## Locally magnifying imager |

Optics Express, Vol. 19, Issue 6, pp. 5676-5689 (2011)

http://dx.doi.org/10.1364/OE.19.005676

Acrobat PDF (1173 KB)

### Abstract

We present a new optical system capable of changing in real-time, anywhere in the field of view, the magnification of the image, while potentially keeping the total field of view constant. This is achieved by using an active optic element to change the direction of some selected rays, thus creating controlled distortion. A mathematical description of such a system is presented, along with the fundamental limits on the amplitude of the active surface and on the F/# to keep the image quality. Experimental results obtained with a simple prototype using a ferrofluidic deformable mirror as the active surface are also presented. The local magnifications obtained are in agreement with the developed mathematical model.

© 2011 OSA

## 1. Introduction

1. J. J. Kumler and M. L. Bauer, “Fish-eye lenses designs and their relative performance,” Proc. SPIE **4093**, 360–369 (2000). [CrossRef]

3. J. Parent and S. Thibault, “Spatial dependence of surface error slopes on tolerancing panoramic lenses,” Appl. Opt. **49**(14), 2686–2693 (2010). [CrossRef]

4. J. Parent and S. Thibault, “Active Imaging Lens with Real-Time Variable Resolution and Constant Field of View,” Proc. SPIE **7652**, 76522F, 76522F-12 (2010). [CrossRef]

5. D. V. Wick and T. Martinez, “Adaptive optical zoom,” Opt. Eng. **43**(1), 8 (2004). [CrossRef]

6. V. Laude and C. Dirson, “Liquid-crystal active lens: application to image resolution enhancement,” Opt. Commun. **163**(1-3), 72–78 (1999). [CrossRef]

7. G. Curatu and J. E. Harvey, “Lens design and system optimization for foveated imaging,” Proc. SPIE **7060**, 70600P, 70600P-9 (2008). [CrossRef]

## 2. Spatio-temporally adjustable magnification lens

### 2.1 Concept of the locally magnifying imager

### 2.2 Mathematical description of the system

3. J. Parent and S. Thibault, “Spatial dependence of surface error slopes on tolerancing panoramic lenses,” Appl. Opt. **49**(14), 2686–2693 (2010). [CrossRef]

3. J. Parent and S. Thibault, “Spatial dependence of surface error slopes on tolerancing panoramic lenses,” Appl. Opt. **49**(14), 2686–2693 (2010). [CrossRef]

_{0}, an approximation valid with systems having FFOV under 30 degrees and no original distortion. For systems with larger FFOV, having initial distortion, more complex equations have to be used.

4. J. Parent and S. Thibault, “Active Imaging Lens with Real-Time Variable Resolution and Constant Field of View,” Proc. SPIE **7652**, 76522F, 76522F-12 (2010). [CrossRef]

10. www.zemax.com

_{curv}. Similarly, if the reflecting mirror is replaced by a refracting surface, RoM(r) is given by Eq. (7), where the + sign means that a positive slope on the active surface bends the rays away from the optical axis contrary to the reflecting case.This is again for small angles θ, but also for small incident angles with the normal. If this is not the case, the impact of the sin(θ) term in the Snell-Descartes refraction law has to be taken into account.

### 2.3 Fundamentals limits of this system

#### 2.3.1 Amplitude of the active surface

_{0}between the mirror and the entrance pupil of the imager is a function of the deformable mirror half-diameter and of the ratio H/f = tan(θ

_{max}), the ratio between the half image plane length H and the original focal length

*f*of the imager (without the deformable mirror). For this example, a circular zone of increased magnification having a diameter of a fraction α of the full mirror diameter D is considered. From the equation of a parabola of curvature Z”, the required amplitude just for the zone of increased magnification (from r = 0 to r = a at Fig. 3) is then given by Eq. (9).If continuous image sampling is required, a zone of settling back is also needed (from r = a to r = b) to restore the original magnification as in the zone where the mirror is flat (from r = b to r = D/2). Depending on the shape and width of that zone of settling back, the total amplitude required on the mirror could be considerably higher than the amplitude just for the zone of interest.

