## A high-accuracy and convenient figure measurement system for large convex lens |

Optics Express, Vol. 20, Issue 10, pp. 10761-10775 (2012)

http://dx.doi.org/10.1364/OE.20.010761

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### Abstract

We present a novel optical configuration of a phase-shifting interferometer for high-accuracy figure metrology of large dioptric convex spherical surfaces. The conformation and design considerations according to measurement accuracy, practicability, and system errors analysis are described. More in detail, we show the design principle and methods for the crucial parts. Some are expounded upon with examples for thorough understanding. The measurement procedures and the alignment approaches are also described. Finally, a verification experiment is further presented to verify our theoretical design. This system gives full-aperture and high-precision surface testing while maintaining relatively low cost and convenient operation.

© 2012 OSA

## 1. Introduction

1. Y. Ohmura, “The optical design for microlithographic lenses,” Proc. SPIE **6342**, 63421T (2007). [CrossRef]

2. T. Matsuyama, Y. Ohmura, and D. M. Williamson, “The lithographic lens: its history and evolution,” Proc. SPIE **6154**, 615403 (2006). [CrossRef]

8. S. Chen, S. Li, Y. Dai, L. Ding, and S. Zeng, “Experimental study on subaperture testing with iterative stitching algorithm,” Opt. Express **16**(7), 4760–4765 (2008). [CrossRef] [PubMed]

9. Y. Dai, S. Chen, S. Li, H. Hu, and Q. Zhang, “Stylus profilometry for steep aspheric surfaces with multisegment stitching,” Opt. Eng. **50**(1), 013601 (2011). [CrossRef]

11. L. Ekstrand and S. Zhang, “Three-dimensional profilometry with nearly focused binary phase-shifting algorithms,” Opt. Lett. **36**(23), 4518–4520 (2011). [CrossRef] [PubMed]

12. H. Jing, L. Kuang, T. Fan, and X. Cao, “Measurement of large aspherical mirrors using coordinate measurement machine during the grinding process,” Proc. SPIE **6148**, 61480I (2006). [CrossRef]

13. V. N. Chekal', Y. I. Chudakov, and S. E. Shevtsov, “The use of coordinate-measurement machines to optimize the technology of automatic shaping of optical surfaces,” J. Opt. Technol. **75**(11), 755–759 (2008). [CrossRef]

14. J. H. Burge, “Fizeau interferometry for large convex surfaces,” Proc. SPIE **2536**, 127–138 (1995). [CrossRef]

## 2. Description of the system configuration and the theory principle

15. B. M. Robinson and P. J. Reardon, “Distortion compensation in interferometric testing of mirrors,” Appl. Opt. **48**(3), 560–565 (2009). [CrossRef] [PubMed]

16. N. Bobroff, “Residual errors in laser interferometry from air turbulence and nonlinearity,” Appl. Opt. **26**(13), 2676–2682 (1987). [CrossRef] [PubMed]

17. C. Zhao and J. H. Burge, “Vibration-compensated interferometer for surface metrology,” Appl. Opt. **40**(34), 6215–6222 (2001). [CrossRef] [PubMed]

_{g(x,y)=a0(x,y)+a1(x,y)cos[2πλL(x,y)],}(1)where

*g(x, y)*is the intensity at a pixel positioned at point

*(x, y)*,

*a*

_{0}

*(x, y)*is the background,

*a*

_{1}

*(x, y)*is the modulation, and

*L*is the optical path difference of the testing surface and the reference mirror. If a tunable laser is used by the interferometer to change the wavelength from

*λ*to

*λ-*Δ

*λ*, the intensity of the interferogram becomes

_{m}_{gm(x,y)=a0(x,y)+a1(x,y)cos[2πλ−ΔλmL(x,y)].}(2)

_{gm(x,y)=a0(x,y)+a1(x,y)cos[2πλL(x,y)+2πΔλmλ(λ−Δλm)L(x,y)],}(3)where

*g*is the

_{m}*m*th acquisition of the interferogram and

_{θm=2πL(x,y)Δλmλ(λ−Δλm)}is the

*m*th phase shift. For short interferometric cavity measurement, we assume

_{L=10 mm}and standard 13 steps phase shifting

_{λ−λm≈λ=633 nm},

_{m=13}, and

_{θm=13×π4=134π}. We get that in one phase-shifting period, the shifting wavelength bandwidth is

