## Design of array of diffractive optical elements with inter-element coherent fan-outs

Optics Express, Vol. 12, Issue 23, pp. 5675-5683 (2004)

http://dx.doi.org/10.1364/OPEX.12.005675

Acrobat PDF (543 KB)

### Abstract

Building on the optimal-rotation-angle method, an algorithm for the design of inter-element coherent arrays of diffractive optical elements (DOEs) was developed. The algorithm is intended for fan-out DOE arrays where the individual elements fan-out to a sub-set of points chosen from a common set of points. By iteratively optimizing the array of elements as a whole the proposed algorithm ensures that the light from neighbouring elements is in-phase in all fan-out points that are common to neighbouring DOEs. This is important in applications where a laser beam scans the DOE array and the fan-out intensities constitute a read-out of information since the in-phase condition ensures a smooth transition in the read-out as the beam moves from one DOE to the next. Simulations show that the inter-element in-phase condition can be imposed at virtually no expense in terms of optical performance, as compared to independently designed DOEs.

© 2004 Optical Society of America

## 1. Introduction

2. F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A **5**, 1058–1065 (1988). [CrossRef]

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

6. J. Bengtsson, “Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method,” Appl. Opt. **36**, 8435–8444 (1997). [CrossRef]

7. M. Johansson, B. Löfving, S. Hård, L. Thylén, M. Mokhtari, U. Westergren, and C. Pala, “Study of an Ultrafast Analog-to-Digital Conversion Scheme Based on Diffractive Optics,” Appl. Opt. **39**, 2881–2887 (2000). [CrossRef]

7. M. Johansson, B. Löfving, S. Hård, L. Thylén, M. Mokhtari, U. Westergren, and C. Pala, “Study of an Ultrafast Analog-to-Digital Conversion Scheme Based on Diffractive Optics,” Appl. Opt. **39**, 2881–2887 (2000). [CrossRef]

8. S. Galt, A. Magnusson, and S. Hård, “Dynamic Demonstration of Diffractive Optic Analog-To-Digital Converter Scheme,” Appl. Opt. **42**, 264–270 (2003). [CrossRef] [PubMed]

## 2. The optimal rotation angle method

6. J. Bengtsson, “Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method,” Appl. Opt. **36**, 8435–8444 (1997). [CrossRef]

10. J. Bengtsson and M. Johansson, “Fan-out diffractive optical elements designed for increased fabrication tolerances to linear relief depth errors,” Appl. Opt. **41**, 281–289 (2002). [CrossRef] [PubMed]

6. J. Bengtsson, “Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method,” Appl. Opt. **36**, 8435–8444 (1997). [CrossRef]

*k*to the field in the position of the

*m*:th fan-out can be written

*k*, and

*φ*

_{k}the amount of phase modulation imposed by the DOE, which is the quantity to be optimized.

*A*

_{km}exp(

*jφ*

_{km}) is the complex transfer function from pixel

*k*to target spot

*m*. If the field incident on pixel

*k*is approximated to have a constant amplitude and a linearly varying phase the scalar diffraction integral can be solved and the transfer function is found to be

*L*

_{km}=

*z*

_{m}-

*z*

_{k},

*k*

_{0}=|

*k*|=2

*π*/

*λ*

_{0},

*k*to target spot location

*m*,

*k̃*

_{x}=

*k*

_{x}+

*k*

_{0}(

*x*

_{k}-

*x*

_{m})/

*k̃*

_{y}=

*k*

_{y}+

*k*

_{0}(

*y*

_{k}-

*y*

_{m})/

*a*and

*b*are the side lengths of a pixel and

*k*

_{x/y/z}is the

*x/y/z*-component of the

*k*-vector of the field incident on pixel

*k*. See also fig. 2a. Note that the transfer function does not depend on the DOE phase modulation

*φ*

_{k}. Thus,

*A*

_{km}and

*φ*

_{km}can be pre-calculated before the optimization loop is entered.

