## Multiscale lens design

Optics Express, Vol. 17, Issue 13, pp. 10659-10674 (2009)

http://dx.doi.org/10.1364/OE.17.010659

Acrobat PDF (859 KB)

### Abstract

While lenses of aperture less than 1000λ frequently form images with pixel counts approaching the space-bandwidth limit, only heroic designs approach the theoretical information capacity at larger scales. We propose to use the field processing capabilities of small-scale secondary lens arrays to correct aberrations due to larger scale objective lenses, with an ultimate goal of achieving diffraction-limited imaging for apertures greater than 10,000λ. We present an example optical design using an 8 mm entrance pupil capable of resolving 20 megapixels.

© 2009 Optical Society of America

## 1. Introduction

1. G. T. di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. **59**, 799–804 (1969). [CrossRef]

^{2}. Given the long history of this result, one is surprised to note that few modern optical systems approach Shannon-limited performance. The goal of this paper is to demonstrate moderate to large scale optical systems approaching the Shannon limit may be achieved by relaxing traditional constraints on lens system design and field uniformity.

*γ*. As discussed in a pioneering study by Lohmann, performance against this metric saturates as a function of spatial scale, meaning that

*γ*tends to fall in inverse proportion to aperture area [3

3. A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. **28**, 4996–4998 (1989). [CrossRef] [PubMed]

*γ*by increasing lens complexity as aperture area increases. This approach was adopted to great effect in the development of lithographic lens systems [4], but is ultimately unsustainable in imaging applications.

*γ*image formation, one may also suppose that smaller lenses are better at wavefront correction. This is the case for two reasons. First, wavefront correction and image formation both yield geometric solutions with less wavelength-scale error over smaller apertures. Second, manufacturing of complex lens surfaces is much easier in smaller scale systems. Sophisticated small scale manufacturing techniques have been particularly advanced by recent efforts to develop wafer-level cameras for mobile devices [5, 6].

*collector*and the multi-aperture secondary array the

*processor*.

7. J. Tanida, T. Kumagai, K. Yamada, S. Miyatake, K. Ishida, T. Morimoto, N. Kondou, D. Miyazaki, and Y. Ichioka, “Thin observation module by bound optics (TOMBO): concept and experimental verification,” Appl. Opt. **40**, 1806–1813 (2001). [CrossRef]

8. M. Shankar, R. Willett, N. Pitsianis, T. Schulz, R. Gibbons, R. T. Kolste, J. Carriere, C. Chen, D. Prather, and D. Brady, “Thin infrared imaging systems through multichannel sampling,” Appl. Opt. **47**, B1–B10 (2008). [CrossRef] [PubMed]

9. T. Mirani, D. Rajan, M. P. Christensen, S. C. Douglas, and S. L. Wood, “Computational imaging systems: joint design and end-to-end optimality,” Appl. Opt. **47**, B86–B103 (2008). [CrossRef] [PubMed]

10. K. Choi and T. J. Schulz, “Signal-processing approaches for image-resolution restoration for TOMBO imagery,” Appl. Opt. **47**, B104–B116 (2008). [CrossRef] [PubMed]

11. A. V. Kanaev, D. A. Scribner, J. R. Ackerman, and E. F. Fleet, “Analysis and application of multiframe superresolution processing for conventional imaging systems and lenslet arrays,” Appl. Opt. **46**, 4320–4328 (2007). [CrossRef] [PubMed]

12. A. D. Portnoy, N. P. Pitsianis, X. Sun, and D. J. Brady, “Multichannel sampling schemes for optical imaging systems,” Appl. Opt. **47**, B76–B85 (2008). [CrossRef] [PubMed]

13. R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, “Three-dimensional information acquisition using a compound imaging system,” Opt. Rev. **14**, 347–350 (2007). [CrossRef]