_{Δλm≈0.07 nm}, and the bandwidth will be much narrower in the long cavity measurement. For a high-precision test, it is better to correct the chromatic aberration in the system, and it is not too hard to correct chromatic aberration for so narrow a bandwidth. The imperfect performance of hardware, phase-shift errors caused by laser’s nonlinearity, and high-order harmonics caused by the CCD’s nonlinearity usually introduce errors in retrieving information from the test plate. To solve this problem, the laser and CCD must be calibrated, and we plan to use a new phase-shifting algorithm [18

18. Z. Shi, J. Zhang, Y. Sui, J. Peng, F. Yan, and H. Yang, “Design of algorithms for phase shifting interferometry using self-convolution of the rectangle window,” Opt. Express **19**(15), 14671–14681 (2011). [CrossRef] [PubMed]

## 3. Details of optical design and measurement procedure

### 3.1 Diverger lens design

_{sinθ=nsinθ′,}(4)

_{Δθ=θ′−θ.}(5)

*i*surfaces, so the total deflection angle is invariable.

_{Δθtotal=arctan(12×0.8)−0=32°=∑1i(θ′i−θi).}(6)

*θ*and

*θ’*and thus reduces aberrations as the nonlinearity. Here we use 3 pieces of a ZF6 lens with a relatively high refraction index and a piece of K9 lens to correct monochromatic aberration and chromatic aberration (Chinese glass). The optimization merit function is written as “MeritFunction1”.

15. B. M. Robinson and P. J. Reardon, “Distortion compensation in interferometric testing of mirrors,” Appl. Opt. **48**(3), 560–565 (2009). [CrossRef] [PubMed]

*ρ*is the radial position in the image coordinate,

_{c}*ρ*is radial position in the object coordinate,

_{t}*m*is the magnification from object to image, and

*ε*is the distortion coefficient. As a result, the radial displacement caused by distortion is expressed as

_{Δρc=ρ′c−ρc=mερt3.}(7)

_{W(ρ,θ)=∑n,kankρncoskθ+bnkρnsinkθ.}(8)

*ε*is usually very small in a deformed wavefront, let

_{(1+ε)n≈1+nε}, and then we get the following relation:

_{anρ′n→anρn+annερn+2.}(9)

*ε*.

*O*, which is the site of the stop. Plane

*A*plumbs the optical axis and passes through both ends of the object. The distance between plane

*A*and point

*O*is

*L*. The imaging system focus is on plane

*A*, the intersection of principal rays from object points with plane

*A*is

*B*, and the intersection of principal rays with the outermost surface of the diverger is

*C*. Assuming the focal length of the diverger is

*f*, the magnification is

*m*, the angle between principal rays and optical axis is

*θ*, the distance from

*B*to optical axis is

*h’,*and the distance from

*C*to optical axis is

*h,*we get that

_{m=h'h=Lf.}(10)

_{hi'=Lcosθi,}(11)

_{hi=fcosθi,}(12)

_{cosθi=hif.}(13)

_{Merit function=∑1i|cosθi−hif|.}(14)The optimization merit function is written as “MeritFunction2”.

_{Merit function=w1⋅Meritfunction1+w2⋅Meritfunction2.}(15)

*w*and

_{1}*w*are weighting factors.

_{2}*w*can be much larger than

_{2}*w*for different units.

_{1}### 3.2 Measurement procedure

**[Au: Do you mean ‘process’ instead of ‘figure’ here? Please clarify.]**of a large dioptric convex spherical surface with our measurement system is composed of two main steps. In Step 1, the SACS and the lens to be tested are installed and adjusted through monitoring interferograms, which are produced by the back bare surface of the optical flat and the test surface. Computer auxiliary adjustment methods and other similar skills can be used to control the relative positions. The main judgment standard is wavefront slope error. Because this part is not in the cavity in the final measurement stage (Step 2) and can be calibrated, this is not too rigorous. Other judgment standards like distortion are also helpful. In Step 2, the optical flat is moved out of the light path; then, we install and adjust the reference sphere and measure the figure of the test surface by wavelength phase shifting. Figure 5 illustrates that process. The reference surface and the back bare surface of the optical flat must be carefully calibrated before testing, as systematic errors are mostly limited by the ability to figure and measure the concave reference surface. Systematic optical path difference (OPD) error is a combination of an imperfect illumination wavefront, reference surface figure error, imaging aberrations depending on field and pupil, the ITF, and so on. Thus for high-accuracy measurements, system error calibrations are necessary. And, a long thermalization time is a prerequisite to reach nanometer range uncertainty.

