## Phase errors in high line density CGH used for aspheric testing: beyond scalar approximation |

Optics Express, Vol. 21, Issue 10, pp. 11638-11651 (2013)

http://dx.doi.org/10.1364/OE.21.011638

Acrobat PDF (1088 KB)

### Abstract

One common way to measure asphere and freeform surfaces is the interferometric Null test, where a computer generated hologram (CGH) is placed in the object path of the interferometer. If undetected phase errors are present in the CGH, the measurement will show systematic errors. Therefore the absolute phase of this element has to be known. This phase is often calculated using scalar diffraction theory. In this paper we discuss the limitations of this theory for the prediction of the absolute phase generated by different implementations of CGH. Furthermore, for regions where scalar approximation is no longer valid, rigorous simulations are performed to identify phase sensitive structure parameters and evaluate fabrication tolerances for typical gratings.

© 2013 OSA

## 1. Introduction

1. J. C. Wyant and V. P. Bennett, “Using computer generated holograms to test aspheric wavefronts,” omputer Generated Holograms to Test Aspheric Wavefronts,” Appl. Opt. **11**2833–2839 (1972) [CrossRef] [PubMed] .

2. S. M. Arnold, “How to test an asphere with a computer-generated hologram,” Proc. SPIE **1052**191–197 (1989) [CrossRef] .

3. E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, “Investigation of large null-CGH realization,” Proc. SPIE **4440** (2001) [CrossRef] .

5. S. Reichelt, C. Pruss, and H. J. Tiziani, “Specification and characterization of CGHs for interferometrical optical testing,” Proc. SPIE **4778** (2002) [CrossRef] .

7. P. Zhou and J. H. Burge, “Fabrication error analysis and experimental demonstration for computer-generated holograms,” Appl. Opt. **46**657–663 (2007) [CrossRef] [PubMed] .

8. P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. **43**2063–2085 (1996) [CrossRef] .

9. W. Yu, K. Takahara, T. Konishi, T. Yotsuya, and Y. Ichioka, “Fabrication of multilevel phase computer-generated hologram elements based on effective medium theory,” Appl. Opt. **39**3531–3536 (2000) [CrossRef] .

10. I. Richter, P.-C. Sun, F. Xu, and Yeshayahu Fainman, “Design considerations of form birefringent microstructures,” Appl. Opt. **34**2421–2429 (1995) [CrossRef] [PubMed] .

11. N. Bokor, R. Shechter, N. Davidson, A. A. Friesem, and Erez Hasman, “Achromatic phase retarder by slanted illumination of a dielectric grating with period comparable with the wavelength,” Appl. Opt. **40**2076–2080 (2001) [CrossRef] .

*Kley et al.*[3

3. E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, “Investigation of large null-CGH realization,” Proc. SPIE **4440** (2001) [CrossRef] .

*Iff et al.*[12

12. W. Iff, S. Glaubrecht, N. Lindlein, and J. Schwider, “Untersuchung der Abweichungen zwischen skalarer und rigoroser Rechnung an CGHs in Twyman-Green-Interferometern zur Linsenprüfung,” DGaO-Proceedings http://www.dgao-proceedings.de (2010).

## 2. Limitations of Scalar Diffraction Theory

*b*, duty cycle

*D*=

*b/*Λ and height

*h*, the complex amplitude of the far field wavefront function

*U*can be written as shown in Eq. 1 [6

6. Y.-C. Chang, P. Zhou, and J. H. Burge, “Analysis of phase sensitivity for binary computer-generated holograms,” Appl. Opt. **45**4223–4234 (2006) [CrossRef] [PubMed] .

*A*

_{0}and

*A*

_{1}correspond to the output wavefront amplitude from the top and the bottom of the grating.

*ϕ*is defined by the ratio of imaginary and real part of

*U*and can be written as:

*ϕ*is given in radians. For simplicity, we will use the wavefront phase

*W*which is defined as:

^{st}order (

*m*= 1), as this order is mostly used for interferometry. For the scalar approximation an ideal reflection was assumed (

*A*

_{0}=

*A*

_{1}= 1).

*D*of 0.5 and a structure height corresponding to a wavefront phase change

*W*of 0.5. Another interesting point is duty cycle sensitivity. Scalar theory states that the phase of the first order is independent from the duty cycle of the grating; not so the results from the rigorous simulations, see Fig. 2(b).

