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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 5 — Mar. 3, 2008
  • pp: 3484–3489
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High-efficiency rotating point spread functions

Sri Rama Prasanna Pavani and Rafael Piestun  »View Author Affiliations


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

A delicate balance between vortex charge and the Gouy phase of a superposition of beams produces the so-called rotating beams. These beams present an intensity distribution that continuously rotates about the optical axis upon propagation [1–6

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

]. Combinations of Gauss-Laguerre (GL) or Bessel beams have been used for optical manipulation of atoms and microparticles [7–9

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]

]. Because each point of the transverse cross section of a rotating beam describes a spiral trajectory, they are attractive for such applications as well. The same optical elements that generate rotating beams can be used to engineer the point spread functions (PSFs) of imaging systems to rotate with defocus. These rotating PSFs significantly increase the sensitivity of depth estimation and are the foundation of a new passive ranging principle named depth from diffracted rotation [10

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

].

The main disadvantage of existing methods to implement rotating PSF systems is low light efficiency, which makes them inappropriate for photon limited applications. In this paper, we introduce a new type of PSF named high-efficiency rotating point spread function (HER-PSF) that solves this low efficiency problem. HER-PSFs present rotating cross sections only in a predetermined region of space, which allows for additional degrees of freedom to be used towards a phase-only implementation of the transfer function and its optical element.

The rest of this paper is organized as follows: In section 2, we review the properties of rotating PSFs and establish their limitations. Section 3 introduces the new three-dimensional (3D) HER-PSFs. Section 4 describes the design methodology, and section 5 analyzes the spatial and spectral properties of HER-PSFs through modeling and experiment.

2. Rotating point spread functions

The main disadvantage of existing rotating PSFs is their very low transfer function efficiency, which is defined as the ratio of the energy in the rotating PSF main lobes to the energy incident on the mask. This is fundamentally due to two reasons: 1) the amplitude of the rotating PSF transfer function creates highly absorptive masks, 2) part of the energy in the rotating PSF is delivered to the side lobes, which are usually not used. In the example of Fig. 1(a), the transfer function efficiency is only about 1.8%; meaning that only 1.8% of the energy incident on the rotating PSF system from a point source actually makes it to the main lobes of the PSF. Adding to - and independent of - this low efficiency problem is the encoding method of the amplitude and phase components of the transfer function that often results in unused diffraction orders, causing further loss of light. Because of the abovementioned inherent factors, current rotating PSF systems are not suitable for photon-limited applications.

3. High-efficiency rotating point spread functions

HER-PSFs solve the rotating PSFs’ low efficiency problem by presenting the following key features: 1) The rotating response appears only within a limited volume instead of the whole 3D space, 2) The main features of the PSF rotate but the entire cross section is only approximately invariant within the volume of interest.

4. HER-PSF design by iterative optimization

Fig. 1. Transfer functions of (a) rotating PSF, (b) HER-PSF initial estimate, and (c) HER-PSF. (d), (e), and (f) are the GL modal plane decompositions of the transfer functions in (a), (b), and (c), respectively. The transfer functions of the HER-PSF initial estimate and the HER-PSF form a cloud around the line in (d).

4.1 Initial estimate

Fig. 2. Movies of different rotating PSFs: (a) Exact rotating PSF, (b) HER-PSF initial estimate, (c) HER-PSF, and (d) the experimental implementation of HER-PSF. HER-PSF has over 30 times higher efficiency than the exact rotating PSF. [Media 1][Media 2][Media 3][Media 4]

4.2. Optimization constraints

The iterative optimization procedure starts with the above transfer function estimate and repeatedly enforces constraints specifically designed for achieving PSF rotation, maximum energy in the PSF main lobes, and a phase-only transfer function.

d(m,n)=k=1N[(mmk)2+(nnk)2]p,
(1)

As is customary in optimization algorithms, there is some freedom in the selection of the weight functions. The selections described above were particularly efficient in this case.