- 1. Primary arrangement of the SACS and the test lens. The stop hole integrated in diverger lens is used as a coarse coordinate to locate the SACS and the test lens, and then the two elements are adjusted while we observe location changes of the light points in auxiliary adjustment module through a CMOS camera until we get a good result.
- 2. Fine adjustment of the diverger lens, the SACS and the test lens. Here, the two interference surfaces are the back bare surface of the optical flat and the test surface; therefore, we can use interferograms produced by those two surfaces to guide the adjustment. The imaging module is focused on the pupil of the optical system, and a phase-measuring algorithm is used to get quantities of the wavefront. Computer auxiliary adjustment methods may be used for multi-component cases. The object of this section is to generate a fairly perfect wavefront transmitted through and reflected at the test surface, and thus the wavefront slope errors may be the objective function. The weak reflection of the optical flat plate can be ignored for wavefront slope measurement, and the front surface may be slightly tilted but smaller than the wedge angle of ordinary TF for further improvement.
- 3. Move the optical flat out of light path, then install and adjust the reference sphere. In this section, the two interference surfaces are the test surface and the reference sphere surface, while the imaging module is focused on the test surface. The adjustment means can be similar to those in Section 2.
- 4. Measuring figure of the test surface by wavelength phase shifting. Here, the wavelength phase-shifting method and algorithm we used is by wavelength stepping and suffers from fundamental limitations in two-beam interference. There are some other wavelength phase-shifting styles and algorithms based on frequency domain that can also be used in handling multiple-surface interference [21,22
21. L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt.

**42**(13), 2354–2365 (2003). [CrossRef] [PubMed]]. So, the measurement results can be compared. But the cavity lengths are restricted by specific expressions, and thus they are a little inconvenient in practice.22. K. Okada, H. Sakuta, T. Ose, and J. Tsujiuchi, “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable-source phase shifting interferometry,” Appl. Opt.

**29**(22), 3280–3285 (1990). [CrossRef] [PubMed]

### 3.3 Spherical aberration compensation system (SACS)

_{δL'=n1u1sinU1nk'uk'sinUk'δL0−12nk'uk'sinUk'∑1kS−=−12nk'uk'sinUk'∑1kS−,}(16)

_{S−=niLsinU(sinI−sinI')(sinI'−sinU)cos12(I−U)cos12(I'+U)cos12(I+I').}(17)

_{I=0}. Thus it doesn’t introduce any spherical aberration. So, the total spherical aberration of the test lens equals the spherical aberration introduced by the other surface. According to Eq. (16) and Eq. (17), we can simply estimate the spherical aberration quantity introduced by the lens to be tested in our measurement system by comparing the curvature difference of its two surfaces. To illustrate this, we show spherical aberrations introduced by four types of test lenses whose forms are concentric, meniscus, plano-convex, and bi-convex in Fig. 7 . The spherical aberrations are progressively increased.

_{δL′=0}, it can create three cases, called aplanatism.

_{(1)L=0,L′=0,(2)sinI−sinI′=0,I=I′=0,(3)sinI′−sinU=0,I′=U,L=(n+n′)r/n,L′=(n+n′)r/n′.}(18)

_{β=NA′NA=n,}(19)where

*n*is the index of refraction. So, we can test a large NA surface with a smaller NA diverger. For example, an NA = 0.9 convex surface can be test by a diverger lens.