*λ*/100 was introduced to give a more practical estimation of the induced error when using scalar approximation. For a fixed duty cycle of 0.5, without any tolerance assumptions the limit was reached for periods of 2.6 – 2.8 μm for a wavelength of 633 nm depending on the polarisation. As the change in the refractive index for glass in the region of the visible light is smooth, one might estimate more generally, that a phase error of

*λ*/100 is already reached for structures with 4

*λ*period. As for structures with changing

*D*, as seen in Fig. 2(b), even for structures with a period of 5 μm the

*λ*/100 limit can be reached, although this requires a rather drastic change in the duty cycle (values smaller than 0.37 or values larger than 0.7). For smaller periods the range, where an error smaller than

*λ*/100 is induced, is decreasing rapidly. The summary of this section is that even for moderate line densities, one needs to use rigorous methods, when producing structures with high requirements (e.g. when only phase errors smaller than

*λ*/100 are allowed).

## 3. Sensitivity analysis: Binary gratings

14. Y. Sheng, D. Feng, and S. Larochelle, “Analysis and synthesis of circular diffractive lens with local linear grating model and rigorous coupled-wave theory,” J. Opt. Soc. Am. A **14**1562–1568 (1997) [CrossRef] .

*b*), height, side wall angle (

*α*) for transmission plus thickness of the Cr-layer (

*d*) for reflection. The start values for the gratings are: Height

*h*= 694 nm, Side wall angle

*α*= 88°, thickness of the Cr-layer of

*d*= 80 nm. As periods Λ

_{1}= 1 and Λ

_{2}= 2 μm were chosen with a duty cycle of 0.5 resulting in line widths of

*b*

_{1}= 0.5 and

*b*

_{2}= 1 μm. A sketch of the simulated structures can be found in Fig. 3. All simulations were done using the ITO inhousetool MicroSim [15

15. M. Totzeck, “Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields,” Optik - International Journal for Light and Electron Optics **112** (2001) [CrossRef] .

16. L. Li and Gérard Granet, “Field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings,” J. Opt. Soc. Am. A **28**738–746 (2011) [CrossRef] .

17. L. Li, “Field singularities at lossless metaldielectric arbitrary-angle edges and their ramifications to the numerical modeling of gratings,” J. Opt. Soc. Am. A **29**593–604 (2012) [CrossRef] .

*η*of the 1

^{st}order was studied over the truncation order

*M*. This was done for perpendicular and angular (30°) incident light, see Fig. 4. For both gratings the truncation order was set to

*M*= 50.

*W*is then given by the difference between the rigorously calculated and the phase change estimated with scalar approximation:

*W*describes the pure “rigorous effects” that are neglected in a scalar treatment, like polarisation effects and effects due to the angle of incidence.

### 3.1. Transmission gratings

^{th}and ±1

^{st}order were simulated. In Fig. 5 the resulting phase difference Δ

*W*for the binary grating in transmission are shown for both grating periods. The difference was plotted using contours with a separation of

*λ*/100.

*b*, the difference for the +1

^{st}order between rigorous and scalar calculation are reduced from

*λ*/40 to nearly zero when the period Λ is increased from 1 to 2 μm. For the 0

^{th}order the difference stays almost the same. For variations in height, phase differences of up to

*λ*/20 can be seen for all orders and all angles of incidence. The phase shows nearly no sensitivity regarding

*α*, hence the differences between scalar and rigorous calculation can be neglected for sidewall angles close to 90°. Results for TM polarisation can be found in appendix A. For periods of 2 μm (

*λ*= 633 nm) one can observe some angles of incidence in the 1

^{st}order where the phase difference is quite large. This behaviour is due to phase jumps in this region.

### 3.2. Reflection grating

*b*and

*α*for gratings with periods of 1 and 1.5 μm. As the metal layer shows a strong effect on the polarisation, both polarisation directions are shown. Results for height variations and variations in the thickness of the Cr-layer are shown in appendix C. The optical properties of the chromium used in the simulations are:

*n*= 3.136 and

*k*= 3.312 (for

*λ*= 633 nm). In addition, the same calculations have been done using the different n- and k values from

*Palik*[18] (

*n*= 3.5763,

*k*= 4.3615 for

*λ*= 633 nm). No apparent change in the results was observed.

*b*variations for the ±1

^{st}order varied from −

*λ*/30 for perpendicular incidence to +

*λ*/50 for oblique incident light (30°). Also changes in the

*α*show a strong effect on the phase, whereas for gratings in transmission nearly no effect was seen. In addition the changes between scalar and rigorous calculations are varying significantly when the period is increased from 1 to 1.5 μm, as gratings with Λ = 1.5 μm show next to no dependence on the polarisation.

## 4. Sensitivity analysis: Blazed transmission gratings

*π*and periods from 1 to 4 μm. Figure 8(b) shows the deviation between rigorous and scalar calculated wavefront phase. For structure sizes smaller than 3 μm the induced error is larger than

*λ*/100.

