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Advanced Optical Incremental Sensors

Encoders and Interferometers

Suzanne J.A.G. Cosijns¹, Maarten J. Jansen¹, and Han Haitjema² ¹ASML, Veldhoven, The Netherlands ²Mitutoyo Research Center Europe, Best, The Netherlands

Abstract

In this chapter the principles of displacement metrology by optical incremental sensors are given. First the basic concepts of displacement interferometer systems as well as their signal processing and error sources are reviewed. Then the concepts of optical encoder systems and some of their design considerations are discussed.

Keywords

Displacement interferometer; Imaging encoder; Interferential encoder

10.1. Introduction

Within the precision engineering and manufacturing industries, displacement laser interferometers and optical encoders are often used as feedback sensors in numerically controlled systems for reliable and accurate noncontact measurement of linear and rotary motion. Laser interferometers use the laser frequency as a reference. Frequency-stabilized helium-neon (HeNe) lasers are used in many national standards laboratories and measurement institutes for obtaining a practical traceable reference to the SI unit of length, the meter. The laser wavelength is related to the laser frequency through the definition of the meter: “The meter is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.” (CGPM, 1983). This means that for a specified light frequency f the wavelength follows from the definition of the speed of light in vacuum c = 299,792,458 m/s and the refractive index n of the medium, as is illustrated in Eq. (10.1):

λ = \frac{c}{n \cdot f} .

$λ = \frac{c}{n \cdot f} .$

(10.1)

Additionally, the vacuum wavelength λ_v is defined as the wavelength for n = 1 in Eq. (10.1). Laser interferometer systems offer an accurate and effective means of delivering traceability for length and dimensional measurement. With the wavelength of light being used as a standard for length metrology, laser displacement interferometers are mostly suited for traceable calibration and acceptance tests of machine tools and coordinate measurement systems. As the wavelength of the laser light is dependent on the refractive index of the air, interferometers are typically used in well-conditioned metrology laboratories. In high-tech industrial equipment, such as lithography machines and reticle/mask metrology tools for the semiconductor industry, laser interferometers are often applied for displacement measurement with sub-nm resolution. Optical encoders use an optical grating on a substrate as a reference and are available for both low-end and high-end applications where environmental conditions may be less well defined, or where there is a risk that a laser beam would be interrupted. In this chapter, the basics of incremental displacement interferometers and optical encoders are explained, together with some of their advantages and disadvantages.

10.2. Displacement interferometers

10.2.1. Basics of displacement interferometry

The displacement interferometer, first introduced by Albert Michelson in 1881 (Fig. 10.1), has been developed into a measurement system with high accuracy. Because interferometry is based on the interference of light, it is a noncontact measurement method of which the accuracy is influenced by the wavelength of the source and the medium in which the measurement takes place.

In an interferometer, coherent light is directed to a semitransparent mirror that acts as an amplitude-dividing beam splitter. Part of the light is transmitted toward a movable mirror and reflected by this mirror. The other part of the light is reflected at 90 degrees toward a fixed reference mirror, reflected and recombined at the beam splitter where their interference is observed. The electromagnetic waves propagating in the reference and measurement arm can be represented as

\begin{matrix} {\overset{⇀}{E}}_{ref} = E_{r} e^{i} (ω t + k {\overset{⇀}{r}}_{r} - Φ_{r}) {\overset{⇀}{e}}_{r} \\ {\overset{⇀}{E}}_{meas} = E_{m} e^{i} (ω t + k {\overset{⇀}{r}}_{m} - Φ_{m}) {\overset{⇀}{e}}_{m} \end{matrix}

$\begin{matrix} {\overset{⇀}{E}}_{ref} = E_{r} e^{i} (ω t + k {\overset{⇀}{r}}_{r} - Φ_{r}) {\overset{⇀}{e}}_{r} \\ {\overset{⇀}{E}}_{meas} = E_{m} e^{i} (ω t + k {\overset{⇀}{r}}_{m} - Φ_{m}) {\overset{⇀}{e}}_{m} \end{matrix}$

(10.2)

with i the imaginary unit number, E_r and E_m the amplitude of the respective E-fields, ω the angular frequency, t the time, k the circular wavenumber (k = 2π/λ_v),

\overset{⇀}{r}

$\overset{⇀}{r}$

the position vector, ϕ_r and ϕ_m the phase in the reference and measurement arms, respectively, and

{\overset{⇀}{e}}_{r}

${\overset{⇀}{e}}_{r}$

and

{\overset{⇀}{e}}_{m}

${\overset{⇀}{e}}_{m}$

the unit vector in the reference and measurement direction, respectively. After recombination in the beam splitter, the electromagnetic field is the linear superposition of these waves. Assuming the propagation of the field takes place in only one dimension, the irradiance at the detector becomes

I = ε_{0} c 〈 {\overset{⇀}{E}}^{2} 〉 = ε_{0} c (E_{r}^{2} + E_{m}^{2} + 2 E_{r} E_{m} \cos (k (r_{m} - r_{r}) - (ϕ_{m} - ϕ_{r})))

$I = ε_{0} c 〈 {\overset{⇀}{E}}^{2} 〉 = ε_{0} c (E_{r}^{2} + E_{m}^{2} + 2 E_{r} E_{m} \cos (k (r_{m} - r_{r}) - (ϕ_{m} - ϕ_{r})))$

(10.3)

where ε₀ is the vacuum permittivity, c is the speed of light, k is the circular wavenumber, r_m is the traveled optical distance in the measurement arm, and r_r is the traveled optical distance in the reference arm. If the waves were initially in phase (ϕ_r = ϕ_m), the cosine term depends on the difference in optical path length between the reference and measurement arms. This is the case for a monochromatic light source. If the two beams are of equal amplitude, the irradiance is

Figure 10.1 Schematic representation of Michelson's interferometer.

I = 2 I_{0} (1 + \cos (\frac{2 π}{λ_{v}} (r_{m} - r_{r})))

$I = 2 I_{0} (1 + \cos (\frac{2 π}{λ_{v}} (r_{m} - r_{r})))$

(10.4)

where

I_{0} = ε_{0} \cdot c \cdot E_{m}^{2}

$I_{0} = ε_{0} \cdot c \cdot E_{m}^{2}$

. When the measurement mirror is displaced over a distance Δl, while the reference mirror remains fixed, the optical path length changes 2nΔl, with n the refractive index of the medium through which the light travels. The factor 2 is because of the fact that this distance is travelled twice by the light. If a detector is used, the measurement signal will change as follows:

I = 2 I_{0} (1 + \cos (\frac{2 π}{λ_{v}} (2 n Δ l)))

$I = 2 I_{0} (1 + \cos (\frac{2 π}{λ_{v}} (2 n Δ l)))$

(10.5)

With λ_v the vacuum wavelength. If the wavelength of the light source is known then the displacement can be calculated from the change in intensity on the detector. From this, it can also be seen that it is a relative measurement, only displacement can be measured, not distance.

Numerous different versions were derived from Michelson's original displacement interferometer, all working on the principle of measuring displacement with use of interference. Most modern displacement interferometers use a frequency-stabilized monomode HeNe laser as a light source because of the long coherence length and relatively short visible wavelength (red light, λ ≈ 633 nm), which results in comfortable alignment and enables a high resolution. Laser interferometers can be divided into two types: homodyne and heterodyne.

10.2.1.1. Homodyne interferometers (detection)

Most commercial homodyne laser interferometers consist of a stabilized single-frequency laser source, polarizing optics, photodetector(s), and measurement electronics. A homodyne laser source is typically a HeNe laser with a single-frequency beam as the output consisting of either a beam linear polarized at 45 degrees or a circularly polarized beam. The beam is split into a reference arm and measurement arm of the interferometer by a beam splitter. Following reflection from their respective targets, the beams recombine in the beam splitter. To observe interference, both beams should have equal polarizations. This is accomplished by a polarizer oriented at 45 degrees to the beam splitter. The photodetector signal is run through electronics, which count the fringes of the interference signal. A fringe is a full cycle of light intensity variation, going from light to dark to light. Every fringe corresponds to an optical path difference of a wavelength corresponding to Eq. (10.5). Because there is no intrinsic time dependency in the measurement signal, this is also known as a “DC interferometer.” Depending on the detector configuration, direction sensing and insensitivity to power changes can be derived as well as a compensation for periodic deviations such as that developed by Heydemann (1981). In Fig. 10.2, the principle of such a homodyne interferometer is shown.

Signal I₀ is used to normalize the intensity. Signals S₀ and S₉₀ are used for phase quadrature measurement. Signal S₀ is the normal signal of a homodyne interferometer (Eq. 10.5) with signal 2I₀ subtracted:

S_{0} = 2 I_{0} \cos (2 π (\frac{2 n}{λ_{v}}) Δ l) .