_{NA=0.91.5=0.6.}(20)

*k*pieces of the lens in the spherical aberration compensation lens, and we can obtain the spherical aberration for different aperture height

*y*:

_{SphT=∑1k(n−1)l2'2n2y2[(1r1−n+1l1)(1r1−1l1)2y14−(1r2−n+1l2')(1r2−1l2')y24].}(21)

_{SphTtest+SphTcompensation=SphTtest+SphT=0.}(22)

23. G. D. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B **62**(1), 2–8 (1949). [CrossRef]

## 4. Verification experiment

### 4.1 Experimental setup

### 4.2 Experimental results

## 5. Conclusion and discussion

## Acknowledgments

## References and links

1. | Y. Ohmura, “The optical design for microlithographic lenses,” Proc. SPIE |

2. | T. Matsuyama, Y. Ohmura, and D. M. Williamson, “The lithographic lens: its history and evolution,” Proc. SPIE |

3. | |

4. | P. E. Murphy, G. W. Forbes, J. F. Fleig, D. Miladinovic, G. DeVries, and S. O'Donohue, “Recent advances in subaperture stitching interferometry,” in |

5. | M. Bray, “Stitching interferometry–the long and winding road,” in |

6. | P. Zhang, H. Zhao, X. Zhou, and J. Li, “Sub-aperture stitching interferometry using stereovision positioning technique,” Opt. Express |

7. | J. H. Burge, P. Su, and C. Zhao, “Optical metrology for very large convex aspheres,” Proc. SPIE |

8. | S. Chen, S. Li, Y. Dai, L. Ding, and S. Zeng, “Experimental study on subaperture testing with iterative stitching algorithm,” Opt. Express |

9. | Y. Dai, S. Chen, S. Li, H. Hu, and Q. Zhang, “Stylus profilometry for steep aspheric surfaces with multisegment stitching,” Opt. Eng. |

10. | A. Wiegmann, M. Schulz, and C. Elster, “Absolute profile measurement of large moderately flat optical surfaces with high dynamic range,” Opt. Express |

11. | L. Ekstrand and S. Zhang, “Three-dimensional profilometry with nearly focused binary phase-shifting algorithms,” Opt. Lett. |

12. | H. Jing, L. Kuang, T. Fan, and X. Cao, “Measurement of large aspherical mirrors using coordinate measurement machine during the grinding process,” Proc. SPIE |

13. | V. N. Chekal', Y. I. Chudakov, and S. E. Shevtsov, “The use of coordinate-measurement machines to optimize the technology of automatic shaping of optical surfaces,” J. Opt. Technol. |

14. | J. H. Burge, “Fizeau interferometry for large convex surfaces,” Proc. SPIE |

15. | B. M. Robinson and P. J. Reardon, “Distortion compensation in interferometric testing of mirrors,” Appl. Opt. |

16. | N. Bobroff, “Residual errors in laser interferometry from air turbulence and nonlinearity,” Appl. Opt. |

17. | C. Zhao and J. H. Burge, “Vibration-compensated interferometer for surface metrology,” Appl. Opt. |

18. | Z. Shi, J. Zhang, Y. Sui, J. Peng, F. Yan, and H. Yang, “Design of algorithms for phase shifting interferometry using self-convolution of the rectangle window,” Opt. Express |

19. | R. Jóźwicki, “Propagation of an aberrated wave with nonuniform amplitude distribution and its influence upon the interferometric measurement accuracy,” Opt. Appl. |

20. | S. O'Donohue, G. Devries, P. Murphy, G. Forbes, and P. Dumas, “Calibrating interferometric imaging distortion using subaperture stitching interferometry,” Proc. SPIE |

21. | L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. |

22. | K. Okada, H. Sakuta, T. Ose, and J. Tsujiuchi, “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable-source phase shifting interferometry,” Appl. Opt. |

23. | G. D. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B |

24. | Zygo is a registered trademark of Zygo Corporation. |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

(220.1000) Optical design and fabrication : Aberration compensation

(220.4830) Optical design and fabrication : Systems design

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: January 10, 2012

Revised Manuscript: March 30, 2012

Manuscript Accepted: April 5, 2012

Published: April 25, 2012

**Citation**

Zhihui Tian, Wang Yang, Yongxin Sui, Yusi Kang, Weiqi Liu, and Huaijiang Yang, "A high-accuracy and convenient figure measurement system for large convex lens," Opt. Express **20**, 10761-10775 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-10761