14. E. H. Adelson and J. Y. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intel. **14**, 99–106 (1992). [CrossRef]

15. M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Trans. Graphics **25**, 924–934 (2006). [CrossRef]

16. H.-J. Lee, D.-H. Shin, H. Yoo, J.-J. Lee, and E.-S. Kim, “Computational integral imaging reconstruction scheme of far 3D objects by additional use of an imaging lens,” Opt. Comm. **281**, 2026–2032 (2007). [CrossRef]

*plenoptic*or

*lightfield*cameras [15

15. M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Trans. Graphics **25**, 924–934 (2006). [CrossRef]

14. E. H. Adelson and J. Y. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intel. **14**, 99–106 (1992). [CrossRef]

*integral imaging systems*use arrays of microlenses to form multiple images of a scene at slightly different perspectives to perform 3D imaging. Recent work by Lee

*et al*. [16

16. H.-J. Lee, D.-H. Shin, H. Yoo, J.-J. Lee, and E.-S. Kim, “Computational integral imaging reconstruction scheme of far 3D objects by additional use of an imaging lens,” Opt. Comm. **281**, 2026–2032 (2007). [CrossRef]

*Integral field spectrometers*[19

19. R. Bacon, P. Y. Copin, G. Monnet, B. W. Miller, J. R. Allington-Smith, M. Bureau, C. M. Carollo, R. L. Davies, E. Emsellem, H. Kuntschner, R. F. Peletier, E. K. Verolme, and P. T. de Zeeuw, “The SAURON project—I. The panoramic integral-field spectrograph,” Monthly Notices of the Royal Astronomical Society **326**, 23–35 (2001). [CrossRef]

*γ*multiscale design.

## 2. Scale and merit functions

*π*(

*A*/2)

^{2}for pupil diameter

*A*. Letting FOV represent the full angular field of view of the system, the maximum spatial frequency in the coherent field illuminating the aperture is

*λ*f

_{#}for small field angles, where

*f*

_{#}is the f-number of the lens. The diameter of the image is roughly 2

*F*tan(FOV/2) and the number of resolvable spots is therefore

*N*

_{max}is approximately one order of magnitude less than S. The information efficiency of a lens may be equivalently defined as the ratio of the number of resolvable spots or the ratio of the Shannon number achieved by the lens to the corresponding diffraction limited figure. For a lens achieving

*N*resolvable spots, substituting from Eqn. (2) yields

*γ*under this definition may differ somewhat from the Shannon number based value. Of course, we expect

*γ*≤1. In practice,

*γ*decreases monotonically in

*A*because

*N*does not grow in proportion to

*A*

^{2}.

*γ*fall as a function of aperture size, consider the ray tracing diagram for a simple lens shown in Fig. 3. Ray tracing is, of course, commonly used to analyze geometric aberrations. In this case, the on-axis ray bundle comes to reasonable focus, but the off-axis rays are severely blurred due to aberrations. For present purposes, the most interesting feature of the diagram is that it is scale invariant, meaning that the drawing is independent of the spatial units used. If we take a given design and multiply all length parameters by

*M*(

*i.e.*scale the lens by the factor

*M*), we can use the scale parameter to explore the balance between the diffraction-limited (small

*M*) and aberration-limited (large

*M*) regimes of image quality.

*M*. The diffraction-limited spotsize, in contrast, is scale-invariant.

_{1}represents the mean aberration spotsize radius for the default scale,

*M*=1. Defining

*D*

_{1}to be the image diameter on the default scale, the number of resolvable elements varies with

*M*as

*γ*as a function of

*M*for three different values for ξ

_{1}. For ξ1=0, geometric aberration is eliminated and the system achieves

*γ*=1 at all scales. For ξ

_{1}>0, diffraction limited performance is achieved for

*M*<

*λ f*

_{#}/ξ

_{1}, but beyond this limit

*γ*decreases in inverse proportion to

*M*

^{2}and the number of pixels in the image becomes scale invariant.