$S_{0} = 2 I_{0} \cos (2 π (\frac{2 n}{λ_{v}}) Δ l) .$

(10.6)

To enable direction sensing and a constant sensitivity over an entire wavelength, part of the measurement signal is split off and receives a phase shift of 90 degrees; this measurement signal is called S₉₀

S_{90} = 2 I_{0} \sin (2 π (\frac{2 n}{λ_{v}}) Δ l)

$S_{90} = 2 I_{0} \sin (2 π (\frac{2 n}{λ_{v}}) Δ l)$

(10.7)

Using these equations, the displacement can be calculated from both intensity signals using

Δ l = (\frac{λ_{v}}{2 n}) \cdot \frac{1}{2 π} \tan^{- 1} \frac{S_{90}}{S_{0}}

$Δ l = (\frac{λ_{v}}{2 n}) \cdot \frac{1}{2 π} \tan^{- 1} \frac{S_{90}}{S_{0}}$

(10.8)

with (λ_v/2n) being the signal pitch, which is the measured displacement corresponding to a phase cycle of 2π rad. In an ideal interferometer, when both signals are plotted against each other a circle is described.

Figure 10.2 Schematic representation of the principle of a homodyne laser interferometer with power compensation and direction sensing. I₀ is used to eliminate effects of power changes, S₀ and S₉₀ are used for phase quadrature measurement.

10.2.1.2. Heterodyne interferometers (detection)

The basic setup of a heterodyne interferometer is shown in Fig. 10.3.

Generally the light source of a heterodyne laser interferometer is a stabilized HeNe laser whose output beam contains two frequency components (f₁ and f₂), each with a unique linear polarization. Their electromagnetic field is represented by

\begin{matrix} {\overset{⇀}{E}}_{1} = E_{01} e^{i (2 π f_{1} t + ϕ_{01}) {\overset{⇀}{e}}_{1}} \\ {\overset{⇀}{E}}_{2} = E_{02} e^{i (2 π f_{2} t + ϕ_{02}) {\overset{⇀}{e}}_{2}} \end{matrix}

$\begin{matrix} {\overset{⇀}{E}}_{1} = E_{01} e^{i (2 π f_{1} t + ϕ_{01}) {\overset{⇀}{e}}_{1}} \\ {\overset{⇀}{E}}_{2} = E_{02} e^{i (2 π f_{2} t + ϕ_{02}) {\overset{⇀}{e}}_{2}} \end{matrix}$

(10.9)

where E₀₁ and E₀₂ represent the amplitude, and ϕ₀₁ and ϕ₀₂ represent the initial phase of the electromagnetic field. The frequency difference can be generated by using the Zeeman effect. Because of this effect, two frequencies can be generated by applying an axial magnetic field to the laser tube. Another way to generate a frequency shift is by an acousto-optic modulator; for example, a Bragg cell driven by a quartz oscillator. With this technique, the frequency split is limited to a maximum of about 4 MHz. Alternatively an acousto-optic modulator enables a frequency split of 20 MHz or more. Generally, in a heterodyne laser interferometer two linearly polarized beams are used with polarization directions that are orthogonal to each other.

Part of the light emitted by the laser source is split off, passes a combining polarizer, and falls onto a detector with a band-pass filter. The resulting signal is an alternating signal with a beat frequency equal to the split frequency in the laser head. This signal forms the reference measurement I_r, given by

I_{r} = 2 E_{01} E_{02} \cos (2 π (f_{2} - f_{1}) t + (ϕ_{02} - ϕ_{01})) .

$I_{r} = 2 E_{01} E_{02} \cos (2 π (f_{2} - f_{1}) t + (ϕ_{02} - ϕ_{01})) .$

(10.10)

As can be seen from Eq. (10.10), the heterodyne interferometer works with a carrier frequency (f₂ − f₁), therefore it is also known as an “AC interferometer.” The rest of the light emerges from the laser head and enters the interferometer optics (Fig. 10.3), consisting of a polarizing beam splitter and two retroreflectors. In the polarizing beam splitter, the two frequencies are split according to their polarization. Frequency f₁ is reflected by the polarizing beam splitter and enters the reference arm, then is reflected by the reference retroreflector and is again reflected by the polarizing beam splitter. Frequency f₂ is transmitted by the polarizing beam splitter and enters the measurement arm, then is reflected by the moving retroreflector and is again transmitted by the beam splitter. Ideally, both frequencies emerge from the polarizing beam splitter in their own unique polarization orthogonal to each other. To enable interference, the beams are transmitted through a polarizer at 45 degrees with their polarization axes. After the polarizer, the light falls onto a second detector with a band-pass filter resulting in the measurement signal

Figure 10.3 Schematic representation of the principle of a heterodyne laser interferometer. The reference retroreflector is in a steady position, whereas the measurement retroreflector is attached to the object of which the displacement should be measured.

I_{m} = 2 E_{01} E_{02} \cos (2 π (f_{2} - f_{1}) t + (ϕ_{02} - ϕ_{01}) + (ϕ_{meas} - ϕ_{ref}))

$I_{m} = 2 E_{01} E_{02} \cos (2 π (f_{2} - f_{1}) t + (ϕ_{02} - ϕ_{01}) + (ϕ_{meas} - ϕ_{ref}))$

(10.11)

where ϕ_meas − ϕ_ref is the difference in phase between the signal in the measurement arm and the reference arm. It consists of a constant term in the reference arm (ϕ_0r) and a phase in the measurement arm consisting of a constant term (ϕ_0m) combined with a changing term as a result of the moving retroreflector

ϕ_{meas} - ϕ_{ref} = Δ ϕ + ϕ_{0 m} = ϕ_{0 r} .

$ϕ_{meas} - ϕ_{ref} = Δ ϕ + ϕ_{0 m} = ϕ_{0 r} .$

(10.12)

In a practical application, the measurement retroreflector is attached to the object of which the displacement is to be measured. As a retroreflector moves in the measurement arm with velocity v, a Doppler shift is generated for frequency f₂

Δ f = \frac{2 v n f_{2}}{c}

$Δ f = \frac{2 v n f_{2}}{c}$

(10.13)

with v the velocity of the moving retroreflector, n the refractive index of the medium through which the light travels (e.g., air), and c the speed of light in vacuum. From Eq. (10.13), it follows that the maximum traveling speed of the target is limited because of the finite frequency shift between the two frequencies in the laser source. The effective phase change in the interference signal resulting from the Doppler shift equals

Δ ϕ = \int_{t 1}^{t 2} 2 π Δ f d t = \int_{t 1}^{t 2} 2 π \frac{2 v n f_{2}}{c} d t = 4 π \frac{n f_{2}}{c} \int_{t 1}^{t 2} v d t = \frac{4 π n f_{2}}{c} Δ l

$Δ ϕ = \int_{t 1}^{t 2} 2 π Δ f d t = \int_{t 1}^{t 2} 2 π \frac{2 v n f_{2}}{c} d t = 4 π \frac{n f_{2}}{c} \int_{t 1}^{t 2} v d t = \frac{4 π n f_{2}}{c} Δ l$

(10.14)

where Δl is the displacement of the retroreflector. So, by measuring the phase change between the measurement signal (Eq. 10.11) and the reference signal (Eq. 10.10), the displacement of the retroreflector can be determined by using the inverse of Eq. (10.14) with vacuum wavelength λ_2v

Δ l = \frac{λ_{2 v}}{2 n} \cdot (\frac{Δ ϕ}{2 π}) .

$Δ l = \frac{λ_{2 v}}{2 n} \cdot (\frac{Δ ϕ}{2 π}) .$

(10.15)

Note that this equation is principally similar to Eq. (10.8)

Figure 10.4 Schematic comparison of the relevant signals of a displacement (a) measurement with a homodyne (b and c) and a heterodyne interferometer, both the reference signal (d) and the measurement signal (e).

10.2.1.3. Signals

Fig. 10.4 shows the effect of a moving object on a homodyne and a heterodyne interferometer signal.

Comparing the intensity signals for homodyne (Eqs. 10.6 and 10.7) and heterodyne interferometers (Eqs. 10.10 and 10.11) the main difference lies in the intensity signal during standstill of the measurement object; in the case of a homodyne interferometer, the intensity signal remains constant. In the case of a heterodyne interferometer, the intensity is modulated by the split frequency of the laser, resulting in a carrier frequency. Taking the signals into account shows the principal difference between homodyne and heterodyne detection. During standstill the signal-to-noise ratio is easily disturbed for a homodyne interferometer by any change in the individual signal strength.

10.2.2. Interferometer concepts

With the basic principles as described in the previous paragraph, different optical configurations can be used to enhance measurement resolution.

10.2.2.1. Linear interferometer

The 1D distance interferometer (see Fig. 10.5) consists of a polarizing beam splitter and two retroreflectors.

Figure 10.5 Schematic representation of the optics in a linear interferometer.

The advantage of this sensor is the limited sensitivity for angle variations of the retroreflectors. The disadvantage is the limited range in the direction perpendicular to the measurement direction; on the other hand this reduces the risk of cosine errors (see Section 10.3.1). As a result this measurement configuration is limited to a 1D movement of the measurement object only.

10.2.2.2. Plane mirror interferometer

Another commonly used interferometer configuration, known as a “plane mirror interferometer,” is shown in Fig. 10.6.

It uses a plane mirror as the target, which enables movement of the target in a direction perpendicular to the measurement beam. As a result, this interferometer is often used in configurations where 2D measurements of the moving object are required, such as 2D moving stages. Because of its configuration, the beams travel toward the measurement mirror twice and, as a result, the measurement resolution of the system is doubled compared with the linear interferometer.

