## High-efficiency rotating point spread functions

Optics Express, Vol. 16, Issue 5, pp. 3484-3489 (2008)

http://dx.doi.org/10.1364/OE.16.003484

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### Abstract

Rotating point spread functions (PSFs) present invariant features that continuously rotate with defocus and are important in diverse applications such as computational imaging and atom/particle trapping. However, their transfer function efficiency is typically very low. We generate highly efficient rotating PSFs by tailoring the range of invariant rotation to the specific application. The PSF design involves an optimization procedure that applies constraints in the Gauss-Laguerre modal plane, the spatial domain, and the Fourier domain. We observed over thirty times improvement in transfer function efficiency. Experiments with a phase-only spatial light modulator demonstrate the potential of high-efficiency rotating PSFs.

© 2008 Optical Society of America

## 1. Introduction

1. Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E **54**, R50–R53 (1996). [CrossRef]

7. N. B. Simpson, L. Allen, and M. J. Padgett, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. of Mod. Opt. **43**, 2485–2491 (1996). [CrossRef]

10. A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. **31**, 181–183 (2006). [CrossRef] [PubMed]

## 2. Rotating point spread functions

3. R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A **17**, 294–303 (2000). [CrossRef]

10. A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. **31**, 181–183 (2006). [CrossRef] [PubMed]

## 3. High-efficiency rotating point spread functions

## 4. HER-PSF design by iterative optimization

12. R. Piestun, B. Spektor, and J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A **13**, 1837–1848 (1996). [CrossRef]

12. R. Piestun, B. Spektor, and J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A **13**, 1837–1848 (1996). [CrossRef]

13. R. Piestun and J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. **19**, 771–773 (1994). [CrossRef] [PubMed]

### 4.1 Initial estimate

### 4.2. Optimization constraints

*d*(

*m,n*) defined as,

*m*and

*n*are the indices of the GL modal plane,

*m*and

_{k}*n*are the

_{k}*m*and

*n*indices of the

*k*GL mode along the rotating PSF line,

^{th}*N*is the number of modes selected along the rotating PSF line, and p is a parameter that determines the width of the cloud. The weight function

*w*(

_{gl}*m,n*) is directly obtained from

*d*(

*m,n*) as

*w*(

_{gl}*m,n*)=max[

*d*(

*m,n*)]-

*d*(

*m,n*).

## 5. Results

*T*), defined as,

*H*(

_{i}*x,y*) is the complex mask immediately before enforcing the phase-only constraint and

*C*(

*x,y*) represents a clear aperture of the same size.

*λ*) dependence. Wavelength dependence arises from four factors: 1) phase retardation of a mask is 2

*πnt*(

*x, y*)/

*λ*, where

*n*and

*t*(

*x,y*) are the mask’s refractive index and thickness function, respectively; 2) material dispersion; 3) defocus is inversely proportional to

*λ*, and 4) PSF size is inversely proportional to

*λ*. (3) and (4) are the result of diffraction upon wave propagation. As an example, the effect of using a BK7 glass HER-PSF mask designed for a wavelength of 550nm with the wavelengths 500nm, 550nm, and 600nm is shown in Fig. 4. The rotation angles were determined by calculating the 3D PSF for each wavelength from the transmittance function produced by the mask at the corresponding wavelength. For all three wavelengths, the PSF exhibits two main lobes that rotate continuously. The rates of rotation are essentially the same for all three wavelengths [Fig. 4(a)], when plotted as a function of the defocus parameter (

*ψ*), defined in [10

10. A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. **31**, 181–183 (2006). [CrossRef] [PubMed]

^{th}order has the highest energy. In order to avoid on-axis effects due to the SLM’s non-ideal modulation, a linear phase is added to the calculated HER-PSF transfer function phase. A collimated light with wavelength 632.8nm is incident on the SLM and a 0.09 NA lens Fourier transforms 0th order of the SLM. The PSF at different axial distances shows two continuously rotating main lobes [Fig. 2(d)] with 37.5% transfer function efficiency. The experimental efficiency is not as high as the theoretical value (56.8%) because of the non-ideal response of the SLM.

## 6. Conclusion

## Acknowledgments

1. | Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E |

2. | V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. |

3. | R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A |

4. | S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, and J. Turunen, “Generation of rotating Gauss-Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. |

5. | E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys.-Usp. |

6. | A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Angular momentum of a rotating light beam,” Opt. Comm. |

7. | N. B. Simpson, L. Allen, and M. J. Padgett, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. of Mod. Opt. |

8. | M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science |

9. | R. Grimm, M. Weidemuller, and Y. B. Ovchinnikov, “Optical dipole traps for neutral atoms,” Adv. At. Mol. Opt. Phys. |

10. | A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. |

11. | W. H. Lee, “Computer-generated holograms: techniques and applications,” in |

12. | R. Piestun, B. Spektor, and J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A |

13. | R. Piestun and J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. |

**OCIS Codes**

(110.4850) Imaging systems : Optical transfer functions

(110.6880) Imaging systems : Three-dimensional image acquisition

(150.5670) Machine vision : Range finding

(110.1758) Imaging systems : Computational imaging

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: December 21, 2007

Revised Manuscript: February 9, 2008

Manuscript Accepted: February 20, 2008

Published: February 29, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Sri Rama Prasanna Pavani and Rafael Piestun, "High-efficiency rotating point spread functions," Opt. Express **16**, 3484-3489 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-5-3484

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### References

- Y. Y. Schechner, R. Piestun, and J. Shamir, "Wave propagation with rotating intensity distributions," Phys. Rev. E 54, R50-R53 (1996). [CrossRef]
- V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, "An algorithm for the generation of laser beams with longitudinal periodicity: rotating images," J. Mod. Opt. 44, 1409-1416 (1997). [CrossRef]
- R. Piestun, Y. Y. Schechner, and J. Shamir, "Propagation-invariant wave fields with finite energy," J. Opt. Soc. Am. A 17, 294-303 (2000). [CrossRef]
- S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, and J. Turunen, "Generation of rotating Gauss-Laguerre modes with binary-phase diffractive optics," J. Mod. Opt. 46, 227-238 (1999).
- E. G. Abramochkin and V. G. Volostnikov, "Spiral light beams," Phys.-Usp. 47, 1177-1203 (2004). [CrossRef]
- A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Angular momentum of a rotating light beam," Opt. Comm. 249, 367-378 (2005). [CrossRef]
- N. B. Simpson, L. Allen, and M. J. Padgett, "Optical tweezers and optical spanners with Laguerre-Gaussian modes," J. of Mod. Opt. 43, 2485-2491 (1996). [CrossRef]
- M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 1101-1103 (2002). [CrossRef] [PubMed]
- R. Grimm, M. Weidemuller, and Y. B. Ovchinnikov, "Optical dipole traps for neutral atoms," Adv. At. Mol. Opt. Phys. 42, 95-170 (2000). [CrossRef]
- A. Greengard, Y. Y. Schechner, and R. Piestun, "Depth from diffracted rotation," Opt. Lett. 31, 181-183 (2006). [CrossRef] [PubMed]
- W. H. Lee, "Computer-generated holograms: techniques and applications," in Progress in Optics, E. Wolf, ed. (Elsevier, 1978), Chap. 3.
- R. Piestun, B. Spektor, and J. Shamir, "Wave fields in three dimensions: analysis and synthesis," J. Opt. Soc. Am. A 13, 1837-1848 (1996). [CrossRef]
- R. Piestun and J. Shamir, "Control of wave-front propagation with diffractive elements," Opt. Lett. 19, 771-773 (1994). [CrossRef] [PubMed]

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