## Miniaturization of zoom lenses with a single moving element

Optics Express, Vol. 17, Issue 8, pp. 6118-6127 (2009)

http://dx.doi.org/10.1364/OE.17.006118

Acrobat PDF (584 KB)

### Abstract

We present an analysis of single-moving-element zoom lenses in the thin-lens limit and show how the length of these zoom lenses is determined by the zoom-factor, sensor-dimension and the depth-of-focus. By decreasing the sensor size and extending the depth-of-focus, the lengths of these zoom lenses can be reduced significantly. As an example we present a ray-traced design of a miniaturized single-moving-element zoom lens with a 2.3× zoom-factor and show how the exploitation of modern miniaturized detector array combined with wavefront coding enables a reduction in length of almost three orders-of-magnitude to 10mm.

© 2009 Optical Society of America

## 1. Introduction

*variator*, which is responsible for “zooming”, introduces (A) defocus, and (B) variations in optical aberrations [1]. To solve problem A, most modern zoom lenses employ so-called mechanical compensation in which at least one extra moving element, a precisely controlled lens called the

*compensator*, is introduced to compensate defocus introduced by the variator [2]. To solve problem B, most modern zoom lenses are designed with additional lenses which compensate for the variations in optical aberrations introduced by zooming. The complexity introduced by these traditional design approaches tends to prevent miniaturization. In this paper we show how combining extended-depth-of-focus (EDOF) techniques, such as wavefront coding (WCF) [3

3. J. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. **34**, 1859–1866 (1995). [CrossRef] [PubMed]

4. S. Mezouari, G. Muyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A **23**, 1058–1062 (2006). [CrossRef]

14. K. Kubala, E. Dowski, and W. Cathey, “Reducing complexity in computational imaging systems,” Opt. Express **11**, 2102–2108 (2003). [CrossRef] [PubMed]

4. S. Mezouari, G. Muyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A **23**, 1058–1062 (2006). [CrossRef]

3. J. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. **34**, 1859–1866 (1995). [CrossRef] [PubMed]

## 2. Analysis of SME zoom lenses

*ϕ*=

_{AB}*ϕ*+

_{A}*ϕ*-

_{B}*dϕ*, where

_{A}ϕ_{B}*ϕ*and

_{A}*ϕ*are the powers of the two lens elements and

_{B}*d*is the displacement between lenses A and B [2]. The back-focal-length (BFL) of the combined lens system is [2]

*Z*=

*f*

_{max}/

*f*

_{min}. We define

*M*as the ratio of the effective focal lengths of the zoom lens when lenses A and B are in contact to when lenses B and C are in contact. The powers of lenses A and B may then be written in the forms [1,2]:

*R*= √

*M*and

*S*is the distance between lenses A and C.

*M*=

*Z*and the -+ configuration has

*M*= 1/

*Z*. Defocus formed by the combination of lenses A, B and C, depends on the lens displacement

*d*and the lens powers. The focal length of the combined lenses A and B is

*f*=

_{AB}*f*/(

_{A}f_{B}*f*+

_{A}*f*-

_{B}*d*) [2], from which the focal length of a combined system of the 3 lenses, A, B and C, can be found using a re-occurrence procedure. Denoting the focal length of the combined 3-lens system at the widest field-of-view as

*f*

_{min}, the power

*ϕ*can be shown, using (2), to be:

_{C}*S*and lens displacement

*d*[1]. The transfer-matrix of the zoom lens is:

*y*and

*y*′ are the light ray-height and

*ϑ*and

*ϑ*′ are light ray-angles. Focusing at infinity (

*ϑ*= 0, ∀

*y*) and solving for

*y*’=0 yields the non-trivial solution for

*y*≠ 0:

*φ*= 0 and

_{c}*S*=

*d*. By insertion of (2) and (3) into (5), the BFL at

*d*= 0 and

*d*=

*S*is:

*d*=0 and

*d*=

*S*. The maximum defocus from the image plane when focusing at infinity for the two-lens configurations is then:

*F*/#) of the zoom lens is constant during zooming. Equation (7) shows that the defocus is approximately inversely proportional to the lens separation

*S*and highlights the fundamental problem in miniaturizing a SME zoom lens: the overall length of the lens can be reduced only by reducing