The light entering from the source is split by the polarizing beam splitter. One polarization is transmitted toward the measurement arm, and one is reflected toward the reference arm. The transmitted beam (vertical polarization) passes a quarter-wave plate changing the polarization into circularly polarized light. This light travels toward the measurement mirror where it is reflected and the polarization rotated 180 degrees. The light again passes the quarter-wave plate where it is changed to horizontal polarization and, as a result, now is reflected by the polarizing beam splitter. The light travels to the retroreflector and is returned to the beam splitter, where it is reflected again, passes the quarter-wave plate, reflects a second time on the measurement mirror, and passes the quarter-wave plate for the fourth time, resulting in vertical polarized light again and, as a result, is again transmitted by the polarizing beam splitter toward the detector.

Figure 10.6 Schematic representation of the optics in a plane mirror interferometer.

The originally horizontal polarization is reflected by the polarizing beam splitter and enters the reference arm, where it is transmitted by the quarter-wave plate resulting in circularly polarized light, which is reflected by the reference mirror and rotated 180 degrees. It then passes again the quarter-wave plate again, resulting in vertically polarized light, which is transmitted by the beam splitter toward the retroreflector. There, the light is translated, again enters the beam splitter, passes though the quarter-wave plate for the third time, and is reflected a second time by the reference mirror. Finally, it passes the quarter-wave plate a fourth time and is once more horizontally polarized and reflected by the polarizing beam splitter toward the detector.

10.2.3. Phase detection and interpolation

Both incremental displacement interferometers and optical encoders output a signal with a phase that varies periodically with displacement Δl.

Δ ϕ (Δ l) = 2 π (\frac{Δ l}{P}) + α

$Δ ϕ (Δ l) = 2 π (\frac{Δ l}{P}) + α$

(10.16)

where Δϕ represents the phase difference between the measurement beam and the reference beam, P the signal period and α a phase offset that is specific for a certain detector signal. Most industrial displacement interferometer and linear encoder systems at least internally generate analog phase quadrature signals consisting of a cosine and sine signal with a 90 degrees phase offset. This phase offset allows for the unambiguous detection of the direction of motion, and the sinusoidal nature guarantees a measurement sensitivity that is independent of the measurement position. The resolution of an interferometer system can be increased by phase interpolation. With analog quadrature signals, it is possible to interpolate within each quadrature cycle. The measured phase is represented by the vector angle of the data point on the Lissajous curve (Fig. 10.7).

To reduce periodic errors, which repeat every phase cycle, the analog phase quadrature signals are normalized by adjusting the gain, the DC offset and phase offset until a centered circle is obtained when plotting the S₀ and S₉₀ signals against each other in a Lissajous plot. In modern systems, the normalization of the Lissajous curve is performed automatically by means of a correction like the Heydemann correction (Heydemann, 1981). To determine a normalization correction, the measurement signal must, preferably, undergo at least a few phase cycles—typically obtained by introducing a movement of the system in the measurement direction. Correct compensation of DC signal offsets will resolve first-order interpolation errors, whereas proper gain and phase-offset calibration corrects for second-order periodic interpolation errors. Depending on the frequency response of a detector, the calibration of the Heydemann correction parameters may have some level of speed dependency, which, if measurement uncertainties in the order of a nanometer are desired, can make optimal correction under all measurement conditions challenging. Phase interpolation is typically obtained by direct calculation of the arctangent function by a microprocessor.

Figure 10.7 Heydemann correction and Lissajous curve. Left: Phase quadrature signals before and after normalization using the Heydemann correction to compensate for gain, offset, and signal phase errors (Heydemann, 1981). Right: Lissajous curve.

Δ ϕ = \tan^{- 1} (\frac{A}{B})

$Δ ϕ = \tan^{- 1} (\frac{A}{B})$

(10.17)

where parameters A and B are the normalized signals that are 90 degrees out of phase and can be derived by taking a linear combination of the interferometer signals in accordance with Table 10.1.

Table 10.1

Determining two quadrature signals A and B with a 90 degrees phase from a linear combination of sensor signals
Analog sensor output signals	A	B
0°, 90° phase offsets	S₉₀	S₀
0°, 90°, 180°, 270° phase offsets	S₉₀ − S₂₇₀	S₀ − S₁₈₀
0°, 120°, 240° phase offsets	$\sqrt{3} (S_{120} - S_{240})$ $\sqrt{3} (S_{120} - S_{240})$	−S₂₄₀ + S₀ − S₁₂₀

Figure 10.8 Phase interpolation using direct arctangent calculation and the calculation of displacement.

The error sensitivity for laser power and signal amplitude variations may be improved by an optical design that generates three or more analog signals with known phase offsets and of which the average intensity is invariant for laser power fluctuations (i.e., 4 degrees × 90 degrees phase shifts or 3 degrees × 120 degrees phase shifts).

After the interpolated phase is calculated using the arctangent function, the measured displacement can be obtained by unwrapping the measured phase and by multiplication of the unwrapped phase by the signal pitch P = λ_v/2n (see Fig. 10.8).

For time-critical high-speed systems, the interpolated phase may be obtained through a 2D lookup table (Fig. 10.9) (Hagiwara, 1992).

This lookup table may be corrected for known signal offsets or nonlinearities, which may cause the signal to deviate from an ideal sinusoidal signal.

Some encoders have a digital square wave as an output. This can be derived directly by thresholding the analog sinusoidal signals to obtain a resolution of one-quarter of the signal pitch, or the resolution can be enhanced further using interpolating circuits. Additional phase signals with a phase offset α are created by taking a linear combination of the analog cosine (S₀) and sine (S₉₀) signals

\sin (Δ ϕ + α) = \cos (α) \cdot S_{90} + \sin (α) \cdot S_{0}

$\sin (Δ ϕ + α) = \cos (α) \cdot S_{90} + \sin (α) \cdot S_{0}$

(10.18)

Figure 10.9 (a and b) 2D interpolation using a 2D lookup table with precalculated arctangent values or binary transistor–transistor logic signal values.

In the example in Fig. 10.10(b), two additional phase signals are generated (α = 45 and α = 135 degrees) to obtain a 2× interpolated quadrature signal. Using comparators, the signals with various phase offsets are converted into TTL signals. Then, a logic circuit turns these TTL signals into a quadrature signal with increased frequency to obtain a resolution of one-eighth of the signal period. When more additional phase signals are generated, higher levels of interpolation can be obtained by this method (Benzaid and Bird, 1993; Feng et al., 2013). In heterodyne detection, the measured phase is transmitted as a phase-modulated signal on top of the phase of a reference signal. The phase difference between the beat frequency of the reference signal and the beat frequency of the measurement signals can be detected using a phase lock-in amplifier, which typically converts the signal with the carrier frequency into a homodyne type S₀ and S₉₀ signal. The heterodyne detection method makes the transmitted signal insensitive for DC and gain offset variations of the detector signal for static measurements but does not eliminate periodic errors as caused by polarization mixing or misalignment of optical components.

10.3. Sources of error and compensation methods

The theoretical resolution of the system is determined by the degree of interpolation of the sinusoidal signals to determine the value Δϕ, whereas in practice precision is limited by the quality of the measurement signal. In the preceding description of an ideal interferometer (see Section 10.2.1), it was shown that the displacement could be determined by Eq. (10.8) for a homodyne laser interferometer and by Eq. (10.15) for a heterodyne laser interferometer. From these equations, it can be seen that the accuracy of the calculated displacement depends on the accuracy of the determination of the phase change Δϕ, the vacuum wavelength of light used λ_v and the refractive index of the medium n. Apart from these inherent sources of error, there are numerous other error sources depending on the setup used. The errors in the laser interferometer can be divided into three categories: setup dependent, instrument dependent, and environment dependent. In the following paragraphs, a brief explanation is given of these categories and the contributors.

10.3.1. Setup dependent error sources

Depending on the measurement configuration, several errors might occur; for example, cosine errors, Abbe errors, dead path errors, target uniformity, and mechanical stability.

10.3.1.1. Cosine error

The cosine errors result from an angular misalignment between the interferometers measurement beam and the axis of displacement (see Fig. 10.11).

Because of reduced signal efficiency, ultimately a shorter distance is measured than that of the actual displacement; the measurement wavelength is virtually increased. This can be seen on the right-hand side of Fig. 10.11: the detector plane is determined by the wavefront angle of the returning beam. A mechanical path change of Δl_error is measured by optical path length l₁ + l₂.

Figure 10.10 Signal generation examples. (a) Transistor–transistor logic (TTL) signal generation by thresholding the sinusoidal phase quadrature signals resulting in a resolution of one-quarter of the signal pitch. (b) Phase interpolation using additional phase signals and logic circuits to obtain a frequency-doubled TTL quadrature signal with a resolution of one-eighth of the signal period. XOR, exclusive OR.

Figure 10.11 Schematic representation of a cosine error occurring in an interferometer measurement.

Δ l_{error} = Δ l \cdot \cos θ .

$Δ l_{error} = Δ l \cdot \cos θ .$

(10.19)

10.3.1.2. Abbe error

An Abbe error exists if the axis of measurement does not coincide with the axis of movement, as shown in Fig. 10.12.

In this case, the movement of an object in the atomic force microscope is to be measured. The interferometer measurement axis does not, however, coincide with the axis of interest but has an Abbe arm A as the offset, resulting in an error Δl_error if the stage is tilted

Δ l_{error} = A \cdot \tan (θ) .

