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Single-shot freeform surface profiler

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Abstract

We propose a novel and simple method of single-shot freeform surface profiler based on spatially phase-shifted lateral shearing interferometry. By the adoption of birefringent materials, the laterally shearing waves are simply generated without any bulky and complicated optical components. Moreover, the phase maps that lead to the 3D profile of the freeform surface can be instantly obtained by the spatial phase-shifting technique using a pixelated polarizing camera. The proposed method was theoretically described and verified by measuring several samples in comparison to the measurement results with a well-established stylus probe.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Freeform optics [1,2] have been extremely important in various fields of optical imaging, illumination and laser manufacturing from traditional industries to new areas of virtual reality (VR), augmented reality (AR), autonomous vehicles and segmented telescopes. Freeform, typically defined as optical surfaces with no translational or rotational symmetry about axes normal to the mean plane [3], can provide more flexibility to optical engineers who desire to design light-weighted and reliable optical systems. One of the limitations to develop freeform optics, however, is that there are no standard methods to evaluate freeform surfaces until now. Typically, stylus techniques are capable of reconstructing freeform surfaces, but they require long measurement time, much effort and surface damage because of their point-like contact measurement characteristics. Optical techniques have advantages of measuring surfaces at once without surface damage and manifold research works have been reported [411]. Deflectometry [46], which detects the variation of specific patterns reflected on the surface, is appropriate for measuring specular surfaces in the large scale, but its resolution is relatively low in the order of a few micrometer range. Even though microscopic deflectometry has been proposed to improve the measuring resolution [7], deflectometry is always compared to interferometry based on the wavelength of light in order to estimate its accuracy. As more precise techniques with sub-nanometer or nanometer resolutions, interferometry can be used with computer generated holograms (CGHs) [8]. In order to capture interference fringes, however, the reference wavefront should be quite close to the measurement one, which leads to high cost of manufacturing CGHs. Even if this condition is fulfilled, the system may fail to obtain interference fringes when the measurement surface is not preliminary known. On the other hand, stitching interferometry [9,10] is an efficient way to measure freeform surfaces with sub-aperture measurements, but it still needs long measurement time corresponding to the number of sub-aperture measurements.

Lateral shearing interferometry (LSI) [11,12] has been widely used to analyze optical wavefronts. By obtaining the interference pattern between the original and laterally sheared waves, LSI can calculate the wavefront by using optimized integration techniques. Because LSI only uses the measurement wave to generate the interference pattern without the reference wave, it does not require any additional specialized devices such as CGHs. However, LSI needs both of a laterally beam shearing device and a phase shifting mechanism to make the system bulky, increase complexity and lower measurement speed, which can restrict its ability to measure freeform optics. In order to overcome the limitations of LSI, a 2D-Fourier transformation (2D-FT) based LSI was proposed to reduce the system size by the use of 2D gratings [13] or masks [14], but the calculation errors caused by the 2D-FT including window functions can lower the measurement accuracy. Furthermore, the 2D gratings and masks should be also designed and manufactured, which is challenging for optical engineers.

In this investigation, we propose a novel single-shot lateral shearing interferometry as a simple freeform surface profiler. The proposed system uses a birefringent material to generate two laterally shifted beams and the interference fringe is analyzed by the spatial phase shifting technique using a pixelated polarizing camera (PPC), which contains a polarizer array on the imaging sensor [15,16].

2. Single-shot freeform surface profiler

Figure 1 shows the single-shot freeform surface profiler proposed in this investigation. As a light source, a monochromatic laser is used and coherent noises are minimized by using a rotating diffuser (RD). With the aid of collimating lens (CL), a plane wave is incident to a measurement target after passing through a 45° rotated linear polarizer (LP) and a beam splitter (BS1). Then, the reflected wave from the target is transmitted or reflected by the BS2, and each wave goes to the lateral shearing part (LSP I or LSP II).

 figure: Fig. 1.

Fig. 1. Optical configuration of the proposed single-shot lateral shearing interferometry; RD: Rotating Diffuser, CL: Collimating Lens, LP: Linear Polarizer (45°), BS1,2: Beam Splitter, BD1,2: Beam Displacer, QWP1,2: Quarter Wave Plater (45°), IL1,2: Imaging Lens, PPC1,2: Pixelated Polarizing Camera.

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In each LSP, the incident wave is divided into two waves by a beam displacer (BD), a kind of birefringent materials. In this case, the fast axes of BD1 and BD2 are set as 0° and 90° for generating x- and y-sheared extraordinary waves. In each of x- and y- shearing parts, then, an ordinary wave and an extraordinary wave have orthogonal polarizations, and they become circularly polarized by a 45° rotated quarter wave plate (QWP) in front of the PPC. In the PPC, an individual pixel has its own polarizer, oriented by 0°, 45°, 90°, and 135° and arranged in repeating a (2×2) unit cell block as seen in Fig. 2(a). Figures 2(b)–2(e) show four kinds of phase shifted interference fringes obtained by the PPC1 in a single-shot measurement.

 figure: Fig. 2.

