**2.2 Laser interferometry**

Since practical realization of meter is closely related with the radiation of stabilized frequency lasers, laser interferometers are utilized for precise and traceable length measurements. Currently the detection principles of laser interferometer systems can be distinguished between homodyne and heterodyne techniques (Webb & Jones, 2004; Chapman, 2002). Homodyne interferometers utilize one frequency laser, and heterodyne two frequencies laser respectively. Heterodyne interferometry is inherently more resistant to noise due to its heterodyne frequency and the design of common-mode rejection which cancels out common noises coming from both reference and measurement signals, and though heterodyne techniques are susceptible to larger nonlinearity errors a large number of commercial systems uses namely this technique. Nevertheless the main parameters that determine the quality of laser interferometric systems are (Bobroff, 1993):


A homodyne laser source is typically a He-Ne laser with a single frequency beam as output consisting of either a single polarization under 45° or a circularly polarized beam. The beam is split into the reference arm and measurement arm of the interferometer by a beam splitter. Following a reflection off their respective targets, the beams recombine in the beam splitter. In order to observe interference the two beams must have equal polarizations. This is accomplished using a linear polarizer oriented at 45° to the beam splitter. The photo detector signal is run through electronics which count the fringes of the interference signal. Every fringe corresponds to a path difference of half a wavelength. After superposition of measurement and reference beams a polarizing beam splitter is used to generate two 90° phase shifted signals. The direction of movement is determined at zero crossings of the interference signal using the other signal. Counting of the zero crossings of both interference signals provides a resolution of /8 which is not sufficient for precision length measurements and therefor it has to be enhanced by interpolation techniques. In homodyne interferometers the amplitudes of the interference signals are used; the phase of the signal can be determined from intensities of perpendicular polarized signals. Manufacturers of homodyne interferometers are Renishaw, Heidenhain, Sios and recently Interferomet.

The effect of the re-definitions and advances in measurement of the frequencies of recommended radiations was to decrease the relative uncertainty attainable in realization of

Measurements of dimensions of material goods are most often referenced to the SI unit of length through material artifacts calibrated as dimensional standards. The meter, the basic unit for length, is usually transferred to measurement standards in the form of line scales or photoelectrical incremental encoders by length measuring machines that typically use a laser interferometer in air as reference measuring system. The measurement results are

Since practical realization of meter is closely related with the radiation of stabilized frequency lasers, laser interferometers are utilized for precise and traceable length measurements. Currently the detection principles of laser interferometer systems can be distinguished between homodyne and heterodyne techniques (Webb & Jones, 2004; Chapman, 2002). Homodyne interferometers utilize one frequency laser, and heterodyne two frequencies laser respectively. Heterodyne interferometry is inherently more resistant to noise due to its heterodyne frequency and the design of common-mode rejection which cancels out common noises coming from both reference and measurement signals, and though heterodyne techniques are susceptible to larger nonlinearity errors a large number of commercial systems uses namely this technique. Nevertheless the main parameters that

A homodyne laser source is typically a He-Ne laser with a single frequency beam as output consisting of either a single polarization under 45° or a circularly polarized beam. The beam is split into the reference arm and measurement arm of the interferometer by a beam splitter. Following a reflection off their respective targets, the beams recombine in the beam splitter. In order to observe interference the two beams must have equal polarizations. This is accomplished using a linear polarizer oriented at 45° to the beam splitter. The photo detector signal is run through electronics which count the fringes of the interference signal. Every fringe corresponds to a path difference of half a wavelength. After superposition of measurement and reference beams a polarizing beam splitter is used to generate two 90° phase shifted signals. The direction of movement is determined at zero crossings of the interference signal using the other signal. Counting of the zero

for precision length measurements and therefor it has to be enhanced by interpolation techniques. In homodyne interferometers the amplitudes of the interference signals are used; the phase of the signal can be determined from intensities of perpendicular polarized signals. Manufacturers of homodyne interferometers are Renishaw,

/8 which is not sufficient

traceable to the meter due to the use of the wavelength of the laser interferometer.

determine the quality of laser interferometric systems are (Bobroff, 1993):

crossings of both interference signals provides a resolution of

Heidenhain, Sios and recently Interferomet.

the meter by five orders of magnitude (Swyt, 2001).

**2.2 Laser interferometry** 

resolution,

 measurement accuracy, repeatability of results,

measurement speed.

dynamic and measurement range,

In heterodyne interferometers double frequency radiation source is required since the interfering measuring and reference beams must have slightly different frequencies and photo detectors detect the phase shift between these two beams, see Fig. 1.