$Δ l_{error} = A \cdot \tan (θ) .$

(10.20)

An example of a measurement setup corrected for this Abbe error is shown in Fig. 10.13(a). In the interferometer-based nano–coordinate measuring machine (CMM) configuration, the measurement beams of three plane mirror interferometers are all pointing toward the work point (stylus tip) to eliminate Abbe errors for x, y, and z displacement measurements (Peggs et al., 1999; Ruijl, 2001).

Figure 10.12 Schematic representation of an Abbe error in an atomic force microscope measurement.

Figure 10.13 Ultraprecision machines without Abbe errors, (a) using three plane mirror interferometers, all pointing to the work point (Ruijl, 2001). The system has a thermally stable measurement frame of Invar and Zerodur mirror table. (b) Laser tracker. CMM, coordinate measuring machine.

Another example of a laser interferometer that adheres to the Abbe principle is the laser tracker shown in Fig. 10.13(b), where the laser beam automatically follows a reflector attached to the probe of a CMM or to the tool head of a production machine (Schwenke et al., 2009).

10.3.1.3. Dead path error

Dead path errors are errors caused by uncompensated path length of the interferometer as shown in Fig. 10.14, if L_m represents the smallest distance to the measurement target and L_r is the fixed reference arm length, then the dead path is represented by L_m − L_r. Because of environmental changes, the optical path length in the dead path changes and creates an error

Δ l = Δ n \cdot (L_{m} - L_{r}) .

$Δ l = Δ n \cdot (L_{m} - L_{r}) .$

(10.21)

Figure 10.14 Schematic representation of dead path error (L_m − L_r) (left) and target uniformity (right).

10.3.1.4. Target uniformity

If the position of the measurement beam shifts over the measurement target, target uniformity also plays a role. In a situation where a flat mirror is used with flatness λ/20, without calibration this error can contribute to an error of 32 nm (λ = 633 nm).

10.3.1.5. Mechanical stability

As an interferometer will only measure the position of the reflecting target, the stability of the measurement target also contributes to the uncertainty budget of the system. As a result, the stiffness and thermal stability of the target and intermediate body should also be taken into account.

10.3.2. Instrument dependent error sources

Based on the choice of the instrument, several instrument-dependent errors add to the uncertainty of the measurement as there are (split) frequency stability, beam walk-off, electronics, data age, data age jitter, periodic deviation, and ghost reflection and stray light.

10.3.2.1. (Split) frequency

In interferometers, the measurement reference is the wavelength of the light, which is used, as can be seen in Eqs. (10.8) and (10.15). From these equations it becomes clear that, with a frequency change of the interferometer source, a measurement uncertainty is introduced. Usually, the relative stabilities of the laser sources are in the range of 2 × 10⁻⁹ and 5 × 10⁻⁸.

It can be shown that for heterodyne interferometers not only the absolute frequency stability is of importance but also the stability of the split frequency as shown in the following equation:

Δ l = \frac{1}{n} \frac{(L_{m s} - L_{r s})}{p} \frac{f_{s}}{f_{m}} + (L_{m} - L_{r}) \frac{f_{a}}{f_{m}} with f_{a} = \frac{f_{m} + f_{r}}{2}, f_{s} = f_{m} - f_{r}

$Δ l = \frac{1}{n} \frac{(L_{m s} - L_{r s})}{p} \frac{f_{s}}{f_{m}} + (L_{m} - L_{r}) \frac{f_{a}}{f_{m}} with f_{a} = \frac{f_{m} + f_{r}}{2}, f_{s} = f_{m} - f_{r}$

(10.22)

where L_ms represents the optical path of the entire measurement signal and L_m represents the optical path of the measurement arm as shown in Fig. 10.15, f_m and f_r represent the frequencies nominally in the reference arm and in the measurement arm, and p represents the optical resolution (number of passes) of the interferometer optics.

10.3.2.2. Beam walk-off

Beam walk-off is the effect of a rotation of the moving target outside its plane in combination with a different path length of reference and measurement arm (see also Fig. 10.11). As a result, a phase error is generated and the wavefront of the reference arm and measurement arm no longer completely coincide over the full measurement range. Consequentially, the signal-to-noise ratio deteriorates and eventually the measurement range is limited by this effect. This especially holds for stage rotation of plane mirror interferometers where the amount of walk-off scales with

Figure 10.15 Schematic representation of relevant beam parts for frequency stability effects.

WO = 4 ϕ \cdot Δ l .

$WO = 4 ϕ \cdot Δ l .$

(10.23)

10.3.2.3. Electronics and data age

Typically the noise and linearity of the electronics will contribute to the uncertainty of the measurement. The actual contribution in uncertainty depends on the optical resolution of the interferometer (number of passes to the measurement target).

Depending on the velocity (v), the aging of the measurement data (δ) may become a critical part in the measurement error

Δ l = δ v .

$Δ l = δ v .$

(10.24)

For a movement at a speed of 1 m/s, a data delay of 10 ns will generate a direction-dependent offset of 10 nm. The variation of the data age (jitter) contributes to the uncertainty of the measurement.

10.3.2.4. Periodic deviation

As displacement interferometers are often based on polarizing optics, the mixing of polarization will introduce periodic deviations in the measurement as shown in Fig. 10.16.

These periodic deviations are also often referred to as nonlinearities. Quenelle (1983) first mentioned the existence of periodic deviations as a result of alignment errors between the laser and the optics, Bobroff (1987) showed the existence of periodic deviations as a result of alignment of the polarizing beam splitter together with the quality of the coating. These errors had a frequency of one cycle per one wavelength optical path change. In the same year, Sutton (1987) proved the existence of a periodic deviation with two cycles per wavelength optical path change. First- and second-order periodic deviations were defined at this time. These deviations were comprehensively modeled by Cosijns et al. (2002). Basically, periodic deviations depend on the polarization quality and birefringence of the optics as well as the antireflection coating. The effects can be reduced in a setup by aligning the optical source to the polarizing optics.

Figure 10.16 Schematic representation of measurement results with a periodic deviation. The magnitude is exaggerated for clarity.

Commercial interferometers may use the compensation methods mentioned in Section 10.2.3 to compensate for these periodic deviations as they do show up as harmonics on the measurement signal (Heydemann, 1981; Chu and Ray, 2004). On the other hand, there are also interferometer configurations, which minimize the periodic deviations by splitting the reference and measurement paths completely (Wu et al., 1999; Joo et al., 2009, 2010; Kim et al., 2010; Meskers et al., 2014).

10.3.3. Environment dependent error sources

Thermal influence on the interferometer is relevant in this category, as well as the refractive index of the medium through which the light travels (mostly air).

10.3.3.1. Thermal effects on the interferometer

Thermal expansion of the interferometer optics or mechanics may cause unwanted effects in the measurements. Depending on the setup, these may introduce drift or pointing effects in the measurement.

10.3.3.2. Refractive index of air

As the wavelength of the system defines the length reference, the stability of the refractive index of the medium through which the measurement occurs determines the accuracy of the measurement. Usually this effect is compensated for using Edlén's updated equation (Edlén, 1966; Birch and Downs, 1993; Bönsch and Potulski, 1998; Ciddor, 1996, 2002) and measuring temperature, pressure, humidity, and the carbon dioxide component. The individual effects are shown in Table 10.2.

Table 10.2

Sensitivity of the refractive index of air
Parameter	Unit	Effect in n Birch and Downs	Effect in n Bönsch and Potulski	Typical daily variation (in measurement laboratory)	Effect in n
Temperature	K⁻¹	−9.298 × 10⁻⁷	−9.299 × 10⁻⁷	0.1°C	−9.3 × 10⁻⁸
Pressure	Pa⁻¹	2.684 × 10⁻⁹	2.683 × 10⁻⁹	20 hPa	5.36 × 10⁻⁶
Humidity	Pa⁻¹	−3.63 × 10⁻¹⁰	−3.706 × 10⁻¹⁰	1 hPa	−3.66 × 10⁻⁸
Carbon dioxide	ppm⁻¹	0	1.447 × 10⁻¹⁰	100 ppm	1.45 × 10⁻⁸

The relative uncertainty of this compensation method is usually limited to 2 × 10⁻⁸, resulting in a measurement uncertainty of 20 nm/m. Another way to compensate for refractive index effects is by means of a refractometer. In such a system, a reference of vacuum is measured as well as the medium through which the interferometer is measuring in a mechanically stable setup with equal length (Schellekens et al., 1986). In some applications, compensation of the refractive index variations is sufficient, and a wavelength tracker can be used. This tracker usually consists of an extra interferometer measurement with a mechanically stable path length (generally constructed of a low expansion material), which is representative for the nominal measurement length.

10.4. Optical encoders

The existing linear encoders encompass various detection techniques, based on brush, magnetic, inductive, capacitive, and optical principle. Optical encoders are popular because of their noncontact, high-resolution measurement characteristics and because their output signal is easily converted into an electronic position or displacement signal. The range of applications of optical encoders spans from measurement of angular or linear motion in low-cost consumer products (such as computer mice and inkjet printers) to advanced industrial systems (such as machine tools, robots, and stage positioning in lithographic projection tools). An optical encoder contains a detection system that can detect light emanating from a patterned movable member. The patterned member is usually referred to as the “optical scale.” For a linear encoder, the scale typically contains one or more sets of parallel lines with a predetermined constant or varying pitch. Rotary encoders, also known as “shaft encoders,” have radial patterns. The detected light is converted into an electrical signal and may be processed as digital square waves or as analog signals, where analog signals allow for resolution enhancement by means of interpolation.

