## Focal modulation using rotating phase filters

Optics Express, Vol. 18, Issue 8, pp. 7820-7826 (2010)

http://dx.doi.org/10.1364/OE.18.007820

Acrobat PDF (704 KB)

### Abstract

We describe a simple method of refocusing optical systems that is based on the use of two identical phase filters. These filters are divided in annuli and each annulus is divided into sectors with a particular phase value. A controlled focus displacement is achieved by rotating one filter with respect to the other. This displacement is related with the filter parameters. Transverse responses are studied as a function of filters relative position. Furthermore, the experimental set up shows that theoretical prediction fit well with experimental results. The main advantage of this system is the ease of fabrication so that it could be useful in different applications requiring small size, light weight or thin systems, like mobile phone cameras, microscopy tomography, and others.

© 2010 OSA

## 1. Introduction

1. N. A. Riza, “Axial scanning confocal microscopy with no moving parts,” Opt. Photon. News **19**(12), 33 (2008). [CrossRef]

2. E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. **281**(4), 880–887 (2008). [CrossRef]

3. C. A. López and A. H. Hirsa, “Fast focusing using a pinned-contact oscillating liquid lens,” Nat. Photonics **2**(10), 610–613 (2008). [CrossRef]

4. L. Dong, A. K. Agarwal, D. J. Beebe, and H. Jiang, “Adaptive liquid microlenses activated by stimuli-responsive hydrogels,” Nature **442**(7102), 551–554 (2006). [CrossRef] [PubMed]

5. S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. **85**(7), 1128–1130 (2004). [CrossRef]

6. D. Graham-Rowe, “Liquid lenses make a splash,” Nat. Photonics **sample**, 2–4 (2006). [CrossRef]

7. H. Ren, S. Xu, Y. J. Lin, and S. T. Wu, “Adaptive-focus lenses,” Opt. Photon. News **19**(10), 42–47 (2008). [CrossRef]

9. A. W. Lohmann and D. P. Paris, “Variable fresnel zone pattern,” Appl. Opt. **6**(9), 1567–1570 (1967). [CrossRef] [PubMed]

10. S. Bara, Z. Jaroszewicz, A. Kolodziejczyk, and V. Moreno, “Determination of basic grids for subtractive moire patterns,” Appl. Opt. **30**(10), 1258–1262 (1991). [CrossRef] [PubMed]

11. S. Bernet and M. Ritsch-Marte, “Adjustable refractive power from diffractive moiré elements,” Appl. Opt. **47**(21), 3722–3730 (2008). [CrossRef] [PubMed]

## 2. Basic theory of segmented filters

_{4}Zernike polynomial, given by Z

_{4}(

*ρ*) = 2

*ρ*

^{2}-1, where

*ρ*is the normalized radial coordinate. Hence, our goal is to obtain a final phase distribution as similar as possible to the phase described by the Z

_{4}Zernike polynomial. Obviously, since we can only create annuli, we have to approximate the Z

_{4}Zernike polynomial using a stepped function. There are different possibilities for the value of the step height. The step value has to take values in the interval (0, 2π). It has been shown that step values that are not a multiple of π produce focus displacement [12

12. M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun. **241**(4-6), 249–253 (2004). [CrossRef]

*n*π where

*n*= 1, 2… Then, in each filter we use a step value of π/2.

_{4}Zernike polynomial provided the radii of the annuli are

*i*= 1, 2, 3. A drawback of this procedure is that with no rotation it yields a central focus (formed by two opposite quadrants with null phase) and two smaller spots in front and after the central spot (focused by the 0-π Fresnel lens in the two other quadrants). This disadvantage is overcome in the device shown in Fig. 2 .

_{4}Zernike polynomial, provided the correct radii (

_{4}(Fig. 2d). In this case the focus displacement provided by the resulting filter is the opposite of that obtained from Fig. 2b.

## 3. Design of the filter

*ρ,θ*) and F2(

*ρ,θ*), placed very close to the lens, and, finally, focuses in a certain spatial light distribution at the focal region. To estimate the effect produced by the filters we analyze the axial behavior of the Point Spread Function (PSF) of the optical system.

## 4. Analysis of focus displacement

*P*(

*ρ,θ*) = exp[

*j ϕ*(

*ρ,θ*)], where

*j*is

*ρ*is the normalized radial coordinate over the pupil plane,

*θ*is the azimuthal angle and

*ϕ(ρ,θ)*is the phase function of the filter. For a converging monochromatic spherical wave front passing through the center of the pupil, the normalized field amplitude

*U*in the focal region may be written aswhere

*v*and

*u*are radial and axial dimensionless optical coordinates with origin at the geometrical focus, given by

*v*=

*k NA r*and

*u*=

*k NA*

^{2}

*z*.

*NA*is the numerical aperture,

*k =*2π/

*λ*and

*r*and

*z*are the usual radial and axial distances. The pupil function is

*P*(

*ρ,θ*) = F1(

*ρ,θ*)F2(

*ρ,θ + θ*). Hence, the pupil function will vary as a function of the relative angular position (

_{0}*θ*

_{0}) between both filters. The PSF will be obtained as the squared modulus of Eq. (1).