_{max}), as in wide-angle lenses, the required amplitude can become very high. For lenses with longer focal length, the amplitude is small, but the distance L

_{0}between the mirror and the imager is large. For increasing values of RoM, the increase in amplitude is significant from 1 to 2, but gets slower with ever increasing RoM, an effect of the (RoM-1)/RoM term in Eq. (9). This figure is for given values of α and D, but looking at Eq. (9) shows that if α is kept constant and D doubled, the amplitude required is also doubled. Also, if D is constant and α is doubled, the amplitude is 4 times higher.

^{1/2}σ as Z = Aexp(-r

^{2}/2σ

^{2}), taking the second derivative, inserting it in Eq. (6) and rearranging the terms gives Eq. (10) for the amplitude as a function of the RoM at the center of the mirror (r = 0), again using from Eq. (8) that L

_{o}= Df/2H.

#### 2.3.2 Entrance pupil diameter

11. D. Malacara-Hernandez, “Some parameters and characteristics of an off-axis paraboloid,” Opt. Eng. **30**(9), 1277–1280 (1991). [CrossRef]

12. J. Roberts, A. H. Bouchez, R. S. Burruss, R. G. Dekany, S. R. Guiwits, and M. Troy, “Optical characterization of the PALM-3000 3388-actuator deformable mirror,” Proc. SPIE **7736**, 77362E, 77362E-8 (2010). [CrossRef]

_{x}RoM

_{y}if both dimensions are considered), but also spreading it on RoM times more pixels, a consequence of conservation of radiance. For an initial plane wave, the resulting PV OPD in the Z direction after reflection on the mirror, an approximation of the PV OPD in the direction of the beam for small angles, is then given by Eq. (11). This is for a parabola of curvature Z” and it includes a factor 2 for the reflection on the mirror. If the angle is not small, a term 1/cos(θ) = ((r

^{2}+ L

_{0}

^{2})

^{1/2})/L

_{0}must be added.

^{−1}on a distance L

_{o}and Eq. (6). The PV of a spherical wavefront being given by PV = (r

^{2}/R)/(1 + (1-r

^{2}/R

^{2})

^{1/2}), the PV OPD is also scaled by 1/RoM. Since the imager adds few aberrations at a high F/# like the one used here, it is then considered that the additional PV OPD is equal in the entrance pupil and in the exit pupil, finally giving Eq. (12).This means that the F/# of the imager, with the effect of the curvature Z” of the mirror included in the RoM terms, by combining Eq. (12) with Eq. (6) and Eq. (8), must be higher than what is prescribed by Eq. (13).As a numerical example, for a RoM of 2, H of 2.88 mm (half long-axis of a 1/2.5” CCD), f = 12.5 mm, D = 100 mm and λ = 550 nm, the local F/# must be higher than 25.6.

## 3. Experimental verification of the system

### 3.1 Experimental prototype

14. D. Brousseau, E. F. Borra, M. Rochette, and D. B. Landry, “Linearization of the response of a 91-actuator magnetic liquid deformable mirror,” Opt. Express **18**(8), 8239–8250 (2010). [CrossRef] [PubMed]

_{0}between the entrance pupil and the center of the mirror is set to 217 mm according to Eq. (8). The lens is a f = 12.5 mm Fujinon HF12.5SA with an entrance pupil measured to be 25 ± 1 mm behind the metallic frame on the object side. It has a manually adjustable focus ranging from 0.1 m to infinity and a manually adjustable F/# ranging from 1.4 to 22. For all the following images, a F/# of 16 is used. According to Eq. (13), this is the limit for a RoM of 1.4 and some defocus is expected when higher RoM are produced. As for the camera, a 2592x1944 pixels (5 megapixels) AVT Guppy F-503 having pixels of 2.2 µm is used. Figure 6 shows the reference image obtained with the deformable mirror in a flat position. The test pattern is a ceiling light diffuser, the plastic cover under a neon light, because of its brightness and regularly spaced pattern. Each small square in the regular pattern, as seen in Fig. 6, is used as a target for measurements. All the images are cropped in the same way to show the zone of interest and the region around it.