There are two types of optical encoders: absolute position and incremental displacement encoders. Most common absolute position encoders use gray-coded or pseudorandom coded scales, or scales with subsets of scales with a different pitch (see Fig. 10.17).

The absolute encoders can be used in both linear and rotary configurations. Gray-coded scales have parallel measurement channels for measuring the bits of the gray-coded position information. A pseudorandom coded scale has a unique single track stripe pattern for each absolute position. This unique stripe pattern is read by a line camera and decoded by a microprocessor (Gribble and Robert, 2011). Absolute encoders may be obtained by the combination of phase information of two or more incremental encoders, which have a different pitch and are integrated on the same scale. In the example in Fig. 10.17(c), the phase differences between the measured phases provide coarse absolute position information, which is refined by the interpolated phase of one of the phase quadrature signals. Absolute scales with different pitches are also available as hybrid scales where the coarse absolute position is derived from subsets of capacitance-based scale signals, which are then refined by the signal from a fine pitch incremental optical scale (Mitutoyo Catalog, 2016).

Figure 10.17 Schematic representation of absolute linear encoder scales. (a) Gray code, (b) pseudorandom code, (c) parallel readout of subsets of scales with a different pitch, (d) example of an absolute gray-coded optical rotary encoder disk.

The most commonly used encoders are of the incremental type, which are relatively easy to interface and typically obtain better speed and resolution performance than absolute encoders.

Incremental displacement encoders usually are supplemented with additional reference marks. The measurement of one or multiple uniquely spaced reference marks incorporated in the scale can be used to construct an absolute position signal, see Fig. 10.18.

If directional information is required, phase quadrature detection is used where at least two (typically 90 degrees) out-of-phase signals are generated. Linear encoders may use transmissive glass scales or reflective scales, employing amplitude gratings or phase gratings to suppress undesired diffraction orders. Scale materials include chrome on glass, metal (stainless steel, gold-plated steel, Invar), ceramics (Zerodur, ultra low expansion glass, Clearceram), and plastics. The scale may be self-supporting, mounted via an adhesive film on the backside of the scale, or mounted in a carrier, which allows the scale to expand freely at both ends to ensure a defined thermal behavior.

Figure 10.18 Incremental encoders with incorporated zeroing reference mark: (a) linear scale, (b) rotary encoder disk.

In this chapter, the focus will be on incremental linear encoders as these are the most commonly applied type of optical encoder for high-precision applications. The imaging and interferential grating encoders are the two most common incremental encoder types that create an electrical signal from the relative movement between a scale grating and a reading head.

10.4.1. Imaging incremental encoder

In imaging encoders, light is typically passed through two periodic patterns forming a moiré pattern. Light is periodically obscured at a photo detector when one pattern is moved relative to the other (see Fig. 10.19). The resolution of optical encoders scales with the grating period.

An imaging incremental encoder consists of a light source generating a collimated light beam and a stationary index grating g_i, which transmits 50% of the light. This transmitted light is imaged onto a moving scale grating g_s either via imaging optics (Fig. 10.20(a)) or via shadow projection (Fig. 10.20(b)).

Note that the encoder detects relative movement between the index grating and the scale grating. It is therefore not important which grating is identified as the index grating and which the scale grating. If the index grating is imaged perfectly on top of the transmissive section of the scale grating, then all light is transmitted by the scale grating. If the moving grating is displaced by half a pitch, all light will be obscured by the opaque part of the moving grating. The light that is passed by the scale grating forms a moiré pattern that is detected by a photodetector, which produces an electrical signal that is proportional to the light intensity.

Figure 10.19 Imaging Moiré encoder: (a) triangular detector signal, (b) sinusoidal detector signal.

Figure 10.20 Imaging incremental encoder with source S, detector D, collimator lens L. The index grating g_i is imaged onto a scale grating g_s: (a) using an imaging lens L_imag, (b) Close contact shadow projection, (c) using an imaging grating g_imag and adopting the Talbot effect, and (d) Fresnel diffraction.

A configuration with an imaging lens allows operation within the depth of field of the imaging optics. A light source, preferably monochromatic, is used to limit the effect of color dispersion. Color dispersion can be avoided when using an imaging grating with a pitch p/2 positioned at one-quarter of the Talbot distance Z_T (see Fig. 10.21) (Patorski, 1986; Dürschmid, 1993).

The Talbot effect is a near-field diffraction effect. Based on Fresnel diffraction, the Talbot effect causes an image of a periodic grating to be reproduced at the Talbot distance Z_T = 2p²/λ, where p is the grating pitch and λ is the wavelength of the light incident on the grating. At one-quarter of the Talbot length, the self-image is halved in size and appears with half the period of the imaged grating. If in Fig. 10.20(c) the imaging grating at Z_T/4 is used as the moving grating, then the signal period at the detector is also reduced by a factor of 2. If both the static index grating and the moving scale grating have a square pattern and the static grating is imaged sharply onto the moving grating, then the detected intensity will vary triangularly with displacement.

Figure 10.21 Talbot effect: self-imaging of a periodic structure that is illuminated by a collimated coherent beam.

Figure 10.22 Encoder with four-phase zones with 90 degrees phase offsets. For signal robustness, the phase quadrature signal is obtained by taking the difference signals S₀ − S₁₈₀ and S₉₀ − S₂₇₀.

For accurate detection using phase quadrature interpolation, the optical system should be designed to generate a sinusoidal signal. The captured diffraction orders of the imaging system, the spectral width, the size and shape of the source, and the distance between the gratings affect the sharpness of the imaged grating. With a proper encoder design a grating can be imaged as sinusoidal image such that, effectively, a sinusoidal modulation of the detector signal is obtained (Dürschmid, 1993). To obtain various phase signals in parallel from the same scale, the reticle mask is equipped with multiple grating areas that have predefined offsets. In the case of an incremental encoder, a second head would read from the same moving grating but be displaced by n + 1/4∙pitch (n ∈

ℕ

$ℕ$

) to produce a second signal with a phase offset of 90 degrees. In practice, often four-phase quadrature detectors signals are generated. By taking the difference of signals that have an opposite phase (S₀ − S₁₈₀) and (S₉₀ − S₂₇₀) the sensitivity for transmission losses of the electrical signal, causing DC offset variations, can be reduced (see Fig. 10.22).

As an alternative to using a single detector for each phase zone, the grating pitch of the index grating may be designed to have a pitch that is slightly different from the pitch of the scale grating. A segmented detector with binned pixels or a line camera is used to capture the moiré pattern that is formed by the light that has interacted with both the index grating and the scale grating (Fig. 10.23).

This encoder mask layout also generates directional information and has an improved robustness against local obscuration by dirt spots because the effect of dirt is more equally distributed over all phase signals.

Figure 10.23 Optical encoder with an index grating and a scale grating with a small pitch difference.

Figure 10.24 (a) Transmission type encoder, (b) reflective type encoder, (c) front view, indicating phase zones.

Note that imaging encoders may be designed in transmission (Fig. 10.24(a)) or reflection (Fig. 10.24(b)). Transmission scales are typically etched chrome on glass gratings, whereas reflective scales often are gold on metal. Reflective scales are often preferred as they are relatively easy to integrate into a mechanical design. Especially for long-stroke actuation, the reflective scales can be easily mounted to any rigid surface and do not need to be self-supporting.

10.4.2. Interferential encoders

The resolution of an optical encoder scales directly with the scale period. With a scale period smaller than 10 μm—for example as small as 0.5 μm—diffraction effects become dominant and can be exploited in interferential encoders, also known as “grating interferometers,” to achieve the highest levels of precision. The detector signals are sinusoidal with grating displacement and largely free of harmonics. With proper electronics these signals can be interpolated more than 1000 times, practically being limited by the signal noise.

The optical path lengths of the grating interferometer arms are matched and short compared with those of displacement interferometers. This is to reduce the sensitivity to wavelength variations as caused by refractive index variations or source frequency variations.

Interferential grating encoders use coherent light-emitting diodes or semiconductor lasers to generate a coherent and collimated light beam. In the example in Fig. 10.25, the light beam is split into diffraction orders (m) by the scale grating with a pitch p.

Figure 10.25 Basic principle of interferential grating encoder. (a) Direct detection of the spatial phase of a spatial interference pattern of the +1 and −1 diffracted order, (b) detection of the interference signal after both interferometer arms are made parallel.

Typically the −1 and +1 diffraction orders are selected to form the interferometer arms. When the +1 and the −1 diffracted orders are combined to interfere at a photodetector, a sinusoidal signal with a signal period of half the grating pitch is created:

Δ ϕ (Δ x) = Δ ϕ_{m = + 1} - Δ ϕ_{m = - 1} = 2 π (+ 1) \frac{Δ x}{p} - 2 x (- 1) \frac{Δ x}{p} = 4 x \frac{Δ x}{p} .

$Δ ϕ (Δ x) = Δ ϕ_{m = + 1} - Δ ϕ_{m = - 1} = 2 π (+ 1) \frac{Δ x}{p} - 2 x (- 1) \frac{Δ x}{p} = 4 x \frac{Δ x}{p} .$

(10.25)

As a result, for in-plane displacements the signal period is directly related to the grating pitch of the scale and not related to the wavelength of the light source.

