## Backward iterative quantization methods for designs of multilevel diffractive optical elements

Optics Express, Vol. 13, Issue 13, pp. 5052-5063 (2005)

http://dx.doi.org/10.1364/OPEX.13.005052

Acrobat PDF (277 KB)

### Abstract

Four types of backward iterative quantization (BIQ) methods were proposed to design multilevel diffractive optical elements (DOEs). In these methods, the phase values first quantized in the early quantization steps are those distant from the quantization levels, instead of the neighboring ones that the conventional iterative method began with. Compared with the conventional forward iterative quantization (FIQ), the Type 4 BIQ achieved higher efficiencies and signal-to-noise ratios for 4-level unequal-phase DOEs. For equal-phase DOEs, the Type 4 BIQ performed better when the range increment of each quantization step was large (>15°), while the FIQ performed better when the range increment was small (<15°).

© 2005 Optical Society of America

## 1. Introduction

3. L.B. Lesem, P.M. Hirsch, and J.R. Jordan Jr., “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. **13**, 150–155 (1969). [CrossRef]

7. J.R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

8. N. Yoshikawa and T. Yatagai, “Phase optimization of a kinoform using simulated annealing,” Appl. Opt. **33**, 863–868 (1994). [CrossRef] [PubMed]

9. N. Yoshikawa, M. Itoh, and T. Yatagai, “Quantized phase quantization of two-dimensional Fourier kinoforms by a genetic algorithm,” Opt. Lett. **20**, 752–754 (1995). [CrossRef] [PubMed]

10. M.A. Seldowilz, J.P. Allebach, and D.W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. **26**, 2788–2798 (1987). [CrossRef]

5. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A **7**, 961–969 (1990). [CrossRef]

*π*] have been commonly used, multilevel DOEs with the quantization levels unequally spaced in the same range are able to achieve even higher diffraction efficiencies [11

11. U. Levy, N. Cohen, and D. Mendlovic, “Analytic approach for optimal quantization of diffractive optical elements,” Appl. Opt. **38**, 5527–5532 (1999). [CrossRef]

13. W.-F. Hsu and C.-H. Lin, “An optimal quantization method for uneven-phase diffractive optical elements by use of a modified iterative Fourier-transform algorithm,” Appl. Opt. (to be published). [PubMed]

14. U. Levy, E. Marom, and D. Mendlovic, “Simultaneous multicolor image formation with a single diffractive optical element,” Opt. Lett. **26**, 1149–1151 (2001). [CrossRef]

15. K. Ballüder and M. R. Taghizadeh, “Optimized quantization for diffractive phase elements by use of uneven phase levels,” Opt. Lett. **26**, 417–419 (2001). [CrossRef]

13. W.-F. Hsu and C.-H. Lin, “An optimal quantization method for uneven-phase diffractive optical elements by use of a modified iterative Fourier-transform algorithm,” Appl. Opt. (to be published). [PubMed]

5. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A **7**, 961–969 (1990). [CrossRef]

13. W.-F. Hsu and C.-H. Lin, “An optimal quantization method for uneven-phase diffractive optical elements by use of a modified iterative Fourier-transform algorithm,” Appl. Opt. (to be published). [PubMed]

5. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A **7**, 961–969 (1990). [CrossRef]

6. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. **28**, 38643–3870 (1989). [CrossRef]

## 2. Backward iterative quantization (BIQ) methods

_{k}| of the kth iteration by the target amplitude |F| [7

7. J.R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

_{k}(u,v) is inverse Fourier transformed and results in a complex-valued function g’

_{k}(x,y) that is generally complex-valued, i.e., g’

_{k}(x,y)=|g’

_{k}(x,y)|exp(jϕ’

_{k}(x,y)). In the signal domain, the new complex amplitude g

_{k+1}(x,y) is subject to the amplitude constraint of

**7**, 961–969 (1990). [CrossRef]

6. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. **28**, 38643–3870 (1989). [CrossRef]

^{(n,q)}=(ϕ

^{(n,q)}- ϕ

^{(n-1,q)})/2 and the quantization ratio is given by 0<

*ε*

^{(1)}<…<

*ε*

^{(q)}<…<

*ε*

^{(Q)}=1. Since ϕ is cyclically periodic, Δ

^{(N+1,q)}=Δ

^{(1,q)}. The parameters of the FIQ used for designing 4-level DOEs are illustrated in Fig. 2(a). Note that for designing equal-phase DOEs the quantization levels ϕ

