**5.1 Angle calibration system based on trigonometric function**

Sine bar and tangent bar methods can be used to precisely measure small angles.

## **5.1.1 Sine bar**

The sine bar is a simple and effective method in calibrating small angles. Sine bar system contains one solid bar and two fixed-distance cylinders. When one cylinder B is lifted in a height (h), while another cylinder A is kept in contact with the base surface (Fig. 17), the bar rotation angle will be:

$$\mathbf{a} = \mathbf{ar}\mathbf{c}\sin\left(\mathbf{h}/\mathbf{L}\right)\tag{3}$$

If the roundness and diameter of two cylinders are very precise, and their geometric positions and moving height are accurate, the angle can be calculated very accurately. The distance error of two cylinders is a systematic error, which can be eliminated easily by calibration. The sine bar technology is able to reach nano-radian accuracy level.

The sine bar system known as the small angle generator with a length of about 523 mm at National Physical Laboratory (NPL) of Britain is used for autocollimator and small angle calibration (NPL, 2010). Its uncertainty is estimated to be ± 0.03 second of arc for angles in the range ± 10 minutes of arc. It is suitable for nano-radian profiler calibration in this angle range. For larger angle calibration the accuracy needs to improve.

The height lifting of a traditional sine bar is made by inserting a precise gauge block under one of the sine bar cylinders, or by use of micrometer. However, the process of cleaning and wringing of gauge blocks and support surface is time consuming and is not very reliable for reaching nano-accuracy. The gauge block accuracy is 50nm. Generally the average wringing film thickness between the gauge blocks is about 10 nm, but some wrings will be over 25 nm. For a non-professional operator the error could be much larger. A 30 nm possible height error with 250mm bar length will produce 0.12 rad error. If this error is combined with several other errors, it will impact the final nano-accuracy.

Fig. 17. Sine bar calibration scheme: = arc sin (h/L)

A computer controlled height lifting device and nanometer accuracy encoder are desired in order to reach nano-accuracy and to maintain temperature stability during the operation. The accuracy of mechanical and contacted Heidenhain length gauge is 30 nm. But its contact characteristic and insufficient accuracy will restrict nano-accuracy development of the sine bar system.

#### **5.1.2 Tangent bar**

98 Modern Metrology Concerns

As described above, now the required measurement accuracy is nano-radian and nanometer. If the instrument cannot be calibrated precisely, nano-accuracy metrology achievement is meaningless. Plane mirror measurement by use of traditional phase shift interferometer is difficult to reach /50-/100 accuracy because of the reference surface accuracy limitations. The absolute flatness test of the plane mirror by use of three-flat test method can reach /100 or better, but it is very difficult for large mirror calibration (for example for 500mm to 1000mm dimension). Pencil beam profilers can solve this problem, but it is only in 1-D. When the tested slope range increases, the angle related systematic error of the measurement instrument increases significantly. Though the pencil beam profilers have potential power to reach nano-radian and nanometer accuracy or less in principle, they still need to be calibrated. Calibration equipment should have the accuracy of three times better than the instrument to be calibrated. This is a great challenge to optical

**5.1 Angle calibration system based on trigonometric function** 

Sine bar and tangent bar methods can be used to precisely measure small angles.

calibration. The sine bar technology is able to reach nano-radian accuracy level.

range. For larger angle calibration the accuracy needs to improve.

several other errors, it will impact the final nano-accuracy.

The sine bar is a simple and effective method in calibrating small angles. Sine bar system contains one solid bar and two fixed-distance cylinders. When one cylinder B is lifted in a height (h), while another cylinder A is kept in contact with the base surface (Fig. 17), the bar

 = arc sin (h/L) (3) If the roundness and diameter of two cylinders are very precise, and their geometric positions and moving height are accurate, the angle can be calculated very accurately. The distance error of two cylinders is a systematic error, which can be eliminated easily by

The sine bar system known as the small angle generator with a length of about 523 mm at National Physical Laboratory (NPL) of Britain is used for autocollimator and small angle calibration (NPL, 2010). Its uncertainty is estimated to be ± 0.03 second of arc for angles in the range ± 10 minutes of arc. It is suitable for nano-radian profiler calibration in this angle

The height lifting of a traditional sine bar is made by inserting a precise gauge block under one of the sine bar cylinders, or by use of micrometer. However, the process of cleaning and wringing of gauge blocks and support surface is time consuming and is not very reliable for reaching nano-accuracy. The gauge block accuracy is 50nm. Generally the average wringing film thickness between the gauge blocks is about 10 nm, but some wrings will be over 25 nm. For a non-professional operator the error could be much larger. A 30 nm possible height error with 250mm bar length will produce 0.12 rad error. If this error is combined with

**5. Calibration of nano-accuracy** 

metrology.

**5.1.1 Sine bar** 

rotation angle will be:

Another relative angle metrology method uses the tangent function by means of two precise fixed distance sensors to measure the bar's tilt angle:

$$a = \text{arc}\,\tan\left(\text{h}\,/\text{ L}\right) \tag{4}$$

where the h is the difference between the two distance-sensor readings and L is the distance between the two sensors (Fig. 18).

Fig. 18. Tangent bar calibration scheme: = arc tan (h/L)

A vertical angle comparator (VAC) was developed at the BESSY-II Optics laboratory for the characterization and calibration of angle measuring sensors (Siewert et al., 2010). The VAC applies the tangent bar principle with a 1.3m long aluminum bar. The axis of the instrument is formed by a structure of a crossing flexure joint. A linear stepper motor actuator and two linear encoders (Heidenhain CertoCP 60 K) enable a controlled tilting of the VAC. The linear encoder provides a constant uncertainty of ±50nm over a range of 60mm. The measuring resolution of the Certo is ±5 nm; this corresponds to a tilting resolution of about 10 nrad. The achievable angular resolution of the VAC is about 0.015 rad, limited by the performance of the linear stepper motor. The error budget of the VAC is estimated to 50nrad rms.

Recent developments in precise interferometric distance sensors has improved measurement accuracy to the one nanometer level, which significantly increases the reliability for trigonometric function calibration. The displacement interferometer can be used for small angle metrology. Fig. 19 is the example that uses a ZYGO system (ZYGO, 2003). The difference of two distance sensor readings divided by the distance between two points is the tilt angle. Its advantages are non-contact and nanometer accuracy. The National Research Council Canada uses this principle in their angle calibration systems (Pekelsky & Munro, 2005).

Fig. 19. Displacement Measuring Interferometers, Courtesy of ZYGO, http://www.zygo.com/?/met/markets/stageposition/zmi/

The fiber-based Fabry-Perot interferometer (FFPI) is a typical multiple-beam interferometer that can be used as a non-contact distance sensor. The space separating the reflecting surface is called the cavity length. The reflected light in the FFPI is wavelength modulated in exact accordance with the cavity length (Pullteap, 2010). The attocube's ultra-high precision displacement FPSsensor has a repeatability of 1 nm at 20 mm cavity length, making it is suitable for small angle calibration with sine bar or tangent bar due to the advantages of high accuracy and non contact measurement characteristic (Attocube, 2010). Assuming its accuracy is 3 nm and it is used with a 250 mm sine bar, the error will be 0.012 rad. In addition, its very compact size is very attractive. However, larger test angles will impact the test accuracy significantly.