In Fig. 10.25, two schematic representations of interferential encoders in their simplest form are presented. In Fig. 10.26(a), the beams of the +1 and −1 diffracted orders are made to interfere when recombined at a photodetector to form a spatial sinusoidal pattern. The period of the spatial interference pattern is equal to half the grating pitch, and the lines are parallel to the angle bisector of the angle between the two diffracted orders. In Fig. 10.26(b), the beams of the +1 and −1 diffracted orders are made parallel such that no spatial pattern emerges and the signal that varies periodically with grating displacement can be detected by a single photosensitive detector.

Figure 10.26 (a–c) Light diffraction of a coherent light beam by an amplitude grating resulting in multiple diffracted beams.

10.4.2.1. Diffraction physics

In Fig. 10.26, the interaction between a periodic grating structure and a coherent light beam is shown. The Huygens–Fresnel principle is used for a simplified understanding when a beam that is diffracted by a grating may be thought of as an array of coherent point sources, each emitting a spherical wave. In specific propagation directions, the spherical waves from all point sources interfere constructively and will propagate as a plane wavefront. This diffracted wavefront is indicated graphically by the straight lines that connect the spherical wavefronts of neighboring slits.

For constructive interference to occur, the optical path length from an input wavefront to an output wavefront must differ by integer multiples of λ for adjacent slits of the scale grating (see Fig. 10.27). This integer multiple m is called the order of diffraction.

The relationship as visualized in Fig. 10.27 between the entrance angle θ_i and the output angle θ_m of a diffracted beam of order m, wavelength λ, and grating pitch p is then described by the general grating equation:

\sin (θ_{i}) + \sin (θ_{m}) = \frac{m λ}{p}

$\sin (θ_{i}) + \sin (θ_{m}) = \frac{m λ}{p}$

(10.26)

When the input beam is perpendicular to the grating (θ_i = 0), this equation simplifies to

\sin (θ_{m}) = \frac{m λ}{π} .

$\sin (θ_{m}) = \frac{m λ}{π} .$

(10.27)

The grating equation applies for diffraction by both transmissive and reflective gratings as can be seen in the examples in Fig. 10.28.

Diffraction at the Littrow angle is considered a special case where, for a reflected beam, the diffracted beam travels in the direction opposite to the input beam.

Figure 10.28 Reflective (a) and transmissive (b) diffraction angles according to the general grating equation. Left: General. Center: Normal incidence (θ_i = 0). Right: Littrow angle (sin(θ_m)) = (sin(θ_i)).

10.4.2.2. Sensitivities

To understand the phase shift sensitivity as a function of the diffraction angles, let us consider the situation of Fig. 10.29. Here, the optical path length differences are plotted for a diffracted beam under the influence of an in-plane and an out-of-plane displacement.

By observing the change in optical path length from an input wavefront to an output wavefront for different grating positions, the phase sensitivity for both the in-plane displacement (Δx in Fig. 10.29) and out-of-plane (Δz in Fig. 10.29) displacement of the grating can be derived geometrically. The beams depicted by the solid line and the dashed line in Fig. 10.29 are of equal length.

Using the length difference of the entrance beam (Δx sin(θ_i)) and the length difference of the reflected beam (Δx sin(θ_m)), the optical path length difference can be calculated

\frac{Δ OPL}{Δ x} = (\sin (θ_{t}) + \sin (θ_{m}))

$\frac{Δ OPL}{Δ x} = (\sin (θ_{t}) + \sin (θ_{m}))$

(10.28)

\frac{Δ OPL}{Δ z} = (\cos (θ_{t}) + \cos (θ_{m})) .

$\frac{Δ OPL}{Δ z} = (\cos (θ_{t}) + \cos (θ_{m})) .$

(10.29)

As, finally, a phase difference between two interferometer arms is measured, the sensitivities for the diffraction on a moving scale are better expressed in phase terms

\frac{d ϕ}{d x} = \frac{2 π}{λ} \cdot \frac{Δ OPL}{Δ x} = \frac{2 π}{λ} (\sin (θ_{t}) + \sin (θ_{m})) = 2 π \frac{m}{p}

$\frac{d ϕ}{d x} = \frac{2 π}{λ} \cdot \frac{Δ OPL}{Δ x} = \frac{2 π}{λ} (\sin (θ_{t}) + \sin (θ_{m})) = 2 π \frac{m}{p}$

(10.30)

\frac{d ϕ}{d z} = \frac{2 π}{λ} \cdot \frac{Δ OPL}{Δ z} = \frac{2 π}{λ} (\cos (θ_{t}) + \cos (θ_{m})) .

$\frac{d ϕ}{d z} = \frac{2 π}{λ} \cdot \frac{Δ OPL}{Δ z} = \frac{2 π}{λ} (\cos (θ_{t}) + \cos (θ_{m})) .$

(10.31)

The resulting sensitivity vector for a single diffracted beam can be defined in complex notation as

Figure 10.29 Phase sensitivity of a diffracted wavefront. For (a) in-plane grating displacement and (b) out-of-plane grating displacement.

\bar{S} = \frac{d ϕ}{d x} + i \frac{d ϕ}{d z} .

$\bar{S} = \frac{d ϕ}{d x} + i \frac{d ϕ}{d z} .$

(10.32)

The sensitivity vector of an interferential encoder sensor is obtained by taking the difference between the sensitivity vectors of each interferometer arm. Examples of a sensitivity vector for reflective and transmissive scale interactions at various output angles are shown in Figs. 10.30 and 10.31.

In the case of a multipass encoder, the sensitivity for each interferometer arm is obtained by summing the sensitivities of every interaction between the beam in the interferometer arm and the moving scale grating. It can be seen in Eq. (10.30) that the in-plane sensitivity dϕ/dx only depends on the scale pitch and that there is no wavelength dependency. The minimum z-sensitivity of an interferometer arm for out-of-plane displacements (Eq. 10.32) is found for transmissive diffraction at the largest possible Littrow angle. A large Littrow angle is obtained for a long wavelength and/or a small scale pitch (see Figs. 10.31(a) and 10.32(b)).

A symmetric design of the optical paths of the interferometer arms will balance the sensitivity for out-of-plane displacements, whereas the sensitivity for in-plane grating displacement is enhanced by interfering beams of opposite diffraction orders.

Figure 10.30 Example of the sensitivity vectors of a reflective scale grating. For the diffraction orders (a) m = −1, (b) m = 0, (c) m = +1.

Figure 10.31 Example of the (normalized) sensitivity vectors of a transmissive scale grating. For the diffraction orders (a) m = −1, (b) m = 0, (c) m = +1.

10.4.2.3. Schematic setups

Many different designs of high-resolution optical encoders can be conceived. Figs. 10.32 and 10.33 show some typical configurations. The ratio between the pitch of the grating scale and the signal period depends on the selected diffraction orders of the interfering beams and the number of times a beam has been diffracted by the scale grating. If the light beams of the interferometer arms are formed by the +1 and −1 diffraction orders and only diffracted once by the scale grating, then the resulting signal period will be half the grating pitch. In case of a double-pass grating interferometer, the signal pitch will be reduced to one-quarter of the grating pitch. In most designs a phase grating is used instead of an amplitude grating, such that the zeroth order is suppressed and the optical power is directed toward the +1 and −1 diffraction orders.

Figure 10.32 Schematic presentation of transmissive diffraction encoders and the sensitivities. (a) Basic, (b) Littrow configuration for small pitch grating, (c) double-pass for increased resolution. PBS, polarizing beam splitter.

In Fig. 10.32(a), a most basic interferential grating encoder is shown. A coherent beam is diffracted into a +1 and −1 diffraction order by a transmissive phase grating, forming the two interferometer arms with opposite phase offsets as a function of grating displacement. When the two interferometer arms are combined, a sinusoidal interference signal as a function of grating displacement is obtained with a period of half the grating pitch. For the design of a high-resolution optical encoder, it is beneficial to have a high in-plane sensitivity, which is obtained by choosing a small grating pitch. At the same time, the out-of-plane sensitivity is preferably reduced by choosing large diffraction angles and the longest possible wavelength. These benefits are combined in the example in Fig. 10.32(b), where the beams are diffracted under the Littrow angle (sin(θ_i) = sin(θ_m)). The example in Fig. 10.32(c) shows a schematic presentation of a double-pass transmission grating encoder. Here, the phase shift, which is introduced when the beams are first diffracted by the moving scale grating, is doubled during the diffraction that occurs in the second pass. With one interferometer arm receiving a positive phase shift twice and the other interferometer arm receiving a negative phase shift twice, a total of four-phase cycles are obtained for every grating displacement equal to the grating pitch.

Figure 10.33 Schematic presentation of interferential encoders using reflective scales. (a) Basic, (b) Littrow configuration for small pitch grating, (c) double-pass for increased resolution.

The interferential encoder concepts in Fig. 10.32, using transmissive scale gratings, are depicted as reflective interferential grating encoders in Fig. 10.33. The cat's eye retroreflection in the schematic representation of a double-pass reflective encoder layout of Fig. 10.33(c) improves tolerances for grating tilts and makes the optical path length from the source to the detector constant across the beam, lowering the requirements on the source coherence.

