**4.4.2 The slope error produced by inhomogeneity**

The optical path difference (OPD) produced by glass inhomogeneity is

$$\text{OPDh} = \text{\\$n\*} \text{ path length} \tag{2}$$

where n is the refractive index error. Commercial optics typically use Grade H2 of SCHOTT glass with n=5e-6 (SCHOTT, 2004). Assuming the refractive index variation has a sinusoidal distribution in 20mm length and 2mm thickness layer, it will produce a 1.57rad slope error. So grade H2 is not suitable for the nano-radian accuracy application. Grade H5 glass or Grade 0AA of Corning HPFS® glass (Corning, 2008) has an index variation of n=5e-7, which will produce 0.157rad slope error. Actual error needs to verify with practical test.

Most non-contact optical profilers that use optical systems for measuring angle variation will confront with these difficulties if there is BLM. Examples of such systems are the optical head of the LTP, the autocollimator of the NOM and optical system of the PTB's profilers.

As a result of the above simulation analysis, it is necessary to apply the best quality glass Grade H5 and /100 figured surfaces to achieve a nano-accuracy profiler. An actual error estimation using the highest quality optics is underway.

#### **4.4.3 Surface figure error in the mid-spatial frequency range and its metrology**

Optical surface errors are general divided into three categories: a) low-spatial frequency (LSF): as surface figure error; b) mid-spatial frequency (MSF): as ripple, and c) high frequency: as surface finish roughness (Youngworth & Stone, 2000; Youngworth et al., 2008). The low-spatial frequency surface error is defined over the spatial period range from 5-10 mm to the entire surface dimension, while the mid-spatial-frequency (MSF) surface error is roughly defined in the range between 0.1 to 5-10 mm spatial periods. The impacts to the optical system quality of both LSF and MSF can be analyzed with ray-based model. MSF could also be considered as surface figure.

MSF will have a more severe impact on surface slope error than LSF error. The following sinusoidal wave-front simulations describe this effect. Fig. 14 shows simulated sinusoidal wave-fronts and their slopes (derivative of the wave-front) in a 20 mm surface length. The wave-fronts have identical amplitudes of ±1nm, equal to /317, but with different

frequencies. It is obvious that higher frequency waves have a much larger slope error than low frequency ones.

Fig. 14. Higher frequency wave-front has much larger slope error even though the amplitude of the wave-front is the same

Traditional surface figure specification is generally defined as height deviation from a nominal surface figure in number of wavelengths. Traditional optical manufacturing uses long pitch or large lap polishing, which will produce relatively small MSF error. It means fewer ripples on surface. Due to new deterministic polishing technology and metrology accuracy developments, LSF height error can be reduced significantly by removing local material precisely according to precise surface test results. However, local figure correction polishing by use of deterministic polishing may leave small ripples on surface. Though the ripple height may be smaller than 100, the short ripple distance (MSF) will have a larger slope than the traditionally polished surface, which could degrade the slope error accuracy. For example, after processing half of the part with fluid jet polishing, the surface flatness P-V is reduced to /100, but the MSF ripples (Fig.15 a) could increase the slope error. Fig. 15 b) displays the residual ripples on the surface (Lightmachinery, 2011).

Fig. 15. Fluid jet polishing: a) After processing half of the part with fluid jet polishing. The surface flatness P-V is reduced to /100; b) After fluid jet polishing the 4" silicon mirror has a surface flatness of 1/100 wave (P-V). Courtesy of Light machinery, Fluid Jet Polishing, http://www.lightmachinery.com/Fluid-Jet-Polishing.html

frequencies. It is obvious that higher frequency waves have a much larger slope error than

**Same amplitude waves have different slope errors: 1hz is 0.314urad, 7hz is 2.20urad**

**same height error of 1nm**

Fig. 14. Higher frequency wave-front has much larger slope error even though the

displays the residual ripples on the surface (Lightmachinery, 2011).

http://www.lightmachinery.com/Fluid-Jet-Polishing.html

a b

Traditional surface figure specification is generally defined as height deviation from a nominal surface figure in number of wavelengths. Traditional optical manufacturing uses long pitch or large lap polishing, which will produce relatively small MSF error. It means fewer ripples on surface. Due to new deterministic polishing technology and metrology accuracy developments, LSF height error can be reduced significantly by removing local material precisely according to precise surface test results. However, local figure correction polishing by use of deterministic polishing may leave small ripples on surface. Though the ripple height may be smaller than 100, the short ripple distance (MSF) will have a larger slope than the traditionally polished surface, which could degrade the slope error accuracy. For example, after processing half of the part with fluid jet polishing, the surface flatness P-V is reduced to /100, but the MSF ripples (Fig.15 a) could increase the slope error. Fig. 15 b)