Fig. 1. Layout of heterodyne laser interferometer

Two frequencies are separated by their polarization state, so that a polarization beam splitter can generate a measurement beam with the frequency *f*1 and a reference beam with *f*<sup>2</sup> . The movement of the measurement reflector with velocity *v* causes a frequency shift *f* in the measurement beam due to Doppler effect. This shift will increase or decrease depending on the movement direction of the measurement reflector. When counting the periods of reference and measurement signals simultaneously their difference is proportional to displacement. The direction of movement can be determined directly from the sign of this difference. Interpolation in heterodyne systems is equivalent to the measurement of the actual phase difference between fixed reference frequency and measurement frequency (Webb & Jones, 2004).

Heterodyne principle has a certain advantage, particularly for He-Ne lasers due to good signal to noise ratio enabling multi-axis measurement with high measurement speeds. In practice, the maximum speed of the laser interferometer will be limited by the primary beat frequency of a laser source, which is usually in the range of few MHz for Zeeman stabilization technique where high power magnetic field is required for separation of the laser frequencies, and 20 MHz for the acousto-optical modulators and from 600 to 1000 MHz for stabilized two-mode lasers that would correspond to more than 200 m/s speed to be measured. However, increase of the beat frequency will decrease the measurement resolution of the laser interferometer in return and therefore the bandwidth of phase detection is usually fixed in the absolute time scale in most phase measuring techniques (Yim et al., 2000).

The main advantage of heterodyne systems is that information about the measured displacement is obtained in form of variable signal and therefore measurement circuits are not sensitive to variations of the measured signal level due to various disturbances. Since information about the displacement is gained from the signal frequency only one photo detector is necessary, and adjustment of optical elements becomes more simple, see comparison of both systems in Table 1.


Table 1 Comparison of heterodyne and homodyne systems

Resolution of interferometers primarily depends on accuracy of phase detection of interference signal. Currently due to rapid development of information technologies and electronics it increased from /8 in 1965 up to / 2048 (the wavelength of radiation source is divided into 2048 parts), and using traditional phase detection techniques the resolution of displacement interferometry amounts to about 0.1 – 10 nm.

Presently some commercially available phase detection systems allow digital signal processing and phase detection accuracy of 0.01 and which in combination with Michelson type heterodyne interferometer correspond to better than 10 pm system measurement accuracy.

The uncertainty sources in the laser interferometry can be grouped in three categories: setup dependent (cosine, Abbe, deadpath errors, mechanical stability), instrument dependent (stability of laser frequency, electronics, periodic deviations, etc.) and environment dependent (refractive index, turbulence, thermal sources) that are inherent in such systems and generally limit the relative uncertainty to 2×10−8, resulting in an error of 20 nm per meter. For measurements over large displacements in air the last group is predominant in the uncertainty budget.

Thus linear measurement scale and accurate and adequate control and stabilization of environmental conditions are the main precision criteria for the modern interferometric displacement measurement systems. The use of digital data processing techniques enables to minimize nonlinearities of laser interferometers and increase the measurement resolution up to 10 pm (Webb & Jones, 2004). But in order to achieve the required relative length measurement uncertainty below 10-7 in practical applications, the measurement accuracy of air refractive index must be not less than 10-8.

not sensitive to variations of the measured signal level due to various disturbances. Since information about the displacement is gained from the signal frequency only one photo detector is necessary, and adjustment of optical elements becomes more simple, see

Continous measurement Yes No Sensing of moving direction Quadrature Always Quadrature output signal Yes Yes

Sensitivity to radiation intensity Yes No Sensitivity to enviromental irradience Yes No

Table 1 Comparison of heterodyne and homodyne systems

resolution of displacement interferometry amounts to about 0.1 – 10 nm.

Error detection not defined Unambigious

Frequency band of electronic circuit 0 -2v/λ f1 – f2 ±2v/λ Signal to noise ratio 6 – 12 bit 2 – 3 bit Multi-axis measurements Limited Yes Photo detector Complex Simple Adjustment simplicity No Yes

Resolution of interferometers primarily depends on accuracy of phase detection of interference signal. Currently due to rapid development of information technologies and

source is divided into 2048 parts), and using traditional phase detection techniques the

Presently some commercially available phase detection systems allow digital signal processing and phase detection accuracy of 0.01 and which in combination with Michelson type heterodyne interferometer correspond to better than 10 pm system measurement