10.4.2.4. Phase detection

In most cases the encoder will be designed such that the interferometer arms have an orthogonal polarization state, and the detector can be formed by a homodyne phase analyzer to determine the phase difference between two interferometer arms (Fig. 10.34).

Figure 10.34 Homodyne phase detector. For creating phase quadrature signals representing the phase difference (Δϕ = ϕ_A − ϕ_B) between the orthogonally polarized interferometer arms of the encoder: (a) for two-phase signals with 90 degrees offsets, the measured phase can be read out in a 2D Lissajous plot, (b) for three-phase signals with 120 degrees phase offsets the measured phase can be read out in a 2D projection of a 3D Lissajous plot.

In a typical homodyne phase analyzer, the polarization states of the interferometer arms are made circular and counterrotating using a quarter-wave retarder plate oriented at an angle of 45 degrees. The sum of the two circularly polarized beams is then a linearly polarized beam with a polarization angle that is equal to half the phase difference Δϕ/2. The phase quadrature signals (from which Δϕ is derived using an arctangent calculation) are obtained by splitting the linearly polarized beam and analyze the intensity after passing the light through analyzing polarizers with different angular alignment. By splitting the beam and passing each beam through a polarizer at 0 and 45 degrees, two-phase quadrature signals are generated with a relative phase offset of 90 degrees. When plotted against each other, the measured phase Δϕ can be read out in a 2D Lissajous plot. The beam may be split in three to generate phase signals with an offset of 0, 120, and 240 degrees, allowing for a more robust phase calculation and compensation of periodic errors. For three-phase signals with 120 degrees phase offsets, the measured phase can be read out in a 2D projection of a 3D Lissajous plot (see Fig. 10.34).

10.4.2.5. Tilt sensitivity

For a robust design, the tilt sensitivity must be designed such that both interferometer arms are parallel at the detector. To reduce the sensitivity to wavefront deviations, the design should also prevent a large beam walk-off between the interfering arms. To allow the use of a short coherence source, the optical paths from the source to the detector should be identical across the beam diameter. As can be seen in Fig. 10.35, typically good design characteristics can be obtained when using grating beam splitters and grating-based beam deflectors.

10.4.2.6. Practical example

Fig. 10.36(a) shows a double-pass interferential grating encoder. Here, a collimated linearly polarized beam is delivered via a polarization maintaining single-mode fiber and directed toward the grating scale.

The grating scale is a reflective phase grating with a groove depth of λ/4, designed to suppress the zeroth diffraction order while boosting the intensities of the first-order diffracted beams. The beam is diffracted by the grating scale into a positive and a negative diffraction angle (m = +1 and m = −1), giving each refracted beam an opposite phase shift (m·2πΔx/p). Each of the interferometer arms is diffracted toward a Porro prism by a static grating, which gives the beam a shift in y-direction. The linear polarization states of the beams in the interferometer arms are then made circular and counterrotating using λ/4 retarder plates with an angular alignment of −45 and +45 degrees. On return, the static phase grating directs the beam toward the scale grating where the beams of both interferometer arms are recombined to receive an additional phase shift on diffraction by the grating scale. The phase difference between the interferometer arms (Δϕ) is now equal to 8πΔx/p, meaning that four-phase cycles are detected for a grating displacement Δx that is equal to the scale pitch p. With the polarization state of the interferometer arms being circular and counterrotating, the phase difference between the interferometer arms is detected using a homodyne phase analyzer as depicted in Fig. 10.34.

Figure 10.35 (a) Design requiring spatially coherent source and with low tilt tolerance, (b) improved design.

10.5. Design considerations

Highly stable optical encoders require a thermally and dynamically stable design as well as high-quality optics and electronics. The design of an encoder should reduce the sensitivity to manufacturing and mounting tolerances, as well as reduce the influence of environmental conditions such as temperature gradients and temperature offsets. Errors should be counterbalanced by design where possible.

10.5.1. Stability

The optical path lengths of the grating interferometer arms are matched and short compared with those of displacement interferometers; this is to reduce the sensitivity to wavelength variations as caused by refractive index variations or source frequency variations. For applications requiring sub-nm stability, a thermally stable design is obtained by using compact and symmetric design with a short distance between the scale and the reading head and by using low-thermal expansion materials. The light source and detector signals may be fiber-coupled to reduce the heat dissipation in the encoder head. This is typically important in vacuum applications where heat cannot be dissipated by means of natural convection. If heat is dissipated inside the sensor, there should be a good thermal interface with its surrounding. For a deterministic behavior, the reading head should be mounted with kinematic mounts and the scale should be allowed to expand freely to make sure that no forces are exerted on the scale as a result of thermal expansion of the carrier material (Breyer et al., 1991).

Figure 10.36 (a and b) Double-pass interferential grating encoder with a signal period of one-quarter of the grating period (Holzapfel and Linnemann, 2006).

10.5.2. Grating scale errors

Measurement uncertainty is further determined by the manufacturing uncertainty of the grating scale. Low- and mid-frequency deviations must be calibrated against an interferometer in vacuum or in air (Sawabe et al., 2004; Holzapfel, 2008; Flügge et al., 2014). When measured in air the refractive index variations due to pressure, temperature, and humidity variations are monitored and compensated for using the Edlén equation (Edlén, 1966; Birch and Downs, 1993; Bönsch and Potulski, 1998; Ciddor, 1996, 2002).

10.5.3. Periodic errors

Where imaging encoders must be carefully designed to obtain a purely sinusoidal pattern, the scanning signals of interferential encoders are always sinusoidal and largely free of harmonics. Periodic errors (sometimes referred to as “cyclic errors”) may still be present as a result of variations in light intensity, polarization mixing of nonideal optics, mechanical misalignment of components in the reading head, and electronic gain errors. Most of the periodic deviations may be compensated for by a phase error lookup table or Heydemann correction (Heydemann, 1981). However, there are some factors that are difficult to correct for fully because the periodic deviations may be speed, position (scale alignment errors/dirt on the scale), and time (laser power variations, mechanical stability) dependent. With proper electronics the encoder signals can be interpolated as much as a few thousand times, practically being limited by signal noise. The impact of periodic errors may be reduced by adopting inline software correction and phase gratings to suppress the effect of unused diffraction orders.

10.5.4. Abbe correction

According to Abbe's principle, the measured point on the object and the effective measuring point of the encoder should be aligned along the measuring direction. An example of an encoder-based nano-CMM working according to the Abbe principle in the x,y plane is presented in Fig. 10.37 (Vermeulen, 1999; Vermeulen et al., 1998; Van Seggelen, 2007; Van Seggelen et al., 2005).

Here, the measurement directions of the x and y scales are aligned to intersect at the work point. Rather than moving orthogonally and independently of each other, as is the case for most CMMs, the x and y axes are connected together at right angles and move as a single unit. To follow the motion of the unit in the direction orthogonal to the measuring directions of the encoder reading heads, the reading head of the x-encoder is mounted on an intermediate body (y-slider) and the y-encoder is mounted on an x-slider. Here, both encoder scales are aligned to intersect at the work point, where in this case a tactile stylus probe is mounted, thus eliminating Abbe errors in the 2D plane.

Figure 10.37 Encoder-based system with scales oriented to be collinear with the work point to eliminate Abbe errors (Vermeulen, 1999; Vermeulen et al., 1998; Van Seggelen, 2007; Van Seggelen et al., 2005).

Figure 10.38 (a) x, y, rz encoder configuration with a thermal center. 3DOF scale arrangement with grating lines pointing toward the work point in the thermal center to reduce sensitivity for thermal expansion; additional rz measurement allows for a mathematical Abbe offset compensation. (b) Redundant x, y, rz systems allowing for a mathematical Abbe offset and thermal expansion compensation (Kuhn, 2010).

In configurations that do not strictly adhere to the Abbe principle of alignment, the Abbe error may be reduced to a practical minimum by, for example, a phantom scale arrangement or a cross-grid scale. With a phantom scale arrangement, additional stage rotation is measured using another scale with a different Abbe offset, such that the Abbe error as induced by stage rotation can be determined and compensated for mathematically (see also Fig. 10.38).

Alternatively, a 2DOF (two degrees of freedom) grid encoder uses a cross-grid and may be used to measure close to the work point to minimize the Abbe errors. In an Eppenstein configuration the Abbe error is compensated automatically. This is applied to a measuring microscope by Haitjema (1996).

10.5.5. Thermal expansion

If the effect of thermal expansion cannot be ignored and it is not possible or impractical to position an encoder directly at the work point and thermal center, then a single 6DOF sensor positioned outside the thermal center will suffer from Abbe measurement errors, and no correction for thermal expansion between the work point and the measurement positions can be applied. It may then be necessary to add measurement redundancy to be able to measure and correct for thermal expansion effects.

Also, multiple encoders may be used in a scale layout such that the sensitivity to thermal expansion effects is minimized, as is depicted in the example of Fig. 10.38(a). Here, a scale configuration is shown for measuring the x-displacement and guide errors in y and rz direction. The rz measurement allows for Abbe arm correction at the work point. For each encoder, the grating lines of the scale are pointing toward the thermal center of the stage, such that the measured position in the thermal center position is not affected by uniform thermal expansion of the stage. The combined measurement sensitivity for each degree of freedom may be tuned by choosing appropriate grid orientations and encoder readhead positions. In Fig. 10.38(b), x, y, and rz can also be measured and corrected for Abbe offsets. Here, the thermal expansion of the scale can be deduced from the difference in the redundant y-measurements and corrected for mathematically. Note that with the use of 2DOF “xz” encoders like these are presented in Fig. 10.40, the configuration of Fig. 10.38 may be used to measure 6 degrees of freedom of a moving stage.