^{(n,q)}are fixed for all q’s. For unequal-phase DOEs, quantization levels ϕ

^{(n,q)}are calculated by minimizing the mean-squared error of the object field at step q, which is defined as [12

12. W.-F. Hsu and I-C. Chu, “Optimal quantization by use of an amplitude-weighted probability-density function for diffractive optical elements,” Appl. Opt. **43**, 3672–3679 (2004). [CrossRef] [PubMed]

^{(n,q)}are given by

^{(N+1,q)}=C

^{(1,q)}+2π. The function p

_{aw}(ϕ) is the product of the probability density function p(ϕ) and the mean of the amplitude |g

_{k}(x,y)| with respect to ϕ.

^{(n,q)}are given by

^{(n,q)}in which phase values will not be quantized.

_{k}(x,y) is given by

^{(n,q)}at

*q*=Q. The Type 2 BIQ method linearly projects the phase values onto new phase values in

*β*

^{(n,q)}=(

*-*C up ( n , q )

*)/(*C low ( n , q )

*C*

^{(n+1,q)}-

*C*

^{(n-q)}is a compression ratio of the quantization step

*q*. At

*q*=Q,

*β*

^{(n,q)}=0 means the free windows close and ϕ(x,y)=ϕ

^{(n,Q)}. In the Type 3 BIQ, the phase values outside

^{(n,q)}sequentially shift toward ϕ

^{(n,q)}, the quantization level. The scroll-back projection of the phase values is given by

^{(1,q)}and ϕ

^{(2,q)}.

## 3. Simulation results

_{k}(u,v), given by

*w*

_{1}and

*w*

_{2}are real constants. Here, η

_{k}and SNR

_{k}are the relative diffraction efficiency and the signal-to-noise ratio of the DOE in iteration cycle k, respectively, defined as

*f*

_{k}becomes constant. The solution of the multilevel DOE is selected when the largest

*f*

_{k}results in the final quantization step.

### 3.1. Four-level equal-phase DOEs

*π*], i.e., [0,

*π*/2,

*π*, and 3π/2]. Eleven quantization steps, Q=[1, 2, 3

3. L.B. Lesem, P.M. Hirsch, and J.R. Jordan Jr., “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. **13**, 150–155 (1969). [CrossRef]

**7**, 961–969 (1990). [CrossRef]

6. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. **28**, 38643–3870 (1989). [CrossRef]

7. J.R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

9. N. Yoshikawa, M. Itoh, and T. Yatagai, “Quantized phase quantization of two-dimensional Fourier kinoforms by a genetic algorithm,” Opt. Lett. **20**, 752–754 (1995). [CrossRef] [PubMed]

12. W.-F. Hsu and I-C. Chu, “Optimal quantization by use of an amplitude-weighted probability-density function for diffractive optical elements,” Appl. Opt. **43**, 3672–3679 (2004). [CrossRef] [PubMed]

15. K. Ballüder and M. R. Taghizadeh, “Optimized quantization for diffractive phase elements by use of uneven phase levels,” Opt. Lett. **26**, 417–419 (2001). [CrossRef]

*ε*

^{(q)}=q/6 and the quantization range increases by a value of π/12, or 15°, at each step. Note that the direct iterative quantization is conducted when Q=1. In order to achieve a higher diffraction efficiency, the amplitude constraint Eq. (3a) was used for all designs of equal-phase DOEs and the designs of the unequal-phase DOEs using the FIQ method. Table 1 lists the overall descriptive statistics of the diffraction efficiencies and the SNRs of 220 DOEs (20 initial arrays and 11 Q’s) with 4 discrete phase levels designed by the above-described methods. Among the BIQ methods, only the Type-4 BIQ performs competitively with the FIQ method (η

_{avg}=0.8165 and SNR

_{avg}=0.7774 vs. η

_{avg}=0.8162 and SNR

_{avg}=0.7882). Figure 4 shows the details of η‘s and SNRs of the 4-level DOEs with respect to Q and the individual DOE. As shown in Fig. 4(a), the FIQ method achieved a higher η for large Q’s, Q≥7, indicating small increments of quantization range. The Type-4 BIQ method, however, performed better when

### 3.2. Four-level unequal-phase DOEs when ϕ(1)=0

^{(1)}, we set ϕ

^{(1)}=0 here. In the optimization of the discrete phase levels of the unequal-phase DOE, only two variables, i.e.,