The uncertainty sources in the laser interferometry can be grouped in three categories: setup dependent (cosine, Abbe, deadpath errors, mechanical stability), instrument dependent (stability of laser frequency, electronics, periodic deviations, etc.) and environment dependent (refractive index, turbulence, thermal sources) that are inherent in such systems and generally limit the relative uncertainty to 2×10−8, resulting in an error of 20 nm per meter. For measurements over large displacements in air the last group is predominant in

Thus linear measurement scale and accurate and adequate control and stabilization of environmental conditions are the main precision criteria for the modern interferometric displacement measurement systems. The use of digital data processing techniques enables to minimize nonlinearities of laser interferometers and increase the measurement resolution up to 10 pm (Webb & Jones, 2004). But in order to achieve the required relative length measurement uncertainty below 10-7 in practical applications, the measurement accuracy of

/8 in 1965 up to

Homodyne Heterodyne

/ 2048 (the wavelength of radiation

comparison of both systems in Table 1.

electronics it increased from

accuracy.

the uncertainty budget.

air refractive index must be not less than 10-8.

A good example for that is the comparison of static stability of He-Ne laser interferometer and linear encoder presented in Fig. 2. Experiments conducted have shown that users can expect fewer measurement fluctuations of the position display from linear encoders than from laser interferometers (Kaušinis et al., 2004).

Fig. 2. Stability of laser interferometer and reference encoder

A long-term stability of the laser interferometer was mainly influenced by the temperature and refractive index. The refractive index of air was compensated by Edlen's formula. Both measurement systems were simultaneously read out in a static mode and actually, in comparatively good ambient conditions the interferometer has revealed clearly higher variances than the linear encoder. The temperature measurement system due to its data acquisition time and remoteness of the laser beam sensor is not able to compensate these variations.

The analysis of the dynamic mode of operation displayed results of the same order as variations of the laser interferometer measured in a static mode. Repeated measurements at different positions of the line scale displayed quite similar results. It is evident that, even in the finely air-conditioned laboratory environment, the low heat capacity of the air causes quick changes in temperature that can lead to relatively large fluctuations in measured values obtained from the laser interferometer. Comparatively short distances between encoder's scale and index grating minimize sensitivity to environmental factors.

## **2.3 Limits and challenges in length metrology**

The future of length and dimensional metrology is being shaped by theoretical and practical limits to attainable uncertainties in measurement, by continuing trends in industry. There are two drivers that force the achievement of ever-smaller uncertainties in length and dimensional measurements. These are, first, the continuing industrial trend to tighter tolerances—represented in the microelectronics domain by Moore's Law—and, second, the continuing scientific trend to explore the limits of understanding through physical measurement.

The latter is bounded by a dimensional equivalent of Johnson, or thermal, noise that places an ultimate limit on the uncertainty of measurement of dimensional features. Thermal length fluctuations of a solid artifact, the spatial equivalent of electronic Johnson noise, are due to thermal agitation of the atoms of the material. In a measuring machine, such thermal noise places an ultimate limit on the ability to set the location of the origin of the axes of the machine and, therefore, on the uncertainty of position measurements the machine can attain. Thermal noise similarly limits the uncertainty with which the length of an object can be measured. Therefore thermal fluctuation *l* in the length *l* of the side of the artifact is given:

$$
\Delta l = \left(\frac{k \cdot T}{\mathfrak{B}} \cdot \mathcal{B} \cdot l\right)^{1/2} \tag{1}
$$

where *k* is the Boltzmann constant and *T* is the thermodynamic temperature, *B* is the bulk modulus of the material of the cube.

For example, for an object with a bulk modulus of that of fused silica, 3.5×1010 N/m2, and a temperature of 300 K, the rms fluctuation in dimension of a 1 m cube is 0.2 fm (10–15 m) or, fractionally, 2×10–16. Besides this factor is inversely proportional to the measured length and therefore by decreasing the geometrical dimensions of measured objects the length measurement uncertainty will increase proportionally (Swyt, 2001).

Generally the limit for length measurement uncertainty is firstly determined by realization capabilities of time unit (second) following the definition of the meter and which is in the uncertainty of second is in the order of 1.5 ×10-15 presently and, secondly, practical realization of temperature unit (Kelvin) as major disturbance parameter which influences the accuracy of dimensional measurements. However the actual precision of length measurements is limited by the other standard closely related with second that is the frequency standard – CH4-stabilized He-Ne laser. Satisfying the requirements of International Committee for Weights and Measures and the most common realization of primary length standard is He-Ne laser with the wavelength of = 632.99 nm and relative standard uncertainty of 2.5×10-11 (Webb & Jones, 2004).