^{st}and 0

^{th}order are shown. The difference in the phase change between rigorous and scalar calculations has values of

*λ*/20 up to

*λ*/10. These values are approximately the same for both periods. As both polarisations show quite similar results, the TE polarisation can be found in appendix B.

### 4.1. Multilevel binary gratings

*x*on the phase was investigated for a 4-layer system, produced by a two-mask-process. Gratings with periods from 1 to 4 μm were analysed. Variations of the misalignment in the range of Δ

*x*=

*p*·0.1 are studied. Examples of the modelled structures are shown in Fig. 10.

^{st}order. It can be seen that the generated phase along the whole range of the observed angles varies strongly. Also even small misalignments cause a great deviation from the predicted phase. This might be expected, as for small Δ

*x*isolated structures occur, which have a strong effect on the generated phase.

### 4.2. Grey scale process

*σ*of the Gaussian function, that we used as an approximation of our writing spot profile. For a continuously scanning writing spot, the resulting intensity pattern can be described as a convolution between perfect grating profile and spot profile, see Eq 6 where * denotes the convolution operator.

*σ*of the Gaussian function was varied between 5 and 100 nm and the effect on the generated phase was observed for oblique and perpendicular incident light. Figure 13 shows the resulting structure for a 1 μm period grating with

*σ*set to 50 nm.

20. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A **19** (2002) [CrossRef] .

*W*for 1% variation in height over sigma are shown.

## 5. Conclusion and outlook

21. P. Zhou and J. H. Burge, “Optimal design of computer-generated holograms to minimize sensitivity to fabrication errors”Opt. Express **15**15410–15417 (2007) [CrossRef] [PubMed] .

*λ*/100 from what would be expected in scalar approximation, if the local grating periods are smaller than 4.5

*λ*for a wavelength of 633 nm. This is true for binary and blazed structures. In addition to that, fabrication tolerances introduce additional phase errors. To identify structure parameters, that have a significant effect on the generated phase, rigorous sensitivity analysis have been done for typical gratings. A variation of 1 % was introduced for selected parameters. The following results were found for grating periods of 1 μm at 633 nm wavelength.

*λ*/20. Phase changes in line width

*b*showed values of

*λ*/30 for TE polarisation. If metallised CGH are used in reflection with TM polarised light, phase changes of up to

*λ*/10 were observed for changes in

*b*. Also the side wall angle

*α*caused changes of up to

*λ*/30. Increasing the period to 1.5μm typically decrease the phase changes for most parameters significantly as well as the dependence on polarisation.

## Acknowledgments

## References and links

1. | J. C. Wyant and V. P. Bennett, “Using computer generated holograms to test aspheric wavefronts,” omputer Generated Holograms to Test Aspheric Wavefronts,” Appl. Opt. |

2. | S. M. Arnold, “How to test an asphere with a computer-generated hologram,” Proc. SPIE |

3. | E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, “Investigation of large null-CGH realization,” Proc. SPIE |

4. | J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “Systematic analysis of the measurement of cone angles using high line density computer-generated holograms,” Opt. Eng. |

5. | S. Reichelt, C. Pruss, and H. J. Tiziani, “Specification and characterization of CGHs for interferometrical optical testing,” Proc. SPIE |

6. | Y.-C. Chang, P. Zhou, and J. H. Burge, “Analysis of phase sensitivity for binary computer-generated holograms,” Appl. Opt. |

7. | P. Zhou and J. H. Burge, “Fabrication error analysis and experimental demonstration for computer-generated holograms,” Appl. Opt. |

8. | P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. |

9. | W. Yu, K. Takahara, T. Konishi, T. Yotsuya, and Y. Ichioka, “Fabrication of multilevel phase computer-generated hologram elements based on effective medium theory,” Appl. Opt. |

10. | I. Richter, P.-C. Sun, F. Xu, and Yeshayahu Fainman, “Design considerations of form birefringent microstructures,” Appl. Opt. |

11. | N. Bokor, R. Shechter, N. Davidson, A. A. Friesem, and Erez Hasman, “Achromatic phase retarder by slanted illumination of a dielectric grating with period comparable with the wavelength,” Appl. Opt. |

12. | W. Iff, S. Glaubrecht, N. Lindlein, and J. Schwider, “Untersuchung der Abweichungen zwischen skalarer und rigoroser Rechnung an CGHs in Twyman-Green-Interferometern zur Linsenprüfung,” DGaO-Proceedings http://www.dgao-proceedings.de (2010). |