Fig. 2. (a) Laterally sheared interference fringe pattern directly obtained by the PPC1 and four sub- images with the polarization angle of (b) 0°, (c) 45°, (d) 90° and (e) 135°.

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In order to mathematically analyze the operation of the proposed system, the reflected wave (Et) from the target and BS1 can be described with a Jones vector as

$${E_t} = {E_0}{e^{jw(x,y)}}\left( {\begin{array}{{c}} 1\\ 1 \end{array}} \right)$$
where w(x,y) is the wavefront distorted by the target and E0 is the amplitude of the wave. Then, Et is divided into two by BS2 for the x- and y- shearing parts of LSP and each wave is also divided into the ordinary and the extraordinary waves by the BD. In the x-shearing part, for example, the two waves can be represented as
$${E_{xo}} = \frac{{{E_0}}}{2}{e^{jw(x,y)}}\left( {\begin{array}{{c}} 1\\ 0 \end{array}} \right)$$
$${E_{xe}} = \frac{{{E_0}}}{2}{e^{jw(x + \varDelta x,y)}}\left( {\begin{array}{{c}} 0\\ 1 \end{array}} \right)$$
where Exo and Exe indicate the ordinary and the extraordinary waves in the x-shearing part and Δx means lateral wave shift along the x-axis. It is noted that j is the imaginary number ($\sqrt {-1}$) and the uniform phase change over the wavefront is ignored. When Exo and Exe pass through QWP1, these two linearly polarized waves are converted into two circularly polarized waves as
$${E_{xo}} = \frac{{{E_0}}}{4}{e^{jw(x,y)}}\left( {\begin{array}{c} {1 + j}\\ {1 - j} \end{array}} \right),$$
$${E_{xe}} = \frac{{{E_0}}}{4}{e^{jw(x + \Delta x,y)}}\left( {\begin{array}{{c}} {1 - j}\\ {1 + j} \end{array}} \right)$$
Then, four kinds of rotated polarizers in the PPC1 can generate the phase shifted interference fringes, respectively, as follows.
$${I_0} = {|{{E_{xo,0}} + {E_{xe,0}}} |^2} = A(1 + \gamma \sin \Delta {w_x}),$$
$${I_1} = {|{{E_{xo,45}} + {E_{xe,45}}} |^2} = A(1 + \gamma \cos \Delta {w_x}),$$
$${I_2} = {|{{E_{xo,90}} + {E_{xe,90}}} |^2} = A(1 - \gamma \sin \Delta {w_x}),$$
$${I_3} = {|{{E_{xo,135}} + {E_{xe,135}}} |^2} = A(1 - \gamma \cos \Delta {w_x})$$
where I1, I2, I3 and I4 are the intensities detected by four different pixel sets of the PPC1, which correspond to sub-array pixels with the polarization angles of 0°, 45°, 90°, 135°, respectively, and A is the mean intensity of the interference fringe denoted as (E02/4). Δwx means the phase difference between two laterally sheared wavefronts as [w(x+Δx,y)-w(x,y)] and γ indicates the visibility of the interference fringes. Because the interference fringes in Eqs. (6)–(9) are shifted as π/2 successively, Δwx can be extracted by the well-known 4-step phase shifting algorithm as
$$\Delta {w_x} = w(x + \Delta x,y) - w(x,y) = {\tan ^{ - 1}}\left( {\frac{{{I_0} - {I_2}}}{{{I_1} - {I_3}}}} \right)$$
In the meanwhile, Δwy is also obtained in the y-shearing part with the same manner and the measurement results of Δwx and Δwy can be used for the reconstruction of w(x,y) by the slope integration technique of Southwell’s zonal method [17,18].

3. Experimental results

In order to verify the capability of the proposed interferometer, spherical/aspherical concave mirrors were measured and compared with the reference measurement results by a commercial contact probe (UA3P-5) with 0.1um level total uncertainty.

Figure 3 presents the single-shot lateral shearing interferometer constructed in this investigation. As seen in Fig. 3, a single mode He-Ne laser with 632.8 nm wavelength was used as a light source and a rotating diffuser was able to remove coherent noises caused by speckle and diffraction. Two calcite (CaCO3) plates (PDC12005-AR400-800@Newlight Photonics Inc.) with 0.5 mm thickness were used as beam displacers for x- and y- wave shearing. In each LSP, 2x telecentric lens (#33-101@Edmund optics) of which the field of view (FOV) was (3.5 mm×3.5 mm) was used as an imaging lens. Commercialized pixelated polarizing cameras (BFS-U3-51S5P-C@FLIR) with the resolution of (2448×2048) and 3.45 µm pixel size were used to capture four phase shifted interference fringes at once. Before measuring specimens, the lateral wavefront shift was estimated with a USAF resolution target by measuring how many pixels are laterally shifted in the horizontal or vertical direction. The width of 111.36 µm on the target was allocated as 31 camera pixels and the images by the ordinary wave and the extraordinary wave are laterally shifted as 17 pixels in both of x- and y- directions. Then, the actual wavefront shift was calculated as (17×111.36 µm / 31 = 61.1 µm).

 figure: Fig. 3.