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### References

- Y. Ohmura, “The optical design for microlithographic lenses,” Proc. SPIE6342, 63421T (2007). [CrossRef]
- T. Matsuyama, Y. Ohmura, and D. M. Williamson, “The lithographic lens: its history and evolution,” Proc. SPIE6154, 615403 (2006). [CrossRef]
- 3. http://www.qedmrf.com/metrology/products/ssi-a .
- P. E. Murphy, G. W. Forbes, J. F. Fleig, D. Miladinovic, G. DeVries, and S. O'Donohue, “Recent advances in subaperture stitching interferometry,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper OFWC2.
- M. Bray, “Stitching interferometry–the long and winding road,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OMA5.
- P. Zhang, H. Zhao, X. Zhou, and J. Li, “Sub-aperture stitching interferometry using stereovision positioning technique,” Opt. Express18(14), 15216–15222 (2010). [CrossRef] [PubMed]
- J. H. Burge, P. Su, and C. Zhao, “Optical metrology for very large convex aspheres,” Proc. SPIE7018, 701818 (2008). [CrossRef]
- S. Chen, S. Li, Y. Dai, L. Ding, and S. Zeng, “Experimental study on subaperture testing with iterative stitching algorithm,” Opt. Express16(7), 4760–4765 (2008). [CrossRef] [PubMed]
- Y. Dai, S. Chen, S. Li, H. Hu, and Q. Zhang, “Stylus profilometry for steep aspheric surfaces with multisegment stitching,” Opt. Eng.50(1), 013601 (2011). [CrossRef]
- A. Wiegmann, M. Schulz, and C. Elster, “Absolute profile measurement of large moderately flat optical surfaces with high dynamic range,” Opt. Express16(16), 11975–11986 (2008). [CrossRef] [PubMed]
- L. Ekstrand and S. Zhang, “Three-dimensional profilometry with nearly focused binary phase-shifting algorithms,” Opt. Lett.36(23), 4518–4520 (2011). [CrossRef] [PubMed]
- H. Jing, L. Kuang, T. Fan, and X. Cao, “Measurement of large aspherical mirrors using coordinate measurement machine during the grinding process,” Proc. SPIE6148, 61480I (2006). [CrossRef]
- V. N. Chekal', Y. I. Chudakov, and S. E. Shevtsov, “The use of coordinate-measurement machines to optimize the technology of automatic shaping of optical surfaces,” J. Opt. Technol.75(11), 755–759 (2008). [CrossRef]
- J. H. Burge, “Fizeau interferometry for large convex surfaces,” Proc. SPIE2536, 127–138 (1995). [CrossRef]
- B. M. Robinson and P. J. Reardon, “Distortion compensation in interferometric testing of mirrors,” Appl. Opt.48(3), 560–565 (2009). [CrossRef] [PubMed]
- N. Bobroff, “Residual errors in laser interferometry from air turbulence and nonlinearity,” Appl. Opt.26(13), 2676–2682 (1987). [CrossRef] [PubMed]
- C. Zhao and J. H. Burge, “Vibration-compensated interferometer for surface metrology,” Appl. Opt.40(34), 6215–6222 (2001). [CrossRef] [PubMed]
- Z. Shi, J. Zhang, Y. Sui, J. Peng, F. Yan, and H. Yang, “Design of algorithms for phase shifting interferometry using self-convolution of the rectangle window,” Opt. Express19(15), 14671–14681 (2011). [CrossRef] [PubMed]
- R. Jóźwicki, “Propagation of an aberrated wave with nonuniform amplitude distribution and its influence upon the interferometric measurement accuracy,” Opt. Appl.20, 229–252 (1990).
- S. O'Donohue, G. Devries, P. Murphy, G. Forbes, and P. Dumas, “Calibrating interferometric imaging distortion using subaperture stitching interferometry,” Proc. SPIE5869, 156–158 (2005).
- L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt.42(13), 2354–2365 (2003). [CrossRef] [PubMed]
- K. Okada, H. Sakuta, T. Ose, and J. Tsujiuchi, “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable-source phase shifting interferometry,” Appl. Opt.29(22), 3280–3285 (1990). [CrossRef] [PubMed]
- G. D. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B62(1), 2–8 (1949). [CrossRef]
- Zygo is a registered trademark of Zygo Corporation.

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