*M*ξ

_{1}/(

*λ*

*f*

_{#}) from increasing linearly in

*M*. Combinations of the following strategies are chosen to decrease ξ

_{1}/(

*λ*

*f*

_{#}) as

*M*increases:

*MD*

_{1}). Geometric aberration generally increases with field angle, so that reducing FOV allows the existing degrees of freedom in the design to better compensate for the aberrations at remaining field angles.

*f*

_{#}. The extraordinary growth in lithographic lens mass and complexity to achieve this objective is well documented [4].

*f*

_{#}as a function of

*M*. This approach reduces geometric aberration while also increasing the image space diffractive blur size. As illustrated in Fig. 5, Lohmann’s empirical rule [3

3. A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. **28**, 4996–4998 (1989). [CrossRef] [PubMed]

*f*

_{#}∝

*M*

^{1/3}describes this f-number scaling in many systems, with the notable exception of telescopes, which adopt strategy (1) and lithographic lenses which adopt strategy (2).

*M*ξ

_{1}/(

*λ*

*f*

_{#}) as M grows. Its basic advantage derives from the potential to overcome the information capacity saturation effect illustrated in Fig. 4. In a multi-aperture camera, the number of resolvable spots may be expected to increase linearly with the number of apertures. For fixed subaperture size, the scaling parameter is proportional to the square root of the number of apertures and the system information capacity scales in proportion to

*M*

^{2}. Of course, this presupposes that the information collected by each subaperture is independent. This assumption may be valid in “close imaging” applications, such as document scanning and microscopy, but is not valid when subaperture fields of view substantially overlap. The delivery of nonredundant views of distant objects to a multi-aperture array is the purpose of the collector element in a multiscale design.

*λ*may be optimal.

## 3. Design principles and examples

*H*is the normalized field angle,

*ρ*the normalized pupil radius,

*φ*the azimuth angle of the pupil coordinate, and

*W*the wavefront aberration coefficient expressed in units of length.

_{ijk}*W*(

*H*,

*ρ*,

*ϕ*) at targeted field points. Multiscale design deviates from conventional design in that the goal is to correct

*local*rather than global wavefront error. Since the lenslets divide the field into subregions, each lenslet can be used to correct the

*local*aberration centered around its appropriate field angle. In pursuit of this objective, we expand Eqn. (6) in terms of the central field angle

*H*for the

_{n}*n*th lenslet:

*in*dependent aberration, should be fully corrected in the collection optics, since there is no benefit to correcting it in the lenslet processor array. This holds true for any of the higher-order spherical aberration terms (

*W*

_{060},

*W*

_{080}, etc.) as well. From the presence of non-Seidel aberrations in the aberration function local to each lenslet, we can infer that the surfaces of the individual lenslets will need to take on non-cylindrically-symmetric shapes. To get an idea of what these shapes might look like, we can first use the example of field curvature illustrated in Fig. 6. By splitting up the field into subregions, defocus becomes the primary aberration for the off-axis lenslets in this case. This is easy to fix, since we need only change the lenslet’s focal length to introduce a compensating change in focus. The off-axis elements next need to correct for a substantial amount of linear defocus (

*i.e.*image tilt). Prisms are often used for this purpose, and thus we can design a wedge shape into the lenslets to untilt the image. This wedge shape is apparent in the lenslet profiles illustrated in Fig. 7(c). The remaining aberration, after subtracting uniform and linear defocus, is the standard form of field curvature, but greatly reduced from its full-field form.

*f*/8 Wollaston landscape lens as the collector. All of the systems were modeled with Zemax optical design software [22