10.5.6. Multiaxis encoder systems

Complete 6DOF stage position measurements using optical encoders is obtained by combining multiple measurement axes in a single encoder package (Sandoz, 2005; Fan et al., 2008) and/or by combining multiple separate optical encoder systems, which are strategically positioned to obtain desired measurement sensitivity for each measured degree of freedom. Sensor and scale arrangements may be chosen such that potential error sources as thermal expansion or guiding errors do not affect the measurement or can be separated from the measurement signal mathematically by using redundant measurement axes.

An xy encoder may have a scale with separate zones for x and y gratings. In Fig. 10.39(a), a long-stroke main grating is intended for displacement measurement, whereas the short stroke can be used for direct compensation of linear guide straightness. Yaw measurement becomes possible with an additional x-encoders positioned at a different y-position. For 2D contouring, a 2D cross-grating and an integrated xy encoder may be used (Fig. 10.39(b)). The cross-grid grating plate generates diffraction orders in both x origin and y directions (Fan et al., 2008).

Figure 10.39 (a) Separate x and y encoder (Kuhn, 2010), (b) xy cross-grid encoder with x and y-encoder enclosed in a single housing (Huber et al., 1993).

When an encoder is designed to be placed under an angle it becomes sensitive for both in-plane and out-of-plane motion (see Figs. 10.40 and 10.41). The encoder signals E₁(x, z) and E₂(x, z) of two encoders with an opposite tilt are used to obtain an in-plane and out-of-plane displacement signal.

Given the wavelength λ, the grating pitch p and the input angles of the diffracted beams, the encoder signals E₁(x, z) and E₂(x, z) can be derived from the phase shift sensitivities of each of the four interferometer arms with relative phases (ϕ_A, ϕ_B, ϕ_C, ϕ_D) by using the grating Eq. (10.2) and the derived sensitivity Eqs. (10.6) and (10.7)

\begin{matrix} E_{1} (x, z) = ϕ_{A} (x, z) - ϕ_{B} (x, z) = \frac{4 π}{p} \cdot Δ x - \frac{2 π}{λ} (\cos (θ_{m θ_{A}}) - \cos (θ_{m θ_{B}})) \cdot Δ z \\ E_{2} (x, z) = ϕ_{C} (x, z) - ϕ_{D} (x, z) = \frac{4 π}{p} \cdot Δ x + \frac{2 π}{λ} (\cos (θ_{m θ_{D}}) - \cos (θ_{m θ_{D}})) \cdot Δ z . \end{matrix}

$\begin{matrix} E_{1} (x, z) = ϕ_{A} (x, z) - ϕ_{B} (x, z) = \frac{4 π}{p} \cdot Δ x - \frac{2 π}{λ} (\cos (θ_{m θ_{A}}) - \cos (θ_{m θ_{B}})) \cdot Δ z \\ E_{2} (x, z) = ϕ_{C} (x, z) - ϕ_{D} (x, z) = \frac{4 π}{p} \cdot Δ x + \frac{2 π}{λ} (\cos (θ_{m θ_{D}}) - \cos (θ_{m θ_{D}})) \cdot Δ z . \end{matrix}$

(10.33)

From the symmetry in Fig. 10.41(b), it follows that

\cos (θ_{m θ_{A}}) - \cos (θ_{m θ_{D}}) and \cos (θ_{m θ_{C}}) - \cos (θ_{m θ_{B}})

$\cos (θ_{m θ_{A}}) - \cos (θ_{m θ_{D}}) and \cos (θ_{m θ_{C}}) - \cos (θ_{m θ_{B}})$

. The in-plane and out-of-plane displacements are then derived by combining the (single-pass) encoders signals E₁(x, z) and E₂(x, z)

Figure 10.40 (a and b) Simplified representation of an xz encoder concept based on integration of two tilted encoder heads (Holzapfel, 2009).

Figure 10.41 (a) Measurement direction for a single symmetric encoder cos(θ_m=+1) = cos(θ_m=−1), (b) effective measurement direction for two symmetric interferential encoders with nonnormal incidence (transmissive type for easier representation). The vector sum or subtraction of both encoder signals make an x or z measurement signal.

\begin{matrix} E_{1} (x, z) + E_{2} (x, z) = \frac{8 π}{p} Δ x \\ E_{1} (x, z) - E_{2} (x, z) = \frac{4 π}{λ} (\cos (θ_{m θ_{A}}) - \cos (θ_{m θ_{B}})) Δ z . \end{matrix}

$\begin{matrix} E_{1} (x, z) + E_{2} (x, z) = \frac{8 π}{p} Δ x \\ E_{1} (x, z) - E_{2} (x, z) = \frac{4 π}{λ} (\cos (θ_{m θ_{A}}) - \cos (θ_{m θ_{B}})) Δ z . \end{matrix}$

(10.34)

If the xz encoder is designed as a transmissive or a reflective double-pass encoder (Holzapfel, 2009), then also the sensitivities would double in magnitude.

10.6. Current and future trends

For the high-end side, interferential encoder and interferometer systems will still be unsurpassed with respect to the attainable resolution and accuracy. For encoders, miniaturization and cost reduction is seen in the integration of encoder components (source, diffractive optics, and detector) on a single chip (ChipEnc, 2004; Carr et al., 2008; Feng et al., 2013). Miniaturization is also seen in fiber-based interferential encoders with integrated diffractive elements (Tobiason, 2005; Tobiason and Altendorf, 2009). For fiber-based interferometer and encoder systems, the light source may be shared among probes, and the signals of multiple axes may be multiplexed and read out sequentially by a single signal analyzer to reduce cost. Fiber-based short stroke interferometers may partly replace tactile systems for measuring surface structure or encoders for measuring stability (Lindner and Schmidke, 2009). As the cost of image sensors and image processing power has dropped significantly, another trend that can be observed is the development of vision-based image correlation and absolute encoders. An optical mouse is a good example of a low-cost vision-based image correlation sensor. In an optical mouse, a low-resolution camera takes successive images of the surface on which the mouse operates. The surface is illuminated at grazing incidence by a light-emitting diode to enhance the contrast of the imaged texture. Alternatively, optical mice may use infrared lasers to generate a speckle image on the surface. Image correlation is used to detect the displacement with respect to a reference image using a dedicated processor. Based on the principle of an optical mouse, laser speckle correlation sensors are commercially available achieving subnanometer resolution (Jones and Nahum, 2001). In vision-based optical encoders, an image of a moving grid pattern is captured and analyzed for determining the absolute position of the object relative to the camera. The imaged patterns typically consist of high-contrast marks or grid patterns. For absolute sensing, the grid typically contains a unique mark layout for every measurement position (Nilsagård et al., 2011; Gribble and Robert, 2011). When imaging moving patterns, motion blur may be avoided using pulsed illumination or shuttered image acquisition while offering absolute timestamps for the recorded images. The measurement resolution may depend on the magnification of the optical system, and the algorithms used to transform the observed images into a position or displacement signal. To reduce the out-of-plane sensitivity of a vision system, telecentric optics may be used. An example of a 1D absolute encoder (Gribble and Robert, 2011) contains a scale with a uniquely spaced barcode-like line pattern that is recorded by a high-speed line camera and analyzed by a microprocessor to obtain the absolute scale position. In comparison with traditional encoder systems image recognition systems are more flexible in design and often can detect multiple degrees of freedom. The image processing may, however, require complex algorithms, which may make it a challenge to design both accurate and fast algorithms to minimize signal delay.

10.7. Conclusion

The stability of laser interferometer systems depends on the stability of the wavelength which, in turn, is dependent on the laser frequency and the fluctuations of the refractive index of the medium (typically air or vacuum), or how well the wavelength variations can be calibrated inline using an (absolute) refractometer. The stability of an optical encoder mainly depends on the stability of the scale material (e.g., Invar, glass, steel, Zerodur) or how well the thermal expansion can be predicted using information from temperature sensors. Interferometers are traceable to SI standards by comparing the laser frequency with a reference laser, where scales are traceable to the SI standard by comparing the scale lines with a laser interferometer-based comparator, which is ideally positioned in vacuum (Kunzman et al., 1993; Sawabe et al., 2004; Holzapfel, 2008; Flügge et al., 2014). Linearity has two aspects: coarse errors and interpolation errors. Laser interferometers do not have coarse errors, whereas for scales the coarse errors must be calibrated. The differences can be reduced by application of calibrated scales and inline error correction. Interpolation errors of laser interferometers based on different principles are comparable and typically in the range of 1–20 nm. For high-end encoders and interferometers, these periodic deviations can be reduced to subnanometer level by inline calibration. The measuring range is easily scalable to several meters for interferometers, whereas for encoder systems the range is limited to the size at which the scales can be manufactured and mounted to a base with sufficient stability. In most machines with stacked linear actuators, it is easier to integrate a scale than an interferometer. Some of the abovementioned aspects are further discussed and illustrated in an overview paper by Gao et al. (2015).

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Table of Contents for 10. Advanced Optical Incremental Sensors: Encoders and Interferometers

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