### 3.2 Parabola producing a zone of constant magnification

_{0}) and/or using a smaller zone of interest [4

4. J. Parent and S. Thibault, “Active Imaging Lens with Real-Time Variable Resolution and Constant Field of View,” Proc. SPIE **7652**, 76522F, 76522F-12 (2010). [CrossRef]

### 3.3 Gaussian bump producing a localized high magnification

## 4. Discussion of the limits of locally magnifying imagers

_{0}means that for smaller amplitudes on the active surface, it is better to use the deformable surface far from the imager lenses, which means a large active surface diameter D or a small ratio H/f = tan(θ

_{max}) according to Eq. (8). Since using a large active surface diameter is often costly, this seems to limit this kind of imager to a long focal length, or in other words, small FFOV.

_{0}Z”) on the product of Z” and L

_{o}, the relative error ΔRoM is given by Eq. (14). With a numerical example, a relative error of 1% (Δ(L

_{0}Z”) = 0.01) will produce relative RoM errors of respectively 0.10%, 1.01% and 4.17% with RoM of 1.1, 2 and 5, meaning that the larger the RoM is, the more impact on the desired magnification is caused by a given relative error on the surface curvature or the distance L

_{o}. It was not possible to measure it in our experiment because of the limited amplitude that can be measured by the Shack-Hartmann.

## 5. Conclusion

17. D. J. Reiley and R. A. Chipman, “Adjustable distortion corrector,” Proc. SPIE **1690**, 11–19 (1992). [CrossRef]

## Acknowledgements

## References and links

1. | J. J. Kumler and M. L. Bauer, “Fish-eye lenses designs and their relative performance,” Proc. SPIE |

2. | J. Parent and S. Thibault, “Tolerancing panoramic lenses,” Proc. SPIE |

3. | J. Parent and S. Thibault, “Spatial dependence of surface error slopes on tolerancing panoramic lenses,” Appl. Opt. |

4. | J. Parent and S. Thibault, “Active Imaging Lens with Real-Time Variable Resolution and Constant Field of View,” Proc. SPIE |

5. | D. V. Wick and T. Martinez, “Adaptive optical zoom,” Opt. Eng. |

6. | V. Laude and C. Dirson, “Liquid-crystal active lens: application to image resolution enhancement,” Opt. Commun. |

7. | G. Curatu and J. E. Harvey, “Lens design and system optimization for foveated imaging,” Proc. SPIE |

8. | S. Kuthirummal and S. K. Nayar, “Flexible Mirror Imaging,” Proc. IEEE Conf. Comput. Vision |

9. | S. Thibault, J. Gauvin, M. Doucet, and M. Wang, “Enhanced optical design by distortion control,” Proc. SPIE 5962 (2005). |

10. | |

11. | D. Malacara-Hernandez, “Some parameters and characteristics of an off-axis paraboloid,” Opt. Eng. |

12. | J. Roberts, A. H. Bouchez, R. S. Burruss, R. G. Dekany, S. R. Guiwits, and M. Troy, “Optical characterization of the PALM-3000 3388-actuator deformable mirror,” Proc. SPIE |

13. | W. J. Smith, |

14. | D. Brousseau, E. F. Borra, M. Rochette, and D. B. Landry, “Linearization of the response of a 91-actuator magnetic liquid deformable mirror,” Opt. Express |

15. | L. Zhao, N. Bai, X. Li, L. S. Ong, Z. P. Fang, and A. K. Asundi, “Efficient implementation of a spatial light modulator as a diffractive optical microlens array in a digital Shack-Hartmann wavefront sensor,” Appl. Opt. |