22. URL http://www.zemax.com.

*d*wavelength (

*λ*=587 nm), and the freeform lenslet surface parameters are taken to be

*x*-

*y*polynomials of up to sixth order. For the conventional designs, the merit function was chosen to minimize the spotsize at field angles 0°, 20°, and 30°. For the multiscale designs, in order to suppress spherical aberration, and to a lesser extent coma, the merit function weights the field angles with weights 100, 1, and 0.1 respectively. For the design of the lenslets themselves, the merit function is naturally more complicated. In a freeform surface design, one may easily obtain very small spotsizes at the designed field angles while small deviations from the design angles show large spotsizes. (This is sometimes called “overdesigning” the lens.) To prevent this behavior from degrading the lenslet designs, a Cartesian grid of 3×3 field angles is chosen within the angles corresponding to the given lenslet. In practice, the curved multiscale design (Fig. 9(d)) used a subfield of ±1.75° for each lenslet, so that the field angles used to define the merit function were a grid of ±1.25° centered on the lenslet chief ray (see Fig. 8). Additionally, the optimization iterations were truncated early in order to prevent overdesign at those 9 positions.

*m*=-0.9. (This demagnification increases with field angle, reaching

*m*=-0.4 at the highest field angles, due to the inward curving focal surface.) Since this design maps the final image onto a plane, it is compatible for use with either a monolithic detector array or with multiple small arrays fixed behind each lenslet. In both cases the various sub-images will still require post-processing in order to properly register and fuse them, while correcting for varying magnification and distortion as well as vignetting effects.

*N*of resolvable spots for each design is estimated by dividing up the field in 0.25° increments, and a Zemax macro was used to locate the image height corresponding to each angular position. From each of these we form a corresponding annulus on the image plane, whose area divided by the corresponding average spotsize gives a rough estimate of the number of resolvable points. For the planar designs, the area of the diffraction-limited Airy disk is approximately given by

*π*(1.22

*λ*

*f*

_{#})

^{2}/cos

^{4}

*θ*[23

23. M. V. R. K. Murty, “On the theoretical limit of resolution,” J. Opt. Soc. Am. **47**, 667–668 (1957). [CrossRef]

*θ*, whereas for the curved designs cos

^{3}

*θ*appears in the denominator instead, since these designs lack the extra image plane obliquity factor. Fig. 11 shows the result of these calculations at each field angle. Summing the resolvable spots in each annulus, with the combined result summing the geometric and diffraction spotsizes in quadrature, gives the total number of resolvable elements for each system:

megapixels |
Design type
| Geometric | Diff-limit | Combined |
---|---|---|---|---|

conventional, planar | 0.5 | 44.4 | 0.5 | |

conventional, spherical | 8.8 | 40.3 | 8.6 | |

multiscale, locally linear | 8.7 | 39.9 | 8.5 | |

multiscale, planar | 11.4 | 42.3 | 9.6 | |

multiscale, curved | 24.0 | 39.7 | 19.9 |

*γ*=0.45. Note that the diffraction-limited performances of the designs vary due to differences in image size and geometry. The planar multiscale design shows a more erratic performance in spotsize vs. field angle, largely due to the fact that the collector lens chosen is unsuited to the design. A collector lens which is more nearly telecentric in image space can achieve better performance. While the curved multiscale design is able to outperform the conventional planar design by a factor of 40, and the conventional design using a spherical focal surface by a factor of 2, it is a much more complicated optical arrangement. Using a more complex objective lens can of course greatly improve the performance of the conventional design, but the aim of the multiscale approach is to show an alternative means of avoiding the complexity limit in design.

## 4. Conclusion

## Acknowledgments

## References and links

1. | G. T. di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. |

2. | J. Kopf, M. Uyttendaele, O. Deussen, and M. F. Cohen, “Capturing and viewing gigapixel images,” ACM Trans. Graphics |

3. | A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. |

4. | T. Matsuyama, Y. Ohmura, and D. M. Williamson, “The lithographic lens: its history and evolution,” in |

5. | R. Völkel, M. Eisner, and K. J. Weible, “Miniaturized imaging systems,” Microelectron. Eng. |

6. | Y. Dagan, “Wafer-level optics enables low cost camera phones,” in |

7. | J. Tanida, T. Kumagai, K. Yamada, S. Miyatake, K. Ishida, T. Morimoto, N. Kondou, D. Miyazaki, and Y. Ichioka, “Thin observation module by bound optics (TOMBO): concept and experimental verification,” Appl. Opt. |