*U*

_{m}in one target spot location is shown together with the contribution from one pixel,

*U*

_{km}. With Φ

_{km}being the difference in phase between the total field in spot

*m*and the contribution to the same from pixel

*k*, the figure shows that the absolute value of the field in point

*m*is altered by Δ|

*U*|

_{km}=

*A*

_{km}

_{km}-Δ

*φ*

_{k})-

*A*

_{km}

_{km}) when the phase modulation in pixel

*k*is altered by Δ

*φ*

_{k}.

*U*|

_{km}for all spot locations, i.e., finding the Δ

*φ*

_{k}that maximizes ∑

_{m}Δ|

*U*|

_{km}|pixel

*k*. It can be shown that the maximum is found by calculating

_{km}used in the optimization. This can lead to an over-compensation and therefore instability but can simply be remedied; in this work we made the phase change more gentle by using a modified value for the optimized phase modulation simply as Δ

*const*·Δ

*const*≤1.

*S*

_{1}and

*S*

_{2}are modified as

*w*

_{m}is a weighting factor that is updated according to

*I*

_{m}is the intensity in target spot location

*m*and

*p*is a constant that determines the weight adjustment speed. The exact value of

*p*is not very critical but should be set sufficiently low to avoid instability. Typically,

*p*is set to 0.1–0.4. In fig. 3a the flow of the complete algorithm is outlined. The iterative loop is repeated until the desired quality constraints are met.

## 3. Extension of ORA to DOE-arrays

*n*, to the appropriate variables to indicate the

*n*:th DOE. The complex transfer function now has to be calculated from each pixel in each DOE to every target spot location. The target spot locations, as before indexed by

*m*, are the same for all elements in the array although their desired intensities can be different from one DOE to another.

*n*added to account for the set of DOEs. Step 4 is altered to equalize power across the DOE elements and last but not least, a fifth step that connects the phases across elements to satisfy the array-constraints is added. In the ADC case this step makes sure that spots illuminated by neighbouring DOEs are in-phase.

*step 4 - weight update*

*w*

_{mn}, for those

*m*and

*n*for which

*I*

_{n}=∑

_{m}|

*I*

_{mn}|

^{2}is the efficiency of DOE

*n*and

*Ī*the average efficiency.

*b*

_{c}determines the convergence speed and should be chosen small enough to maintain stability, typically

*b*

_{c}≈1. The first step in eq. 6 adjusts the weights to get equal efficiency and the second step renormalizes the smallest beam-dump weight to zero to get the best possible efficiency.

*step 5 - inter-element coherence*

*φ*

_{mn}=arg(

*U*

_{mn}), which can be viewed as the target phases of the ORA algorithm in step 6, do not fulfill the array specific requirements. The proposed method of establishing the desired inter-element coherence is to replace

*φ*

_{mn}by some other phases

*φ′*

_{mn}to make the ORA algorithm “aim” for other phases than the actual ones in a way suitable to the specific application. The determination of target phases

*φ′*

_{mn}from the previously calculated phase values

*φ*

_{mn}is referred to as a “merger”. Taking the ADC application as an example, in the following two different mergers are studied.

*A*exp(

*iφ′*

_{mn})=∑

_{n}exp(

*iφ*

_{mn}), in which case

*φ′*

_{mn}evidently is independent of

*n*. With a 5 bit ADC and inverse word generation this gives 10 phase groups, one for each spot position. The second merger method is a bit more advanced and assigns one phase group for every group of neighbouring elements that illuminates the same spot, see fig. 4. As an example, one can see that the spot location where the most significant bit (MSB) is read out should have a high intensity (a“one”) where any of the DOEs numbered 16–31 is illuminated. The phases of the optical field in this position from these DOEs should then be the same, to avoid negative inter-DOE interference, and these phases thus constitute a phase group labelled “2” in fig. 4. Within each phase group one again finds the target phase value as the average of the calculated phase values, from the preceding iteration, within the group.

## 4. Simulated results

^{5}=32 DOEs that comprise the entire diffractive structure 32×32 pixels were used in the calculations.

*π*for independently designed DOEs, while the merger approach in the modified ORA algorithm reduces it to a mere 1% of its original value. Finally, we notice that the 36-merger design performs slightly better in every respect than does the 10-merger design. This is expected because the 36-merger design uses more degrees of freedom.