## 5. Transversal behavior

*x-y*shown in Fig. 3. Following the previous scheme we will work within the framework of the scalar diffraction theory, using Eq. (1). We analyze the transversal behavior of the filter described in Fig. 1. Figure 6a shows the transversal PSF for a single lens (lens L in Fig. 3) at its focal plane. Figure 6b shows the transversal PSF at the lens focal plane when the two filters are introduced with no rotation between them. It can be seen that the PSF maintains the same axial position and width as for the case without filter. When a rotation angle

*θ*

_{0}= π/2 is introduced between the two filters, the focus is displaced producing a transversal PSF as that shown in Fig. 6c. This PSF is very similar to the previous ones, with a slight resolution loss. If a rotation

*θ*

_{0}= π is performed (Fig. 6d) the same result as for a rotation

*θ*

_{0}= 0 is obtained.

## 6. Experimental checking

*θ*

_{0}= π/2. In this case the phase grows from zero to 2π (from the center to the outside) with steps of π/2. With no rotation, there is no focus displacement, while with π/2 rotation the focus is displaced at a distance of 775μm from the first one, with a high Strehl value. Figure 9 shows the transversal light distribution of both foci. It can be seen that a slight pattern appears as a consequence of the filter structure. Furthermore, with no rotation the two spots focused by the 0-π Fresnel lens in opposite quadrants yield the side-lobes in the Fig. 9a.

## 7. Conclusions

## Acknowledgements

## References and links

1. | N. A. Riza, “Axial scanning confocal microscopy with no moving parts,” Opt. Photon. News |

2. | E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. |

3. | C. A. López and A. H. Hirsa, “Fast focusing using a pinned-contact oscillating liquid lens,” Nat. Photonics |

4. | L. Dong, A. K. Agarwal, D. J. Beebe, and H. Jiang, “Adaptive liquid microlenses activated by stimuli-responsive hydrogels,” Nature |

5. | S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. |

6. | D. Graham-Rowe, “Liquid lenses make a splash,” Nat. Photonics |

7. | H. Ren, S. Xu, Y. J. Lin, and S. T. Wu, “Adaptive-focus lenses,” Opt. Photon. News |

8. | L. W. Alvarez, “Two-element variable-power spherical lens,” U.S. patent 3,305,294 (1967). |

9. | A. W. Lohmann and D. P. Paris, “Variable fresnel zone pattern,” Appl. Opt. |

10. | S. Bara, Z. Jaroszewicz, A. Kolodziejczyk, and V. Moreno, “Determination of basic grids for subtractive moire patterns,” Appl. Opt. |

11. | S. Bernet and M. Ritsch-Marte, “Adjustable refractive power from diffractive moiré elements,” Appl. Opt. |

12. | M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun. |

13. | ITME Institute of Electronic Materials Technology, Warsaw, Poland. |

**OCIS Codes**

(100.2980) Image processing : Image enhancement

(120.2440) Instrumentation, measurement, and metrology : Filters

**ToC Category:**

Image Processing

**History**

Original Manuscript: November 5, 2009

Revised Manuscript: February 15, 2010

Manuscript Accepted: March 9, 2010

Published: March 31, 2010

**Citation**

Pedro J. Valle, Vidal F. Canales, and Manuel P. Cagigal, "Focal modulation using rotating phase filters," Opt. Express **18**, 7820-7826 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-7820

Sort: Year | Journal | Reset

### References

- N. A. Riza, “Axial scanning confocal microscopy with no moving parts,” Opt. Photon. News 19(12), 33 (2008). [CrossRef]
- E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281(4), 880–887 (2008). [CrossRef]
- C. A. López and A. H. Hirsa, “Fast focusing using a pinned-contact oscillating liquid lens,” Nat. Photonics 2(10), 610–613 (2008). [CrossRef]
- L. Dong, A. K. Agarwal, D. J. Beebe, and H. Jiang, “Adaptive liquid microlenses activated by stimuli-responsive hydrogels,” Nature 442(7102), 551–554 (2006). [CrossRef] [PubMed]
- S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85(7), 1128–1130 (2004). [CrossRef]
- D. Graham-Rowe, “Liquid lenses make a splash,” Nat. Photonics sample, 2–4 (2006). [CrossRef]
- H. Ren, S. Xu, Y. J. Lin, and S. T. Wu, “Adaptive-focus lenses,” Opt. Photon. News 19(10), 42–47 (2008). [CrossRef]
- L. W. Alvarez, “Two-element variable-power spherical lens,” U.S. patent 3,305,294 (1967).
- A. W. Lohmann and D. P. Paris, “Variable fresnel zone pattern,” Appl. Opt. 6(9), 1567–1570 (1967). [CrossRef] [PubMed]
- S. Bara, Z. Jaroszewicz, A. Kolodziejczyk, and V. Moreno, “Determination of basic grids for subtractive moire patterns,” Appl. Opt. 30(10), 1258–1262 (1991). [CrossRef] [PubMed]
- S. Bernet and M. Ritsch-Marte, “Adjustable refractive power from diffractive moiré elements,” Appl. Opt. 47(21), 3722–3730 (2008). [CrossRef] [PubMed]
- M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun. 241(4-6), 249–253 (2004). [CrossRef]
- ITME Institute of Electronic Materials Technology, Warsaw, Poland.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.