8. | M. Shankar, R. Willett, N. Pitsianis, T. Schulz, R. Gibbons, R. T. Kolste, J. Carriere, C. Chen, D. Prather, and D. Brady, “Thin infrared imaging systems through multichannel sampling,” Appl. Opt. |

9. | T. Mirani, D. Rajan, M. P. Christensen, S. C. Douglas, and S. L. Wood, “Computational imaging systems: joint design and end-to-end optimality,” Appl. Opt. |

10. | K. Choi and T. J. Schulz, “Signal-processing approaches for image-resolution restoration for TOMBO imagery,” Appl. Opt. |

11. | A. V. Kanaev, D. A. Scribner, J. R. Ackerman, and E. F. Fleet, “Analysis and application of multiframe superresolution processing for conventional imaging systems and lenslet arrays,” Appl. Opt. |

12. | A. D. Portnoy, N. P. Pitsianis, X. Sun, and D. J. Brady, “Multichannel sampling schemes for optical imaging systems,” Appl. Opt. |

13. | R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, “Three-dimensional information acquisition using a compound imaging system,” Opt. Rev. |

14. | E. H. Adelson and J. Y. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intel. |

15. | M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Trans. Graphics |

16. | H.-J. Lee, D.-H. Shin, H. Yoo, J.-J. Lee, and E.-S. Kim, “Computational integral imaging reconstruction scheme of far 3D objects by additional use of an imaging lens,” Opt. Comm. |

17. | J. Duparré, P. Schreiber, A. Matthes, E. Pshenay-Severin, A. Bräuer, A. Tünnermann, R. Völkel, M. Eisner, and T. Scharf, “Microoptical telescope compound eye,” Opt. Express |

18. | J. A. Cox and B. S. Fritz, “Variable focal length micro lens array field curvature corrector,” (2003). US Patent 6556349. |

19. | R. Bacon, P. Y. Copin, G. Monnet, B. W. Miller, J. R. Allington-Smith, M. Bureau, C. M. Carollo, R. L. Davies, E. Emsellem, H. Kuntschner, R. F. Peletier, E. K. Verolme, and P. T. de Zeeuw, “The SAURON project—I. The panoramic integral-field spectrograph,” Monthly Notices of the Royal Astronomical Society |

20. | URL http://www.gmto.org/codrfolder/GMT-ID-01467-Chapter 6 Optics.pdf/. |

21. | |

22. | URL http://www.zemax.com. |

23. | M. V. R. K. Murty, “On the theoretical limit of resolution,” J. Opt. Soc. Am. |

**OCIS Codes**

(100.0100) Image processing : Image processing

(110.0110) Imaging systems : Imaging systems

(220.1000) Optical design and fabrication : Aberration compensation

(220.3620) Optical design and fabrication : Lens system design

(110.1758) Imaging systems : Computational imaging

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: March 20, 2009

Revised Manuscript: June 1, 2009

Manuscript Accepted: June 5, 2009

Published: June 10, 2009

**Citation**

David J. Brady and Nathan Hagen, "Multiscale lens design," Opt. Express **17**, 10659-10674 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-10659

