of the focussed laser beam, which can reach the millimeter range.

3.2.

Translational Tracking Stage Errors

The interferometer depicted in Fig. 5 provides a measurement of a spherical target with the sphere center being tracked in closed loop using the PSD signal. The interferometric data, however, correspond to the relative position of the interferometer’s PBS and CCR to the sphere. While the rotating target position is to be measured interferometrically in the $z$ direction, the interferometer may be moving in the $x/y$ plane for the tracking motion to maintain beam alignment. This tracking motion is an additional source of interferometric measurement error. Mechanical translation systems have translational error motions such that a movement of the interferometer will occur in the $z$ direction when the interferometer is moved in the $x/y$ plane. As an example, a ball-bearing system has typical translational error motions in the micron range, which are partially nonreproducible. This parasitic error motion introduces a large uncertainty in the interferometric measurement, and thus deteriorating the measurement precision by many orders of magnitude from the nanometer to the micron range.

This issue can be solved by implementing an externally fixed plane reference mirror in the interferometer as depicted in Fig. 6. The resulting interferometer provides a differential measurement between the reference mirror and spherical target. A translational movement of the interferometer in the $z$-direction caused by error motions of the tracking system is thereby compensated. The reference mirror, however, needs precise alignment perpendicular to the interferometer beam, as in a single-pass interferometer. In order to obtain interferometer signal, the tilt of the interfering wavefronts has to below a quarter of the wavelength within the diameter of the laser beam. For a 3-mm beam, this results in an acceptable alignment error $\arcsin (150\mathrm{nm}/3\mathrm{mm})=50\mu \mathrm{rad}$. The remaining alignment error will produce a constant slope error when the tracking system is moving in the $x/y$ plane, which could be alleviated by using a corner cube instead of the plane mirror, which however will limit the tracking range drastically as the reflected beam will spatially shift, and reducing the beam overlap in the interferometer output.

Fig. 6

Differential tracking interferometer for compensation of translational error motions of the tracking stage mechanism.

3.3.

Rotational Tracking Stage Errors

In addition to translational error motions, a tracking stage system also displays parasitic angular error motions. A typical ball-bearing system shows angular errors in the tens to hundreds of microradians. Just like the interferometer presented in Fig. 1(b), the tracking interferometer of Fig. 6 is measuring these angular changes and is therefore prone to angular error motions, while it is most sensitive to the rotation around the $y$-axis due to the separation of the beams. For example, a $10\text{-}\mu \mathrm{rad}$ rotation results in an error of the position measurement of 100 nm in case of a small beam separation of 10 mm. Note that these are a separate issue from the alignment of the flat mirror in a single-pass interferometer itself. An optical compensation of these error motions is required in order to use a tracking interferometer for measurements in the 1-nm range.

For a demonstration of the angular error motion sensitivity, the interferometer shown in Fig. 7(a) was built. It corresponds to the interferometer of Fig. 6 with the lens system and sphere target replaced by a plane mirror. The interferometer was installed on a Thorlabs MT1-Z8 translation stage and repeatedly moved back and forth along the $x$-direction. Interferometric position data were obtained at 0.05 mm intervals using a Hewlet Packard laser Model 5517A, an optical receiver module E1709A, and an N1231A PCI interferometer card, both from Agilent Technologies (now Keysight Technologies). The beam separation in the interferometer was 15 mm. The Thorlabs MT1-Z8 stage specification states 200 microrad angular error motions and was operated without backlash correction. From this specification, a geometrical measurement error of $15\mathrm{mm}·\sin (200\text{microrad})=3\mu \mathrm{m}$ can be expected.

Fig. 7

(a) Interferometer from Fig. 6 with the sphere being replaced by a plane mirror for demonstrating the sensitivity to angular error motions of a tracking mechanism. (b) Experimental result.

Figure 7(b) shows in red the interferometer readout when moving in the positive direction, while the blue data points correspond to a movement in the negative direction. In the plot, the blue lines are shifted by $5\mu \mathrm{m}$ to the side for better visibility of the red data points. The blue and red data points were taken at identical nominal stage positions. Many cycles were performed to obtain statistical information. The width of the data distribution is caused by environmental noise and nonrepeatable errors of the translation stage.