*e*

^{2}intensity diameter corresponding to the 128×128 element width.

## 5. Conclusion

**36**, 8435–8444 (1997). [CrossRef]

## Acknowledgments

## References

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

2. | F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A |

3. | J. Turunen, A. Vasara, and J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. |

4. | B. K. Jennison, J. P. Allebach, and D.W. Sweeney, “Efficient design of direct-binary-search computer-generated holograms,” J. Opt. Soc. Am. A |

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

6. | J. Bengtsson, “Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method,” Appl. Opt. |

7. | M. Johansson, B. Löfving, S. Hård, L. Thylén, M. Mokhtari, U. Westergren, and C. Pala, “Study of an Ultrafast Analog-to-Digital Conversion Scheme Based on Diffractive Optics,” Appl. Opt. |

8. | S. Galt, A. Magnusson, and S. Hård, “Dynamic Demonstration of Diffractive Optic Analog-To-Digital Converter Scheme,” Appl. Opt. |

9. | J. Stigwall, S. Galt, and S. Hård, “Experimental evaluation of an ultra-fast free space optical analog-to-digital conversion scheme using a tunable semiconductor laser,” presented at Microwave and Teraherz Photonics, Photonics Europe, Strasbourg, France. Proc. SPIE5466 (2004). |

10. | J. Bengtsson and M. Johansson, “Fan-out diffractive optical elements designed for increased fabrication tolerances to linear relief depth errors,” Appl. Opt. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(090.1760) Holography : Computer holography

(100.5090) Image processing : Phase-only filters

(200.2610) Optics in computing : Free-space digital optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 3, 2004

Revised Manuscript: November 4, 2004

Published: November 15, 2004

**Citation**

Johan Stigwall and Jörgen Bengtsson, "Design of array of diffractive optical elements with inter-element coherent fan-outs," Opt. Express **12**, 5675-5683 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-23-5675

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### References

- R. W. Gerchberg and W. O. Saxton, "Practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-250 (1972).
- F. Wyrowski and O. Bryngdahl, "Iterative Fourier-transform algorithm applied to computer holography," J. Opt. Soc. Am. A 5, 1058-1065 (1988). [CrossRef]
- J. Turunen, A. Vasara, and J. Westerholm, "Kinoform phase relief synthesis: a stochastic method," Opt. Eng. 28, 1162-1167 (1989).
- B. K. Jennison, J. P. Allebach, and D.W. Sweeney, "Efficient design of direct-binary-search computer-generated holograms," J. Opt. Soc. Am. A 8, 652-660 (1991). [CrossRef]
- N. Yoshikawa and T. Yatagai, "Phase optimization of a kinoform by simulated annealing," Appl. Opt. 33, 863-868 (1994). [CrossRef] [PubMed]
- J. Bengtsson, "Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method," Appl. Opt. 36, 8435-8444 (1997). [CrossRef]
- M. Johansson, B. Löfving, S. Hård, L. Thylén, M. Mokhtari, U. Westergren, and C. Pala, "Study of an Ultrafast Analog-to-Digital Conversion Scheme Based on Diffractive Optics," Appl. Opt. 39, 2881-2887 (2000). [CrossRef]
- S. Galt, A. Magnusson, and S. Hård, "Dynamic Demonstration of Diffractive Optic Analog-To-Digital Converter Scheme," Appl. Opt. 42, 264-270 (2003). [CrossRef] [PubMed]
- J. Stigwall, S. Galt, and S. Hård, "Experimental evaluation of an ultra-fast free space optical analog-to-digital conversion scheme using a tunable semiconductor laser," presented at Microwave and Teraherz Photonics, Photonics Europe, Strasbourg, France. Proc. SPIE 5466 (2004).
- J. Bengtsson and M. Johansson, "Fan-out diffractive optical elements designed for increased fabrication tolerances to linear relief depth errors," Appl. Opt. 41, 281-289 (2002). [CrossRef] [PubMed]

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