0 5 10 15 20

**very different slope error vs. spatial frequency**

distance (mm)


0

Slope/1hz(urad)

2

4

Slope/1hz(urad) Slope/3hz Slope/5hz Slope/7hz

6

8

Fig. 15. Fluid jet polishing: a) After processing half of the part with fluid jet polishing. The surface flatness P-V is reduced to /100; b) After fluid jet polishing the 4" silicon mirror has a surface flatness of 1/100 wave (P-V). Courtesy of Light machinery, Fluid Jet Polishing,

low frequency ones.

amplitude of the wave-front is the same





height/1Hz(nm) Height/3Hz Height/5Hz Height/7Hz

0

height/1Hz(nm)

1


Table 2. Slope error of 20 mm period sine wave-front with /100 or /20 amplitude and different spatial frequency

The above table 2 shows that the slope error of a 20 surface without MSF could be better than a 100 surface with MSF. It means that if the MSF can be controlled, the requirement on surface height error can be reduced significantly. Modern sub-aperture and deterministic optical fabrication techniques are more prone to ripple errors. So after deterministic polishing, an additional surface smoothing process may be necessary if there is considerable MSF error. Also, striae are an important material defect that creates MSF and should be considered more carefully. As a result of this analysis, it may be necessary to specify surface error by use of slope error in addition to the use of ambiguous surface height error.

The metrology for the MSF requires higher lateral detection resolution. Point scan surface profilers such as the LTP and NOM apply pencil beam spots of one to several millimeters as a detecting tool, so it is very suitable for the LSF test and will cover partial range of the MSF. Stitching interferometry covers the higher MSF range. However, a new nano-accuracy profiler for large surface test covering the 0.1-2mm MSF range is necessary for high quality optical system, and is currently under development.

## **4.5 A nano-accuracy solution for a new surface profiler in the large slope testing range: Adopting scanning optical-head combined with non-tilted reference beam**

There is increasing demand for nano-accuracy in the testing of strongly curved surfaces, for example, for K–B mirrors used for X-ray synchrotron radiation, for optics for extreme ultraviolet projections lithography and for the new astronomical telescopes.

As described above, there is beam lateral motion (BLM) in the optical system during the beam scan on a curved surface, which will produce significant slope error. If this error cannot be removed or greatly decreased, nano-accuracy will be difficult to reach in the range of large-slope tests.

Good design of the profiler system can reduce BLM significantly and very effectively. The first step is to use as few optical components as possible in order to eliminate unnecessary BLM error. The second step is to design novel optical system of the profiler to minimize BLM. Adopting the mode of scanning optical-head combined with non-tilted reference beam is an effective solution (Qian, 2011) for a nano-accuracy surface profiler (NSP).

The first optical head, fixed on the carriage, is scanned along the air bearing to probe the surface slope, and the second optical head (as used in the LTP-MF or using an autocollimator) is fixed to the granite table for measuring the air-bearing pitch error. In this way its beam can be set without BLM, because there is no problem with sample beam and reference beam spots overlapping. Application of the non-tilted reference will eliminate the BLM completely, so no error will be produced in the reference arm. Other non-tilted reference methods can be used to simplify the system.

The first advantage of applying a scanning optical-head is to create the opportunity to use a short fixed working distance for the sample beam. In this way, the sample beam's BLM can be significantly reduced to ±1mm (50 mm working distance) in comparison with ±20mm BLM in the scanning penta-prism mode for a test range of ±0.01 rad. This significantly lowers slope systematic error.

The second great advantage of applying a scanning optical head is its very simple calibration. Only one error compensation curve will be necessary to correct for all systematic errors for testing various mirrors. In contrast, it is very hard to compensate systematic errors in the penta-prism scanning mode, in which mirrors with different radius of curvature, different dimensions and different scanning start positions will need different compensation curves. Obviously, for a large test range, using the penta-prism scanning mode effectively precludes reaching nano-radian accuracy due to the BLM.