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### References

- G. T. di Francia, "Degrees of freedom of an image," J. Opt. Soc. Am. 59, 799-804 (1969). [CrossRef]
- J. Kopf, M. Uyttendaele, O. Deussen, and M. F. Cohen, "Capturing and viewing gigapixel images," ACM Trans. Graphics 26, 93 (2007). [CrossRef]
- A. W. Lohmann, "Scaling laws for lens systems," Appl. Opt. 28, 4996-4998 (1989). [CrossRef] [PubMed]
- T. Matsuyama, Y. Ohmura, and D. M. Williamson, "The lithographic lens: its history and evolution," in Optical Microlithography XIX, D. G. Flagello, ed., vol. 6154 of Proc. SPIE (2006).
- R. Völkel, M. Eisner, and K. J. Weible, "Miniaturized imaging systems," Microelectron. Eng. 67-68, 461-472 (2003).
- Y. Dagan, "Wafer-level optics enables low cost camera phones," in Integrated Optics: Devices, Materials, and Technologies XIII, J.-E. Broquin and C. M. Greiner, eds., vol. 7218 of Proc. SPIE (2009).
- J. Tanida, T. Kumagai, K. Yamada, S. Miyatake, K. Ishida, T. Morimoto, N. Kondou, D. Miyazaki, and Y. Ichioka, "Thin observation module by bound optics (TOMBO): concept and experimental verification," Appl. Opt. 40, 1806-1813 (2001). [CrossRef]
- M. Shankar, R. Willett, N. Pitsianis, T. Schulz, R. Gibbons, R. T. Kolste, J. Carriere, C. Chen, D. Prather, and D. Brady, "Thin infrared imaging systems through multichannel sampling," Appl. Opt. 47, B1-B10 (2008). [CrossRef] [PubMed]
- T. Mirani, D. Rajan, M. P. Christensen, S. C. Douglas, and S. L. Wood, "Computational imaging systems: joint design and end-to-end optimality," Appl. Opt. 47, B86-B103 (2008). [CrossRef] [PubMed]
- K. Choi and T. J. Schulz, "Signal-processing approaches for image-resolution restoration for TOMBO imagery," Appl. Opt. 47, B104-B116 (2008). [CrossRef] [PubMed]
- A. V. Kanaev, D. A. Scribner, J. R. Ackerman, and E. F. Fleet, "Analysis and application of multiframe superresolution processing for conventional imaging systems and lenslet arrays," Appl. Opt. 46, 4320-4328 (2007). [CrossRef] [PubMed]
- A. D. Portnoy, N. P. Pitsianis, X. Sun, and D. J. Brady, "Multichannel sampling schemes for optical imaging systems," Appl. Opt. 47, B76-B85 (2008). [CrossRef] [PubMed]
- R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, "Three-dimensional information acquisition using a compound imaging system," Opt. Rev. 14, 347-350 (2007). [CrossRef]
- E. H. Adelson and J. Y. Wang, "Single lens stereo with a plenoptic camera," IEEE Trans. Pattern Anal. Mach. Intel. 14, 99-106 (1992). [CrossRef]
- M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, "Light field microscopy," ACM Trans. Graphics 25, 924-934 (2006). [CrossRef]
- H.-J. Lee, D.-H. Shin, H. Yoo, J.-J. Lee, and E.-S. Kim, "Computational integral imaging reconstruction scheme of far 3D objects by additional use of an imaging lens," Opt. Comm. 281, 2026-2032 (2007). [CrossRef]
- J. Duparré, P. Schreiber, A. Matthes, E. Pshenay-Severin, A. Bräuer, A. Tünnermann, R. Völkel, M. Eisner, and T. Scharf, "Microoptical telescope compound eye," Opt. Express 13, 889-903 (2005). [CrossRef] [PubMed]
- J. A. Cox and B. S. Fritz, "Variable focal length micro lens array field curvature corrector," (2003). US Patent 6556349.
- R. Bacon, P. Y. Copin, G. Monnet, B. W. Miller, J. R. Allington-Smith, M. Bureau, C. M. Carollo, R. L. Davies, E. Emsellem, H. Kuntschner, R. F. Peletier, E. K. Verolme, and P. T. de Zeeuw, "The SAURON project—I. The panoramic integral-field spectrograph," Monthly Notices of the Royal Astronomical Society 326, 23-35 (2001). [CrossRef]
- http://www.gmto.org/codrfolder/GMT-ID-01467-Chapter 6 Optics.pdf/.
- http://www2.keck.hawaii.edu/inst/hires/.
- http://www.zemax.com.
- M. V. R. K. Murty, "On the theoretical limit of resolution," J. Opt. Soc. Am. 47, 667-668 (1957). [CrossRef]

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