The bottleneck for calibration of material artifacts is determined by the capabilities of measurement systems to detect the boundaries/edges of geometrical features, requirements on environmental parameters as well as optical wave interferometry. In practice the lower limit for optical interferometry is in the range of 10-7 and mainly restrained by variations of air refractive index which in turn can be computed by Edlen's formula contributing to the length measurement error in amount of 5 ×10-8 and might be considered as ultimate limit for measurements in air. On the other hand, vacuum interferometry might be an option where currently achievable expanded measurement uncertainties are below 5 nm measuring high quality artifacts in lengths below 1 m (Köning et al., 2007).

Typically for calibration of precision line scales and traceability to length standard optical comparators are used where optical or X-ray interferometry is used and such systems enable calibration of length standards with uncertainties ranging from several nanometers up to some tenth of nanometers when measured length is significantly longer in comparison with the wavelength of radiation source. However when geometrical dimensions of the measured structures are of the order of the wavelength the use of such technologies becomes quite complicated and therefore comparative methods using X-ray diffraction or scanning microscopes are more common, particularly for investigation of micro electro-mechanical system (MEMS) structures, lithography and similarly (PTB, 2003; Swiss Federal Office of Metrology, 2004).

machine and, therefore, on the uncertainty of position measurements the machine can attain. Thermal noise similarly limits the uncertainty with which the length of an object can be measured. Therefore thermal fluctuation *l* in the length *l* of the side of the artifact is

*k T l Bl*

3 

where *k* is the Boltzmann constant and *T* is the thermodynamic temperature, *B* is the bulk

For example, for an object with a bulk modulus of that of fused silica, 3.5×1010 N/m2, and a temperature of 300 K, the rms fluctuation in dimension of a 1 m cube is 0.2 fm (10–15 m) or, fractionally, 2×10–16. Besides this factor is inversely proportional to the measured length and therefore by decreasing the geometrical dimensions of measured objects the length

Generally the limit for length measurement uncertainty is firstly determined by realization capabilities of time unit (second) following the definition of the meter and which is in the uncertainty of second is in the order of 1.5 ×10-15 presently and, secondly, practical realization of temperature unit (Kelvin) as major disturbance parameter which influences the accuracy of dimensional measurements. However the actual precision of length measurements is limited by the other standard closely related with second that is the frequency standard – CH4-stabilized He-Ne laser. Satisfying the requirements of International Committee for Weights and Measures and the most common realization of

The bottleneck for calibration of material artifacts is determined by the capabilities of measurement systems to detect the boundaries/edges of geometrical features, requirements on environmental parameters as well as optical wave interferometry. In practice the lower limit for optical interferometry is in the range of 10-7 and mainly restrained by variations of air refractive index which in turn can be computed by Edlen's formula contributing to the length measurement error in amount of 5 ×10-8 and might be considered as ultimate limit for measurements in air. On the other hand, vacuum interferometry might be an option where currently achievable expanded measurement uncertainties are below 5 nm measuring high

Typically for calibration of precision line scales and traceability to length standard optical comparators are used where optical or X-ray interferometry is used and such systems enable calibration of length standards with uncertainties ranging from several nanometers up to some tenth of nanometers when measured length is significantly longer in comparison with the wavelength of radiation source. However when geometrical dimensions of the measured structures are of the order of the wavelength the use of such technologies becomes quite complicated and therefore comparative methods using X-ray diffraction or scanning microscopes are more common, particularly for investigation of micro electro-mechanical system (MEMS) structures, lithography and similarly (PTB, 2003; Swiss Federal Office of

measurement uncertainty will increase proportionally (Swyt, 2001).

primary length standard is He-Ne laser with the wavelength of

standard uncertainty of 2.5×10-11 (Webb & Jones, 2004).

quality artifacts in lengths below 1 m (Köning et al., 2007).

Metrology, 2004).

1/2

(1)

= 632.99 nm and relative

given:

modulus of the material of the cube.

While comparing these different methods it is necessary to consider not only the measurement range and accuracy requirements but also effectiveness of such systems relating to metrological network and needs of particular country, design costs, etc.

Although different calibration methods and equipment, see in Table 2, for dimensional measurement of a material artifact are used, currently, optical comparators combining properties of laser interferometers and optical microscopes represent the lowest relative uncertainty (*U/L*) of dimensional measurements provided in a length calibration of a 1 m line scale, the relative expanded uncertainty (coverage factor *k* = 2) is 7×10-8 m at 1 m.


Table 2. Comparison of different length calibration techniques (PTB, 2003; Swiss Federal Office of Metrology, 2004)