The third advantage of applying a scanning optical head is that small BLM and fixed working distance minimize the operational aperture of the lens, so it simplifies the aberration-reduction design of the lens. Another necessary approach to reduce systematic error is to improve the quality of the optical components including surface quality, optical material inhomogeneity and roughness.


Table 3. Comparison of three scanning modes

Table 3 is the comparison of three scanning modes

Recently, the PTB described new research in *"Scanning deflectometric form measurement avoiding path-dependent angle measurement errors*" to reduce the BLM problem (Fig. 16) (Schulz et al., 2010). In the case of the PTB, the first autocollimator (AC1) beam is scanned through a penta-prism to the mirror under test (MUT). But the MUT is no longer stationary now, and it is tilted by a tilting stage in order to direct the reflected sample beam back along the incoming direction. This means that the MUT is tilted according to the slope at each scanning point. So this measurement arm incurs no BLM. The second stationary

The first advantage of applying a scanning optical-head is to create the opportunity to use a short fixed working distance for the sample beam. In this way, the sample beam's BLM can be significantly reduced to ±1mm (50 mm working distance) in comparison with ±20mm BLM in the scanning penta-prism mode for a test range of ±0.01 rad. This significantly

The second great advantage of applying a scanning optical head is its very simple calibration. Only one error compensation curve will be necessary to correct for all systematic errors for testing various mirrors. In contrast, it is very hard to compensate systematic errors in the penta-prism scanning mode, in which mirrors with different radius of curvature, different dimensions and different scanning start positions will need different compensation curves. Obviously, for a large test range, using the penta-prism scanning mode effectively

The third advantage of applying a scanning optical head is that small BLM and fixed working distance minimize the operational aperture of the lens, so it simplifies the aberration-reduction design of the lens. Another necessary approach to reduce systematic error is to improve the quality of the optical components including surface quality, optical

> Test angle **(mrad)**

Extra

Pentaprism /mirror

optics Comment

±10 N/A Larger test angle +

±5 N/A Suitable for plane &

high accuracy

near plane mirror test

Suitable for plane & near plane mirror test

BLM(mm) /at test angle

 ±0.5/±5mrad ±1/±10mrad

Sample:

Ref: 0

Ref :

Sample: ±0.5/±5mrad

Sample:

±10/±5mrad

±10/±5mrad ±5

Recently, the PTB described new research in *"Scanning deflectometric form measurement avoiding path-dependent angle measurement errors*" to reduce the BLM problem (Fig. 16) (Schulz et al., 2010). In the case of the PTB, the first autocollimator (AC1) beam is scanned through a penta-prism to the mirror under test (MUT). But the MUT is no longer stationary now, and it is tilted by a tilting stage in order to direct the reflected sample beam back along the incoming direction. This means that the MUT is tilted according to the slope at each scanning point. So this measurement arm incurs no BLM. The second stationary

lowers slope systematic error.

precludes reaching nano-radian accuracy due to the BLM.

material inhomogeneity and roughness.

distance **(mm)**

Sample: 50 (fixed) Ref: 100- 1100

Sample: 50 (fixed)

Sample: 300-1300

Ref: 100-1100

Table 3. Comparison of three scanning modes

Table 3 is the comparison of three scanning modes

Scan mode working

Scan OH+ nontilted REF (NSP)

Scan OH+ tilted REF (LTP

Scan Pentaprism (PPLTP,

NOM)

II)

autocollimator (AC2) is used to measure the tilt angle/slope of the MUT with a mirror fixed to the tilting stage. In this way, this tilting angle/slope test arm has a fixed short distance, which will reduce the BLM significantly (as in the case of the scanning optical head with non-tilted reference method). This is a good method to reduce measurement error caused by BLM.

Fig. 16. Principle of operation of the EADS system. AC1: Straightness representation and null instrument, AC2: angle measurement.

Courtesy of M. Schulz et al. Scanning deflectometric form measurement avoiding pathdependent angle measurement errors, JEOS Rapid Publications 5, 10026 (2010)

The main principle and similar strategy in both cases of the scanning optical head with nontilted reference method and the PTB method are: the probe arms to measure large tilted angles are short and fixed in order to reduce the BLM and are combined with another nontilted beam arm in order to eliminate the BLM. As a matter of fact, in the PTB case, the angle tilt test is converted to a new arm by use of a tilting stage. However, a precision tilting stage must be used.