13. | J. D. Gaskill, |

14. | Y. Sheng, D. Feng, and S. Larochelle, “Analysis and synthesis of circular diffractive lens with local linear grating model and rigorous coupled-wave theory,” J. Opt. Soc. Am. A |

15. | M. Totzeck, “Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields,” Optik - International Journal for Light and Electron Optics |

16. | L. Li and Gérard Granet, “Field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings,” J. Opt. Soc. Am. A |

17. | L. Li, “Field singularities at lossless metaldielectric arbitrary-angle edges and their ramifications to the numerical modeling of gratings,” J. Opt. Soc. Am. A |

18. | E. D. Palik, |

19. | J. Turunen and F. Wyrowski, |

20. | E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A |

21. | P. Zhou and J. H. Burge, “Optimal design of computer-generated holograms to minimize sensitivity to fabrication errors”Opt. Express |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: February 27, 2013

Revised Manuscript: April 12, 2013

Manuscript Accepted: April 26, 2013

Published: May 6, 2013

**Citation**

S. Peterhänsel, C. Pruss, and W. Osten, "Phase errors in high line density CGH used for aspheric testing: beyond scalar approximation," Opt. Express **21**, 11638-11651 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-11638

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### References

- J. C. Wyant and V. P. Bennett, “Using computer generated holograms to test aspheric wavefronts,” omputer Generated Holograms to Test Aspheric Wavefronts,” Appl. Opt.112833–2839 (1972). [CrossRef] [PubMed]
- S. M. Arnold, “How to test an asphere with a computer-generated hologram,” Proc. SPIE1052191–197 (1989). [CrossRef]
- E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, “Investigation of large null-CGH realization,” Proc. SPIE4440 (2001). [CrossRef]
- J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “Systematic analysis of the measurement of cone angles using high line density computer-generated holograms,” Opt. Eng.50(2011).
- S. Reichelt, C. Pruss, and H. J. Tiziani, “Specification and characterization of CGHs for interferometrical optical testing,” Proc. SPIE4778 (2002). [CrossRef]
- Y.-C. Chang, P. Zhou, and J. H. Burge, “Analysis of phase sensitivity for binary computer-generated holograms,” Appl. Opt.454223–4234 (2006). [CrossRef] [PubMed]
- P. Zhou and J. H. Burge, “Fabrication error analysis and experimental demonstration for computer-generated holograms,” Appl. Opt.46657–663 (2007). [CrossRef] [PubMed]
- P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt.432063–2085 (1996). [CrossRef]
- W. Yu, K. Takahara, T. Konishi, T. Yotsuya, and Y. Ichioka, “Fabrication of multilevel phase computer-generated hologram elements based on effective medium theory,” Appl. Opt.393531–3536 (2000). [CrossRef]
- I. Richter, P.-C. Sun, F. Xu, and Yeshayahu Fainman, “Design considerations of form birefringent microstructures,” Appl. Opt.342421–2429 (1995). [CrossRef] [PubMed]
- N. Bokor, R. Shechter, N. Davidson, A. A. Friesem, and Erez Hasman, “Achromatic phase retarder by slanted illumination of a dielectric grating with period comparable with the wavelength,” Appl. Opt.402076–2080 (2001). [CrossRef]
- W. Iff, S. Glaubrecht, N. Lindlein, and J. Schwider, “Untersuchung der Abweichungen zwischen skalarer und rigoroser Rechnung an CGHs in Twyman-Green-Interferometern zur Linsenprüfung,” DGaO-Proceedings http://www.dgao-proceedings.de (2010).
- J. D. Gaskill, Linear Systems, Fourier Transforms and Optics, (Wiley, 1978).
- Y. Sheng, D. Feng, and S. Larochelle, “Analysis and synthesis of circular diffractive lens with local linear grating model and rigorous coupled-wave theory,” J. Opt. Soc. Am. A141562–1568 (1997). [CrossRef]
- M. Totzeck, “Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields,” Optik - International Journal for Light and Electron Optics112 (2001). [CrossRef]
- L. Li and Gérard Granet, “Field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings,” J. Opt. Soc. Am. A28738–746 (2011). [CrossRef]
- L. Li, “Field singularities at lossless metaldielectric arbitrary-angle edges and their ramifications to the numerical modeling of gratings,” J. Opt. Soc. Am. A29593–604 (2012). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press, 1991).
- J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications, (Akademie Verlag, 1997).
- E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A19 (2002). [CrossRef]
- P. Zhou and J. H. Burge, “Optimal design of computer-generated holograms to minimize sensitivity to fabrication errors”Opt. Express1515410–15417 (2007). [CrossRef] [PubMed]

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