Fig. 3. Photograph of the proposed single-shot lateral shearing interferometry; RD: Rotating Diffuser, CL: Collimating Lens, LP: Linear Polarizer (45°), BS1,2: Beam Splitter, BD1,2: Beam Displacer, QWP1,2: Quarter Wave Plater (45°), IL1,2: Imaging Lens, PPC1,2: Pixelated Polarizing Camera.

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After calibrating the wavefront shift, several concave mirrors with various radii of curvature were measured. Figure 4 shows the measurement result of the concave mirror with a 150 mm radius of curvature as an example. As shown in Figs. 4(a)–4(d), the four phase shifted interference fringes of laterally sheared waves in both of x- and y- directions were used to calculate their corresponding phase maps. Based on these phase maps, the 3D surface profile of the mirror was reconstructed as Fig. 4(e). In this case, the radius of curvature was estimated to be 149.73 mm by fitting the spherical model to the reconstructed 3D surface profile, and the residual 3D error map was calculated after subtracting a best-fit model as shown in Fig. 4(f). Table 1 summarizes the measurement results of the radii of curvature for several spherical/aspheric mirrors, which shows that the measured values were consistent with the reference values measured by the commercial instrument of UA3P-5 for all cases. The consecutive 10 measurements were implemented and the standard deviations of measurement results were less than 0.02 mm, attributed to the polarization extinction ratio of the polarizers inside of the PPC and alignment errors. It is noted that the results by UA3P-5 as seen in Table 1 were taken by a point-by-point measurements with a spatial resolution of 9 µm and the measurement time for the full 3D scanning was a few minutes. Here, R and K represent the radius of curvature and conic constant of the test sample, respectively.

 figure: Fig. 4.

Fig. 4. Measurement results of spherical mirror with a radius of 150 mm: (a) four phase-shifted interferograms of x-sheared waves, (b) phase map of (a), (c) four phase-shifted interferograms of y-sheared waves, (d) phase map of (c), (e) reconstructed 3D surface profile using two phase maps, and (f) residual error map after subtracting a best-fit spherical surface from (e).

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Tables Icon

Table 1. Summary of the measurement results compared with a commercial instrument (UA3P-5)

Next, the instrument was used to measure a freeform surface object and we chose a F-theta lens as a representative example of freeform surface to validate our proposed interferometer. Figure 5 indicates the F-theta lens and the measurement area in this investigation. It is the injection molded plastic lens for the laser scanning unit used in laser printers and it has the freeform surface and does not have translational or rotational symmetry as seen in Fig. 5. The measurement area was (3.83 mm×0.45 mm) determined by the numerical aperture of the imaging lens.

 figure: Fig. 5.

Fig. 5. Photograph of the F-theta lens measured in this investigation. The measurement area was indicated as the black line on the specimen.

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Figures 6(a)–6(d) present the four phase shifted interference fringes and corresponding phase maps of x- and y- direction, respectively. Figures 6(e)–6(h) show the reconstructed 3D surface profile of F- theta lens and comparison of our measurement result to the reference surface profile obtained by UA3P-5. With regards to accuracy, these two measurement results were very close to each other without any significant difference and it is noted that the maximum deviation between two results was less than 0.2 µm. It is noted that four fiducial marks were located a little bit outside of the measurement area to confirm the measurement areas were the same as seen in Fig. 5.

 figure: Fig. 6.

Fig. 6. Measurement result of F-theta lens as an example of freeform surface: (a) four phase-shifted interferograms of x-sheared wave, (b) measured slope map of (a), (c) four phase-shifted interferograms of y-sheared wave, (d) measured slope map of (c), (e) 3D surface profile map measured by UA3P, (f) 3D surface profile map reconstructed by our proposed method, (g) comparison results of two methods of α-α′ line profile (e) and (h) β-β′ line profile (f).

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4. Discussion

The proposed single-shot surface profiler can instantly measure 3D surface profiles at the frame rate of the camera and its optical configuration is relatively simple. However, several calibrations for successful measurements are necessary before the genuine measurement procedures. The first calibration is the camera pixel matching between two shearing parts. In Fig. 1, two cameras are used for obtaining x- and y- shearing interference fringes and their images by ordinary waves should be exactly the same. For the purpose, the camera pixels were mechanically aligned and the remaining mismatch was calibrated by software in this investigation. As a result, the pixel mismatch was reduced less than the pixel size.