The typical distribution width ranges from 1 to $2\mu \mathrm{m}$ peak to peak and one can clearly see that the stage shows a different behavior when moving in the positive or negative direction. The blue and red lines are separated by up to $3\mu \mathrm{m}$ and the separation strongly depends on the actual stage position. As a consequence, in a tracking interferometer, this angular error motion would limit the measurement accuracy to a few microns instead of the desired nanometer range.

3.4.

Optical Error Compensation

The interferometer depicted in Fig. 8 is a modified version of the tracking interferometer allowing for translational and angular errors of the tracking mechanism to be optically compensated. For this, an optical measurement is performed to two external flat reference mirrors.

Fig. 8

Tracking interferometer optically compensating translational and angular error motions.

The laser beam enters the interferometer and is split by a polarizing beamsplitter PBS1. The reference beam (blue in Fig. 8) propagates to the reference mirror R1 and back to PBS1 while passing a quarter waveplate twice, resulting in a 90-deg rotated polarization. The beam is, therefore, transmitted by PBS1 and PBS2. It is returned again to PBS2 by reference mirror R2, with two passes through another quarter waveplate and is, therefore, then reflected by PBS2. It then passes a half waveplate and is transmitted by PBS4 to recombine with the measurement beam.

The propagation of the measurement beam is similar. It is first transmitted by PBS1, then reflected by PBS3, bounced off the measurement target, and then transmitted by PBS3 and PBS4. After the reflection at the reference mirror R2, it is recombined with the reference beam in PBS4.

Please note that even though here four polarizing beamsplitters are employed, this is not needed. When removing PBS2 and PBS4 from the setup, the interferometer functionality remains and the shift of the output beam compared to the input beam is removed. The measurement results presented were obtained with the four beamsplitter versions of the interferometer.

The propagation lengths within the interferometer are constant such that the effective propagation path for the reference beam is $2·(a+b)$ and for the measurement beam $2·(c+d)$. The optical path difference and measurement signal are therefore given by $2·(a+b)-2·(c+d)$. Error motions caused by a tracking movement of the interferometer between the two reference mirrors are now optically compensated. For the simplest case, let us assume a translational movement of $\mathrm{\Delta }$ of the interferometer toward the measurement target. The optical paths $a$ and $c$ are shortened by $\mathrm{\Delta }$, while the paths at $b$ and $d$ are extended by the same amount. Thus, the interferometer output is unchanged.

This translation can also be understood as a rotation with the point of rotation at infinity. Let us move this rotation point to another extreme case, the center of the interferometer plate, and rotate in a clockwise direction. Here, the optical paths are shortened at $a$ and $d$, but extended at $b$ and $c$. The resulting path difference is, therefore, unchanged $2·(a-\mathrm{\Delta }+b+\mathrm{\Delta })-2·(c+\mathrm{\Delta }+d-\mathrm{\Delta })=2·(a+b)-2·(c+d)$.

Although the interferometer is a single-pass design, the alignment of the external reference mirror R2 is not critical as it represents a common mirror surface. A tilt of that mirror causes a shift of the output beams, but their parallelism is preserved. However, a tilt of R2 is measured by the interferometer, meaning this surface has to be angularly stable.

Also note that compared to the previous design of the tracking interferometer, the interferometer of Fig. 8 is optically balanced and therefore thermally stable. Both beams, the reference and measurement beams, propagate through the same amount of material, apart from the lens which would require a window to be implemented in reference path $a$.

The previous experiment presented in Sec. 3.3 was now repeated using the optically compensating design, while again the spherical target and lens are replaced by a flat mirror. The result is depicted in Fig. 9. The interferometer signal is now independent of the direction of motion, i.e., the blue and red data points overlap. The peak-to-peak data within one position are decreased from approximately $2\mu \mathrm{m}$ in the previous case to approximately 50 nm peak to peak and are attributed to environmental noise, mainly air fluctuations. The remaining slope is caused by the tilt between the reference mirror R1 and the flat measurement mirror that replaced the spherical target.

Fig. 9

Test results for an interferometer optically compensating translational and rotational tracking errors.

The overlap of blue and red data demonstrates the optical compensation of the interferometer compared to the previous case of Fig. 7(b), where their separation was up to $3\mu \mathrm{m}$. The separation in the compensated case is still present but reduced to around 15 nm, by a factor of 200 less. This makes it now possible to apply the tracking scheme for nanometric measurements, while error motions of the tracking mechanism in the 20-microrad regime would be acceptable to introduce an error in the interferometric measurement on the 1 nm scale.