16. | F. Pardo, |

17. | D. J. Reiley and R. A. Chipman, “Adjustable distortion corrector,” Proc. SPIE |

**OCIS Codes**

(080.2740) Geometric optics : Geometric optical design

(120.3620) Instrumentation, measurement, and metrology : Lens system design

(220.1080) Optical design and fabrication : Active or adaptive optics

**History**

Original Manuscript: November 8, 2010

Revised Manuscript: December 21, 2010

Manuscript Accepted: March 7, 2011

Published: March 11, 2011

**Citation**

Jocelyn Parent and Simon Thibault, "Locally magnifying imager," Opt. Express **19**, 5676-5689 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-6-5676

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### References

- J. J. Kumler and M. L. Bauer, “Fish-eye lenses designs and their relative performance,” Proc. SPIE 4093, 360–369 (2000). [CrossRef]
- J. Parent and S. Thibault, “Tolerancing panoramic lenses,” Proc. SPIE 7433, 7433D (2009).
- J. Parent and S. Thibault, “Spatial dependence of surface error slopes on tolerancing panoramic lenses,” Appl. Opt. 49(14), 2686–2693 (2010). [CrossRef]
- J. Parent and S. Thibault, “Active Imaging Lens with Real-Time Variable Resolution and Constant Field of View,” Proc. SPIE 7652, 76522F, 76522F-12 (2010). [CrossRef]
- D. V. Wick and T. Martinez, “Adaptive optical zoom,” Opt. Eng. 43(1), 8 (2004). [CrossRef]
- V. Laude and C. Dirson, “Liquid-crystal active lens: application to image resolution enhancement,” Opt. Commun. 163(1-3), 72–78 (1999). [CrossRef]
- G. Curatu and J. E. Harvey, “Lens design and system optimization for foveated imaging,” Proc. SPIE 7060, 70600P, 70600P-9 (2008). [CrossRef]
- S. Kuthirummal and S. K. Nayar, “Flexible Mirror Imaging,” Proc. IEEE Conf. Comput. Vision ICCV, 1–8 (2007).
- S. Thibault, J. Gauvin, M. Doucet, and M. Wang, “Enhanced optical design by distortion control,” Proc. SPIE 5962 (2005).
- www.zemax.com
- D. Malacara-Hernandez, “Some parameters and characteristics of an off-axis paraboloid,” Opt. Eng. 30(9), 1277–1280 (1991). [CrossRef]
- J. Roberts, A. H. Bouchez, R. S. Burruss, R. G. Dekany, S. R. Guiwits, and M. Troy, “Optical characterization of the PALM-3000 3388-actuator deformable mirror,” Proc. SPIE 7736, 77362E, 77362E-8 (2010). [CrossRef]
- W. J. Smith, Modern Optical Engineering 4th ed., (McGraw-Hill, 2007).
- D. Brousseau, E. F. Borra, M. Rochette, and D. B. Landry, “Linearization of the response of a 91-actuator magnetic liquid deformable mirror,” Opt. Express 18(8), 8239–8250 (2010). [CrossRef] [PubMed]
- L. Zhao, N. Bai, X. Li, L. S. Ong, Z. P. Fang, and A. K. Asundi, “Efficient implementation of a spatial light modulator as a diffractive optical microlens array in a digital Shack-Hartmann wavefront sensor,” Appl. Opt. 45(1), 90–94 (2006). [CrossRef] [PubMed]
- F. Pardo, et al., “Characterization of Piston-Tip-Tilt mirror pixels for scalable SLM arrays,” Proceedings of IEEE Conference on Optical MEMS and Their Applications, (2006), pp. 21–22.
- D. J. Reiley and R. A. Chipman, “Adjustable distortion corrector,” Proc. SPIE 1690, 11–19 (1992). [CrossRef]

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