The second calibration is the wavefront shift by the birefringent material. We used a beam displacer as a wavefront shearing devices and the shift was 61.1 µm. Although this calibration is straightforward and the procedure is not so complicated, the beam displacer needs to be changed into another according to change of the targets because the sensitivity of the interferometer is strongly dependent on the wavefront shift as typical LSI is. In Table 1, the difference between the reference and the measured values was relatively large in the case of the concave spherical mirror with 200 mm radius of curvature. As the wavefront is approaching to plane, the lateral phase shift of the wavefront becomes smaller, which causes in low sensitivity of measuring the wavefront slope. To improve the sensitivity, the lateral wavefront shift needs to be larger, which means the replacement of the wavefront shearing device should be involved and the calibration is also implemented. In case that the surface of the target is enough to be far from the flat surface such as freeform surfaces, however, superfluous change of the device is avoidable. It is worthy to be investigated further for the relationship between the amount of the lateral wavefront shift and the shape of the specimens.

In the meantime, LSI has the fundamental limitation to measure the surface roughness because it is based on the slope measurement. By integrating the slope measurement results, the surface roughness cannot be validated in the reconstructed 3D surface profile. In Figs. 6(e) an (f), small defects were detected in the UA3P-5 measurement result, but they were not shown in the LSI measurement result. Measurement noise, randomly distributed in the phase map, can be also magnified by the integration method. However, LSI is beneficial to extract the shape of the surface with the aid of optimization technique as shown in the measurement results and the proposed system can be applied to inline-manufacturing process because of its single-shot characteristics.

5. Summary

In this investigation, a novel freeform surface profiler was proposed and verified. The proposed system adopted the spatial phase shifting by a pixelated polarizing camera to instantly obtain the phase map and the lateral shearing interferometric principle enabled to measure the surface profiles of freeform surfaces. By using birefringent materials to generate two sheared waves, the whole system was simple and the system size can be minimized. As the experimental results, the surface profiles of several concave mirrors were successfully reconstructed and they were consistent with the reference values obtained by conventional contact method. A F-theta lens was measured with the proposed system as an example of freeform surfaces. It is expected that the proposed system can be effectively applied to measure small size of freeform optics, limited by measuring other techniques, in real time.

Disclosures

The authors declare no conflicts of interest.

References

1. E. Savio, L. D. Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007). [CrossRef]  

2. F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013). [CrossRef]  

3. ISO 17450-1:2011, Geometrical product specifications (GPS).

4. M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free form surfaces,” Proc. SPIE 5457, 366–376 (2004). [CrossRef]  

5. P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49(23), 4404–4412 (2010). [CrossRef]  

6. M. T. Nguyen, Y.-S. Ghim, and H.-G. Rhee, “Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method,” Sci. Rep. 9(1), 3157 (2019). [CrossRef]  

7. G. Häusler, M. C. Knauer, C. Faber, C. Richter, S. Peterhänsel, C. Kranitzky, and K. Veit, “Deflectometry: 3D-metrology from nanometer to meter,” Fringe 2009 Springer, 1–6 (2009).

8. S. Scheiding, M. Beier, U.-D. Zeitner, S. Risse, and A. Gebhardt, “Freeform mirror fabrication and metrology using a high performance test CGH and advanced alignment features,” Proc. SPIE 8613, 86130J (2013). [CrossRef]  

9. C. Chao and J. H. Burge, “Stitching of off-axis sub-aperture null measurements of an aspheric surface,” Proc. SPIE 7063, 706316 (2008). [CrossRef]  

10. K. Pant, D. Burada, M. Michra, M. Bichra, M. Singh, A. Ghosh, G. Khan, S. Sinzinger, and C. Shakher, “Subaperture stitching for measurement of freeform wavefront,” Appl. Opt. 54(34), 10022–10028 (2015). [CrossRef]  

11. J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12(9), 2057–2060 (1973). [CrossRef]  

12. Y.-S. Ghim, H.-G. Rhee, A. Davies, H.-S. Yang, and Y.-W. Lee, “3D surface mapping of freeform optics using wavelength scanning lateral shearing interferometry,” Opt. Express 22(5), 5098–5105 (2014). [CrossRef]  

13. T. Ling, Y. Yang, X. Yue, D. Liu, Y. Ma, J. Bai, and K. Wang, “Common-path and compact wavefront diagnosis system based on cross grating lateral shearing interferometer,” Appl. Opt. 53(30), 7144–7152 (2014). [CrossRef]  

14. J.-C. F. Chanteloup and M. Cohen, “Compact high resolution four wave lateral shearing interferometer,” Proc. SPIE 5252, 282–292 (2004). [CrossRef]  

15. J. E. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004). [CrossRef]  

16. J. E. Millerd and M. North-Morris, “Dynamic interferometry: measurement of space optics and structures,” Proc. SPIE 10329, 103291G (2017). [CrossRef]  

17. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70(8), 998–1006 (1980). [CrossRef]  

18. P. H. Phuc, N. T. Manh, H.-G. Rhee, Y.-S. Ghim, H.-S. Yang, and Y.-W. Lee, “Improved wavefront reconstruction algorithm from slope measurements,” J. Korean Phys. Soc. 70(5), 469–474 (2017). [CrossRef]  

References

  • View by:

  1. E. Savio, L. D. Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007).
    [Crossref]
  2. F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
    [Crossref]
  3. ISO 17450-1:2011, Geometrical product specifications (GPS).
  4. M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free form surfaces,” Proc. SPIE 5457, 366–376 (2004).
    [Crossref]
  5. P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49(23), 4404–4412 (2010).
    [Crossref]
  6. M. T. Nguyen, Y.-S. Ghim, and H.-G. Rhee, “Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method,” Sci. Rep. 9(1), 3157 (2019).
    [Crossref]
  7. G. Häusler, M. C. Knauer, C. Faber, C. Richter, S. Peterhänsel, C. Kranitzky, and K. Veit, “Deflectometry: 3D-metrology from nanometer to meter,” Fringe 2009 Springer, 1–6 (2009).
  8. S. Scheiding, M. Beier, U.-D. Zeitner, S. Risse, and A. Gebhardt, “Freeform mirror fabrication and metrology using a high performance test CGH and advanced alignment features,” Proc. SPIE 8613, 86130J (2013).
    [Crossref]
  9. C. Chao and J. H. Burge, “Stitching of off-axis sub-aperture null measurements of an aspheric surface,” Proc. SPIE 7063, 706316 (2008).
    [Crossref]
  10. K. Pant, D. Burada, M. Michra, M. Bichra, M. Singh, A. Ghosh, G. Khan, S. Sinzinger, and C. Shakher, “Subaperture stitching for measurement of freeform wavefront,” Appl. Opt. 54(34), 10022–10028 (2015).
    [Crossref]
  11. J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12(9), 2057–2060 (1973).
    [Crossref]
  12. Y.-S. Ghim, H.-G. Rhee, A. Davies, H.-S. Yang, and Y.-W. Lee, “3D surface mapping of freeform optics using wavelength scanning lateral shearing interferometry,” Opt. Express 22(5), 5098–5105 (2014).
    [Crossref]
  13. T. Ling, Y. Yang, X. Yue, D. Liu, Y. Ma, J. Bai, and K. Wang, “Common-path and compact wavefront diagnosis system based on cross grating lateral shearing interferometer,” Appl. Opt. 53(30), 7144–7152 (2014).
    [Crossref]
  14. J.-C. F. Chanteloup and M. Cohen, “Compact high resolution four wave lateral shearing interferometer,” Proc. SPIE 5252, 282–292 (2004).
    [Crossref]
  15. J. E. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
    [Crossref]
  16. J. E. Millerd and M. North-Morris, “Dynamic interferometry: measurement of space optics and structures,” Proc. SPIE 10329, 103291G (2017).
    [Crossref]
  17. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70(8), 998–1006 (1980).
    [Crossref]
  18. P. H. Phuc, N. T. Manh, H.-G. Rhee, Y.-S. Ghim, H.-S. Yang, and Y.-W. Lee, “Improved wavefront reconstruction algorithm from slope measurements,” J. Korean Phys. Soc. 70(5), 469–474 (2017).
    [Crossref]

2019 (1)

M. T. Nguyen, Y.-S. Ghim, and H.-G. Rhee, “Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method,” Sci. Rep. 9(1), 3157 (2019).
[Crossref]

2017 (2)

J. E. Millerd and M. North-Morris, “Dynamic interferometry: measurement of space optics and structures,” Proc. SPIE 10329, 103291G (2017).
[Crossref]

P. H. Phuc, N. T. Manh, H.-G. Rhee, Y.-S. Ghim, H.-S. Yang, and Y.-W. Lee, “Improved wavefront reconstruction algorithm from slope measurements,” J. Korean Phys. Soc. 70(5), 469–474 (2017).
[Crossref]

2015 (1)

2014 (2)

2013 (2)

S. Scheiding, M. Beier, U.-D. Zeitner, S. Risse, and A. Gebhardt, “Freeform mirror fabrication and metrology using a high performance test CGH and advanced alignment features,” Proc. SPIE 8613, 86130J (2013).
[Crossref]

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

2010 (1)

2008 (1)

C. Chao and J. H. Burge, “Stitching of off-axis sub-aperture null measurements of an aspheric surface,” Proc. SPIE 7063, 706316 (2008).
[Crossref]

2007 (1)

E. Savio, L. D. Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007).
[Crossref]

2004 (3)

M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[Crossref]

J.-C. F. Chanteloup and M. Cohen, “Compact high resolution four wave lateral shearing interferometer,” Proc. SPIE 5252, 282–292 (2004).
[Crossref]

J. E. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

1980 (1)

1973 (1)

Angel, R. P.