5. Results

T=xyHi(x,y)xyC(x,y),
(2)

where Hi(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.

Fig. 3. Movie of the evolutions of (a) mask phase, (b) GL modal plane, (c) main lobe peak intensity, and (d) mask transparency in the iterative optimization procedure.[Media 5]

Fig. 4. A HER-PSF mask designed for a wavelength of 550nm, when used with the wavelengths 500nm and 600nm, exhibits (a) same rotation rates as function of defocus parameter (ψ) and (b) slightly different rotation rates as a function of depth (NA=0.71).

Although the HER-PSF design used a coherent transfer function model, it applies equally well for an incoherent imaging system because the incoherent PSF is the modulo-squared of its coherent counterpart. However, because the mask is designed for one particular wavelength, it is interesting to analyze its wavelength (λ) dependence. Wavelength dependence arises from four factors: 1) phase retardation of a mask is 2πnt(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]

]. However, as a function of depth, the rotation rate increases when the mask is used with wavelengths smaller than its design wavelength, and decreases when used with wavelengths greater than the design wavelength. This is exemplified in Fig. 4(b) for a unity magnification system with 0.71 numerical aperture (NA).

For experimental demonstration, we implement HER-PSF with a reflective phase-only spatial light modulator (SLM). Because of the sampling of the phase function by the SLM’s pixels, the SLM produces multiple orders, of which the 0th 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

We introduced HER-PSFs, described their design methodology, analyzed their spatial and frequency response, and demonstrated them experimentally. We showed that HER-PSFs can offer over thirty times higher efficiency than the exact rotating PSFs.

Acknowledgments

S. R. P. Pavani thankfully acknowledges support from a CDM Optics - OmniVision Technologies Ph.D. fellowship. This work was funded by the Technology Transfer Office of the University of Colorado and the National Science foundation (award ECS-225533).

1.

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

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. 44, 1409–1416 (1997). [CrossRef]

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]

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. 46, 227–238 (1999).

5.

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys.-Usp. 47, 1177–1203 (2004). [CrossRef]

6.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Angular momentum of a rotating light beam,” Opt. Comm. 249, 367–378 (2005). [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]

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 296, 1101–1103 (2002). [CrossRef] [PubMed]

9.

R. Grimm, M. Weidemuller, and Y. B. Ovchinnikov, “Optical dipole traps for neutral atoms,” Adv. At. Mol. Opt. Phys. 42, 95–170 (2000). [CrossRef]

10.

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

11.

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1978), Chap. 3.

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]

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/oe/abstract.cfm?URI=oe-16-5-3484


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References

  1. Y. Y. Schechner, R. Piestun, and J. Shamir, "Wave propagation with rotating intensity distributions," Phys. Rev. E 54, R50-R53 (1996). [CrossRef]
  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. 44, 1409-1416 (1997). [CrossRef]
  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]
  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. 46, 227-238 (1999).
  5. E. G. Abramochkin and V. G. Volostnikov, "Spiral light beams," Phys.-Usp. 47, 1177-1203 (2004). [CrossRef]
  6. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Angular momentum of a rotating light beam," Opt. Comm. 249, 367-378 (2005). [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]
  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 296, 1101-1103 (2002). [CrossRef] [PubMed]
  9. R. Grimm, M. Weidemuller, and Y. B. Ovchinnikov, "Optical dipole traps for neutral atoms," Adv. At. Mol. Opt. Phys. 42, 95-170 (2000). [CrossRef]
  10. A. Greengard, Y. Y. Schechner, and R. Piestun, "Depth from diffracted rotation," Opt. Lett. 31, 181-183 (2006). [CrossRef] [PubMed]
  11. W. H. Lee, "Computer-generated holograms: techniques and applications," in Progress in Optics, E. Wolf, ed. (Elsevier, 1978), Chap. 3.
  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]

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