_{1}and d

_{2}, are calculated to construct 4 phase levels. As in Section 3.1, 11 Q values were chosen, however, the quantization ratio

*ε*

^{(q)}of a specified Q does not increase linearly. In the designs of unequal-phase DOEs, the amplitude constraint Eq. (3a) and the FIQ method were used, and Eq. (3b) was used with the four BIQ methods. The descriptive statistics of these simulation results are listed in Table 2. The efficiency and SNR of the 4-level DOEs using the Type 4 BIQ method increase significantly (η

_{avg}=0.8242 and SNR

_{avg}=0.8152). Figure 5 shows these performances with respect to the number of quantization steps Q and the individual DOE. The Type 4 BIQ method was able to produce 4-level DOEs with the largest efficiencies in all cases, and higher SNRs in some cases than in others. The largest efficiencies obtained by the FIQ and Type 4 BIQ were 0.8247 (DOE#7, Q=30) and 0.8323 (DOE#14, Q=30), and the highest SNR by the FIQ and Type 4 BIQ were 1.318 (DOE#3, Q=2) and 1.234 (DOE#20, Q=12), respectively. Although the highest SNR of 1.318 was obtained using the FIQ method, the corresponding efficiency was 0.8108 which is lower than the efficiency 0.8272 of DOE#20 at Q=12 obtained using the Type 4 BIQ. On the contrary, the lowest SNRs obtained using the FIQ and Type 4 BIQ were 0.2078 (DOE#17, Q=4) and 0.4401 (DOE#11, Q=12), respectively. These results show that the performance of the DOEs designed by using the Type 4 BIQ method is not only higher but is also more stable than the performance obtained using the FIQ and the other BIQs.

### 3.3. Four-level unequal-phase DOEs when ϕ(1)≠0

^{(1)}≠0. For the unequal-phase DOEs of 4 discrete phase levels, three variables ϕ

^{(1)}, d

_{1}, and d

_{2}were calculated to optimize the 4-level DOE. Equations (3a) and (3b) were used for the amplitude constraint when applying the FIQ and BIQ methods, respectively, to obtain a higher diffraction efficiency. The overall descriptive statistics of the simulation results are listed in Table 3. The means of the efficiencies and SNRs were on the average slightly higher than those of the DOEs when ϕ

^{(1)}=0. The largest efficiency obtained by the FIQ and Type 4 BIQ were 0.8244 (DOE#2, Q=12) and 0.8307 (DOE#4, Q=9), and the highest SNR by the FIQ and Type 4 BIQ were 1.223 (DOE#14, Q=12) and 1.281 (DOE#12,

^{(1)}=0, these results show that the influence of the bias phase ϕ

_{1}is not significant when the etching depths (d

_{1}and d

_{2}in 4-level DOEs) are optimized to obtain multilevel DOEs. Figure 6 shows these performances with respect to the number of quantization steps Q and the individual DOE. Still, the DOEs designed using the Type 4 BIQ method outperformed the DOEs designed using the other methods.

*π*(d

_{1}) and 0.493π (d

_{2}).

## 4. Discussion

*π*, 0.248

*π*, 0.489

*π*, 0.737

*π*, 1.027

*π*, 1.275

*π*, 1.515

*π*, and 1.763

*π*), corresponding to three etching depths 1.027

*π*(d

_{1}), 0.489

*π*(d

_{2}), and 0.248

*π*(d

_{3}). Note that the first three BIQ methods performed competitively with the Type 4 BIQ and FIQ methods for higher phase levels because the quantization ranges are much smaller than those of the 4-level cases. Correspondingly, both the number of pixels changed and the phase changes in each iteration were small, and the performance improved.

^{(1)}≠0 and only one phase level when ϕ

^{(1)}=0, which are dynamically optimized in the iterative process.

## 5. Conclusion

## Acknowledgments

## References and links

1. | J. Turunen and F. Wyrowski ed., |

2. | M.B. Stern, “Binary optics fabrication,” in |

3. | L.B. Lesem, P.M. Hirsch, and J.R. Jordan Jr., “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. |

4. | R.W. Gerchberg and W.O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik |

5. | F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A |

6. | F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. |

7. | J.R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

8. | N. Yoshikawa and T. Yatagai, “Phase optimization of a kinoform using simulated annealing,” Appl. Opt. |

9. | N. Yoshikawa, M. Itoh, and T. Yatagai, “Quantized phase quantization of two-dimensional Fourier kinoforms by a genetic algorithm,” Opt. Lett. |