Bai, J.

Beier, M.

S. Scheiding, M. Beier, U.-D. Zeitner, S. Risse, and A. Gebhardt, “Freeform mirror fabrication and metrology using a high performance test CGH and advanced alignment features,” Proc. SPIE 8613, 86130J (2013).
[Crossref]

Bichra, M.

Brock, N.

J. E. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Burada, D.

Burge, J. H.

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49(23), 4404–4412 (2010).
[Crossref]

C. Chao and J. H. Burge, “Stitching of off-axis sub-aperture null measurements of an aspheric surface,” Proc. SPIE 7063, 706316 (2008).
[Crossref]

Chanteloup, J.-C. F.

J.-C. F. Chanteloup and M. Cohen, “Compact high resolution four wave lateral shearing interferometer,” Proc. SPIE 5252, 282–292 (2004).
[Crossref]

Chao, C.

C. Chao and J. H. Burge, “Stitching of off-axis sub-aperture null measurements of an aspheric surface,” Proc. SPIE 7063, 706316 (2008).
[Crossref]

Chiffre, L. D.

E. Savio, L. D. Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007).
[Crossref]

Cohen, M.

J.-C. F. Chanteloup and M. Cohen, “Compact high resolution four wave lateral shearing interferometer,” Proc. SPIE 5252, 282–292 (2004).
[Crossref]

Davies, A.

Evans, C.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Faber, C.

G. Häusler, M. C. Knauer, C. Faber, C. Richter, S. Peterhänsel, C. Kranitzky, and K. Veit, “Deflectometry: 3D-metrology from nanometer to meter,” Fringe 2009 Springer, 1–6 (2009).

Fang, F. Z.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Gebhardt, A.

S. Scheiding, M. Beier, U.-D. Zeitner, S. Risse, and A. Gebhardt, “Freeform mirror fabrication and metrology using a high performance test CGH and advanced alignment features,” Proc. SPIE 8613, 86130J (2013).
[Crossref]

Ghim, Y.-S.

M. T. Nguyen, Y.-S. Ghim, and H.-G. Rhee, “Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method,” Sci. Rep. 9(1), 3157 (2019).
[Crossref]

P. H. Phuc, N. T. Manh, H.-G. Rhee, Y.-S. Ghim, H.-S. Yang, and Y.-W. Lee, “Improved wavefront reconstruction algorithm from slope measurements,” J. Korean Phys. Soc. 70(5), 469–474 (2017).
[Crossref]

Y.-S. Ghim, H.-G. Rhee, A. Davies, H.-S. Yang, and Y.-W. Lee, “3D surface mapping of freeform optics using wavelength scanning lateral shearing interferometry,” Opt. Express 22(5), 5098–5105 (2014).
[Crossref]

Ghosh, A.

Häusler, G.

M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[Crossref]

G. Häusler, M. C. Knauer, C. Faber, C. Richter, S. Peterhänsel, C. Kranitzky, and K. Veit, “Deflectometry: 3D-metrology from nanometer to meter,” Fringe 2009 Springer, 1–6 (2009).

Hayes, J.

J. E. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Kaminski, J.

M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[Crossref]

Khan, G.

Knauer, M.

M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[Crossref]

Knauer, M. C.

G. Häusler, M. C. Knauer, C. Faber, C. Richter, S. Peterhänsel, C. Kranitzky, and K. Veit, “Deflectometry: 3D-metrology from nanometer to meter,” Fringe 2009 Springer, 1–6 (2009).

Kranitzky, C.

G. Häusler, M. C. Knauer, C. Faber, C. Richter, S. Peterhänsel, C. Kranitzky, and K. Veit, “Deflectometry: 3D-metrology from nanometer to meter,” Fringe 2009 Springer, 1–6 (2009).

Lee, Y.-W.

P. H. Phuc, N. T. Manh, H.-G. Rhee, Y.-S. Ghim, H.-S. Yang, and Y.-W. Lee, “Improved wavefront reconstruction algorithm from slope measurements,” J. Korean Phys. Soc. 70(5), 469–474 (2017).
[Crossref]

Y.-S. Ghim, H.-G. Rhee, A. Davies, H.-S. Yang, and Y.-W. Lee, “3D surface mapping of freeform optics using wavelength scanning lateral shearing interferometry,” Opt. Express 22(5), 5098–5105 (2014).
[Crossref]

Ling, T.

Liu, D.

Ma, Y.

Manh, N. T.

P. H. Phuc, N. T. Manh, H.-G. Rhee, Y.-S. Ghim, H.-S. Yang, and Y.-W. Lee, “Improved wavefront reconstruction algorithm from slope measurements,” J. Korean Phys. Soc. 70(5), 469–474 (2017).
[Crossref]

Michra, M.

Millerd, J. E.

J. E. Millerd and M. North-Morris, “Dynamic interferometry: measurement of space optics and structures,” Proc. SPIE 10329, 103291G (2017).
[Crossref]

J. E. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Nguyen, M. T.

M. T. Nguyen, Y.-S. Ghim, and H.-G. Rhee, “Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method,” Sci. Rep. 9(1), 3157 (2019).
[Crossref]

North-Morris, M.

J. E. Millerd and M. North-Morris, “Dynamic interferometry: measurement of space optics and structures,” Proc. SPIE 10329, 103291G (2017).
[Crossref]

J. E. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Novak, M.

J. E. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Pant, K.

Parks, R. E.

Peterhänsel, S.

G. Häusler, M. C. Knauer, C. Faber, C. Richter, S. Peterhänsel, C. Kranitzky, and K. Veit, “Deflectometry: 3D-metrology from nanometer to meter,” Fringe 2009 Springer, 1–6 (2009).

Phuc, P. H.

P. H. Phuc, N. T. Manh, H.-G. Rhee, Y.-S. Ghim, H.-S. Yang, and Y.-W. Lee, “Improved wavefront reconstruction algorithm from slope measurements,” J. Korean Phys. Soc. 70(5), 469–474 (2017).
[Crossref]

Rhee, H.-G.

M. T. Nguyen, Y.-S. Ghim, and H.-G. Rhee, “Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method,” Sci. Rep. 9(1), 3157 (2019).
[Crossref]

P. H. Phuc, N. T. Manh, H.-G. Rhee, Y.-S. Ghim, H.-S. Yang, and Y.-W. Lee, “Improved wavefront reconstruction algorithm from slope measurements,” J. Korean Phys. Soc. 70(5), 469–474 (2017).
[Crossref]

Y.-S. Ghim, H.-G. Rhee, A. Davies, H.-S. Yang, and Y.-W. Lee, “3D surface mapping of freeform optics using wavelength scanning lateral shearing interferometry,” Opt. Express 22(5), 5098–5105 (2014).
[Crossref]

Richter, C.

G. Häusler, M. C. Knauer, C. Faber, C. Richter, S. Peterhänsel, C. Kranitzky, and K. Veit, “Deflectometry: 3D-metrology from nanometer to meter,” Fringe 2009 Springer, 1–6 (2009).

Risse, S.

S. Scheiding, M. Beier, U.-D. Zeitner, S. Risse, and A. Gebhardt, “Freeform mirror fabrication and metrology using a high performance test CGH and advanced alignment features,” Proc. SPIE 8613, 86130J (2013).
[Crossref]

Savio, E.

E. Savio, L. D. Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007).
[Crossref]

Scheiding, S.

S. Scheiding, M. Beier, U.-D. Zeitner, S. Risse, and A. Gebhardt, “Freeform mirror fabrication and metrology using a high performance test CGH and advanced alignment features,” Proc. SPIE 8613, 86130J (2013).
[Crossref]

Schmitt, R.

E. Savio, L. D. Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007).
[Crossref]

Shakher, C.

Singh, M.

Sinzinger, S.

Southwell, W. H.

Su, P.

Veit, K.

G. Häusler, M. C. Knauer, C. Faber, C. Richter, S. Peterhänsel, C. Kranitzky, and K. Veit, “Deflectometry: 3D-metrology from nanometer to meter,” Fringe 2009 Springer, 1–6 (2009).

Wang, K.

Wang, L.

Weckenmann, A.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Wyant, J. C.

J. E. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12(9), 2057–2060 (1973).
[Crossref]

Yang, H.-S.

P. H. Phuc, N. T. Manh, H.-G. Rhee, Y.-S. Ghim, H.-S. Yang, and Y.-W. Lee, “Improved wavefront reconstruction algorithm from slope measurements,” J. Korean Phys. Soc. 70(5), 469–474 (2017).
[Crossref]

Y.-S. Ghim, H.-G. Rhee, A. Davies, H.-S. Yang, and Y.-W. Lee, “3D surface mapping of freeform optics using wavelength scanning lateral shearing interferometry,” Opt. Express 22(5), 5098–5105 (2014).
[Crossref]

Yang, Y.

Yue, X.

Zeitner, U.-D.