10. | M.A. Seldowilz, J.P. Allebach, and D.W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. |

11. | U. Levy, N. Cohen, and D. Mendlovic, “Analytic approach for optimal quantization of diffractive optical elements,” Appl. Opt. |

12. | W.-F. Hsu and I-C. Chu, “Optimal quantization by use of an amplitude-weighted probability-density function for diffractive optical elements,” Appl. Opt. |

13. | W.-F. Hsu and C.-H. Lin, “An optimal quantization method for uneven-phase diffractive optical elements by use of a modified iterative Fourier-transform algorithm,” Appl. Opt. (to be published). [PubMed] |

14. | U. Levy, E. Marom, and D. Mendlovic, “Simultaneous multicolor image formation with a single diffractive optical element,” Opt. Lett. |

15. | K. Ballüder and M. R. Taghizadeh, “Optimized quantization for diffractive phase elements by use of uneven phase levels,” Opt. Lett. |

16. | J.W. Goodman, |

**OCIS Codes**

(050.1380) Diffraction and gratings : Binary optics

(050.1970) Diffraction and gratings : Diffractive optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 16, 2005

Revised Manuscript: June 17, 2005

Published: June 27, 2005

**Citation**

Wei-Feng Hsu, "Backward iterative quantization methods for designs of multilevel diffractive optical elements," Opt. Express **13**, 5052-5063 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-13-5052

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### References

- J. Turunen and F. Wyrowski ed., Diffractive Optics for Industrial and Commercial Applications, (Akademie Verlag, Berlin, 1997).
- M.B. Stern, �??Binary optics fabrication,�?? in Micro-optics: Elements, Systems and Applications, H.P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 53-85.
- L.B. Lesem, P.M. Hirsch, and J.R. Jordan, Jr., �??The kinoform: a new wavefront reconstruction device,�?? IBM J. Res. Dev. 13, 150-155 (1969). [CrossRef]
- R.W. Gerchberg and W.O. Saxton, �??A practical algorithm for the determination of phase from image and diffraction plane pictures,�?? Optik 35, 237-246 (1972).
- F. Wyrowski, �??Diffractive optical elements: iterative calculation of quantized, blazed phase structures,�?? J. Opt. Soc. Am. A 7, 961-969 (1990). [CrossRef]
- F. Wyrowski, �??Iterative quantization of digital amplitude holograms,�?? Appl. Opt. 28, 38643-3870 (1989). [CrossRef]
- J.R. Fienup, �??Phase retrieval algorithms: a comparison,�?? Appl. Opt. 21, 2758-2769 (1982). [CrossRef] [PubMed]
- N. Yoshikawa and T. Yatagai, �??Phase optimization of a kinoform using simulated annealing,�?? Appl. Opt. 33, 863-868 (1994). [CrossRef] [PubMed]
- N. Yoshikawa, M. Itoh, and T. Yatagai, �??Quantized phase quantization of two-dimensional Fouriern kinoforms by a genetic algorithm,�?? Opt. Lett. 20, 752-754 (1995). [CrossRef] [PubMed]
- M.A. Seldowilz, J.P. Allebach, and D.W. Sweeney, "Synthesis of digital holograms by direct binary search," Appl. Opt. 26, 2788-2798 (1987). [CrossRef]
- U. Levy, N. Cohen, and D. Mendlovic, �??Analytic approach for optimal quantization of diffractive optical elements,�?? Appl. Opt. 38, 5527-5532 (1999). [CrossRef]
- W.-F. Hsu and I-C. Chu, �??Optimal quantization by use of an amplitude-weighted probability-density function for diffractive optical elements,�?? Appl. Opt. 43, 3672-3679 (2004). [CrossRef] [PubMed]
- W.-F. Hsu and C.-H. Lin, �??An optimal quantization method for uneven-phase diffractive optical elements by use of a modified iterative Fourier-transform algorithm,�?? Appl. Opt. (to be published). [PubMed]
- U. Levy, E. Marom, and D. Mendlovic, �??Simultaneous multicolor image formation with a single diffractive optical element,�?? Opt. Lett. 26, 1149-1151 (2001). [CrossRef]
- K. Ballüder and M. R. Taghizadeh, �??Optimized quantization for diffractive phase elements by use of uneven phase levels,�?? Opt. Lett. 26, 417-419 (2001). [CrossRef]
- J.W. Goodman, Introduction to Fourier Optics, 2nd Ed., (McGraw-Hill, New York, 1996), pp. 96-125.

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