S. Scheiding, M. Beier, U.-D. Zeitner, S. Risse, and A. Gebhardt, “Freeform mirror fabrication and metrology using a high performance test CGH and advanced alignment features,” Proc. SPIE 8613, 86130J (2013).
[Crossref]

Zhang, G. X.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Zhang, X. D.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Appl. Opt. (4)

CIRP Ann. (2)

E. Savio, L. D. Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007).
[Crossref]

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

J. Korean Phys. Soc. (1)

P. H. Phuc, N. T. Manh, H.-G. Rhee, Y.-S. Ghim, H.-S. Yang, and Y.-W. Lee, “Improved wavefront reconstruction algorithm from slope measurements,” J. Korean Phys. Soc. 70(5), 469–474 (2017).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Express (1)

Proc. SPIE (6)

S. Scheiding, M. Beier, U.-D. Zeitner, S. Risse, and A. Gebhardt, “Freeform mirror fabrication and metrology using a high performance test CGH and advanced alignment features,” Proc. SPIE 8613, 86130J (2013).
[Crossref]

C. Chao and J. H. Burge, “Stitching of off-axis sub-aperture null measurements of an aspheric surface,” Proc. SPIE 7063, 706316 (2008).
[Crossref]

M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[Crossref]

J.-C. F. Chanteloup and M. Cohen, “Compact high resolution four wave lateral shearing interferometer,” Proc. SPIE 5252, 282–292 (2004).
[Crossref]

J. E. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

J. E. Millerd and M. North-Morris, “Dynamic interferometry: measurement of space optics and structures,” Proc. SPIE 10329, 103291G (2017).
[Crossref]

Sci. Rep. (1)

M. T. Nguyen, Y.-S. Ghim, and H.-G. Rhee, “Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method,” Sci. Rep. 9(1), 3157 (2019).
[Crossref]

Other (2)

G. Häusler, M. C. Knauer, C. Faber, C. Richter, S. Peterhänsel, C. Kranitzky, and K. Veit, “Deflectometry: 3D-metrology from nanometer to meter,” Fringe 2009 Springer, 1–6 (2009).

ISO 17450-1:2011, Geometrical product specifications (GPS).

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Figures (6)

Fig. 1.
Fig. 1. Optical configuration of the proposed single-shot lateral shearing interferometry; RD: Rotating Diffuser, CL: Collimating Lens, LP: Linear Polarizer (45°), BS1,2: Beam Splitter, BD1,2: Beam Displacer, QWP1,2: Quarter Wave Plater (45°), IL1,2: Imaging Lens, PPC1,2: Pixelated Polarizing Camera.
Fig. 2.
Fig. 2. (a) Laterally sheared interference fringe pattern directly obtained by the PPC1 and four sub- images with the polarization angle of (b) 0°, (c) 45°, (d) 90° and (e) 135°.
Fig. 3.
Fig. 3. Photograph of the proposed single-shot lateral shearing interferometry; RD: Rotating Diffuser, CL: Collimating Lens, LP: Linear Polarizer (45°), BS1,2: Beam Splitter, BD1,2: Beam Displacer, QWP1,2: Quarter Wave Plater (45°), IL1,2: Imaging Lens, PPC1,2: Pixelated Polarizing Camera.
Fig. 4.
Fig. 4. Measurement results of spherical mirror with a radius of 150 mm: (a) four phase-shifted interferograms of x-sheared waves, (b) phase map of (a), (c) four phase-shifted interferograms of y-sheared waves, (d) phase map of (c), (e) reconstructed 3D surface profile using two phase maps, and (f) residual error map after subtracting a best-fit spherical surface from (e).
Fig. 5.
Fig. 5. Photograph of the F-theta lens measured in this investigation. The measurement area was indicated as the black line on the specimen.
Fig. 6.
Fig. 6. Measurement result of F-theta lens as an example of freeform surface: (a) four phase-shifted interferograms of x-sheared wave, (b) measured slope map of (a), (c) four phase-shifted interferograms of y-sheared wave, (d) measured slope map of (c), (e) 3D surface profile map measured by UA3P, (f) 3D surface profile map reconstructed by our proposed method, (g) comparison results of two methods of α-α′ line profile (e) and (h) β-β′ line profile (f).

Tables (1)

Tables Icon

Table 1. Summary of the measurement results compared with a commercial instrument (UA3P-5)

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

E t = E 0 e j w ( x , y ) ( 1 1 )
E x o = E 0 2 e j w ( x , y ) ( 1 0 )
E x e = E 0 2 e j w ( x + Δ x , y ) ( 0 1 )
E x o = E 0 4 e j w ( x , y ) ( 1 + j 1 j ) ,
E x e = E 0 4 e j w ( x + Δ x , y ) ( 1 j 1 + j )
I 0 = | E x o , 0 + E x e , 0 | 2 = A ( 1 + γ sin Δ w x ) ,
I 1 = | E x o , 45 + E x e , 45 | 2 = A ( 1 + γ cos Δ w x ) ,
I 2 = | E x o , 90 + E x e , 90 | 2 = A ( 1 γ sin Δ w x ) ,
I 3 = | E x o , 135 + E x e , 135 | 2 = A ( 1 γ cos Δ w x )
Δ w x = w ( x + Δ x , y ) w ( x , y ) = tan 1 ( I 0 I 2 I 1 I 3 )

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