*S*and at a cost of increased defocus, Δ

*z*. The total length of the two zoom lens configurations is

*L*=

*S*+

*BFL*± Δ

*z*where the ± is positive for the +-+ configuration and negative for the -+-, configuration according to the defocus indicated in Fig. 1. Since Δ

*z*is much smaller than

*S*and BFL,

*L*≈

*S*+

*BFL*and using (7), in the thin-lens approximation,

*x*is the horizontal sensor size,

_{sensor}*θ*is the full-angle field-of-view and

*W*

_{20}=

*Δz*/(8(

*F*/#)

^{2}) is the defocus parameter. According to the Hopkins criterion, acceptable image quality requires that the defocus parameter

*W*

_{20}is less than

*λ/6*and in this case the second term in the major brackets in (8) dominates and so for a system with acceptable defocus we can write

*Z*=2.5,

*θ*= 65° at wide field and

*F*/#=2.8, the minimum lengths of the +-+ and -+- configurations are 64 mm and 154 mm respectively. This assumes that, in accordance with Hopkin’s criterion,

*W*

_{20}=λ/6 at 550 nm. The variation of minimum lengths with zoom factor for these systems is shown in Fig. 2(a). This miniaturization of SME zoom lenses, enabled by reduced sensor size alone, gives lengths which are still impractically long for many consumer applications, such as integration into mobile telephones. It can be observed from Eq. (9) that lens length,

*L*, can be further reduced by allowing an increase in

*W*

_{20}and fortunately several techniques exist to achieve EDOF whilst retaining acceptable image quality [3–12

3. J. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. **34**, 1859–1866 (1995). [CrossRef] [PubMed]

4. S. Mezouari, G. Muyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A **23**, 1058–1062 (2006). [CrossRef]

*quid pro quo*for the increased DOF. The reduction in SNR is described as a noise-gain given by [4

**23**, 1058–1062 (2006). [CrossRef]

*F*is the inverse filter in the frequency domain and

*m*,

*n*are the image dimensions. Although some enhancement in DOF can be attained using any of several reported antisymmetric or symmetric masks, the optimal trade of EDOF against suppression of MTF is obtained using a mask with antisymmetry [16

16. G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. **30**, 2715–2717 (2005). [CrossRef] [PubMed]

**34**, 1859–1866 (1995). [CrossRef] [PubMed]

11. S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization or extended focus, aberration corrected imaging systems,” Proc. SPIE **5559**, 335–345 (2004). [CrossRef]

**34**, 1859–1866 (1995). [CrossRef] [PubMed]

*z*(

*x*,

*y*) =

*α*(

*x*

^{3}+

*y*

^{3}) where 2

*α*is the peak-to-peak optical-path-difference introduced and (

*x*,

*y*) are normalized coordinates of the aperture stop. With a simple inverse filter, where the overall system MTF is recovered to that of a near-diffraction-limited system,

*F*(

*u*,

*v*) is approximately inversely proportional to

*α*and the maximum tolerable defocus is ∣

*W*

_{20}∣=3

*α*(1-

*v*) [16

16. G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. **30**, 2715–2717 (2005). [CrossRef] [PubMed]

*v*is the normalized spatial sampling frequency. The noise-gain is then approximately proportional to the maximum defocus for which image recovery is possible.

*W*

_{20}for Z=2.5 and

*x*=3.58 mm as given by Eq. (8). For example reducing the maximum length of the zoom lenses; from the lengths of 64mm and 115mm required to fulfill the Hopkins criteria; to a length of 10mm that is compatible with use in mobile phone cameras results in a maximum defocus of 1.76λ and 4.21λ (λ=550 nm) for the +-+ and -+-configurations respectively. This represents a three orders-orders-of-magnitude reduction compared to the impractical 10m required for a SME zoom lens based on traditional technologies: two orders-of-magnitude reduction due to the use of modern small detector arrays and an additional order-of-magnitude due to the use of WFC to increase DOF.

_{sensor}*quid pro quo*of degradation in the SNR of the recovered image. From the presented examples, it can be seen that the +-+ zoom lens introduces less defocus and hence enables lower levels of noise degradation than the -+-zoom lens configuration.

## 3. Design of miniaturized SME zoom lens

### 3.1 Location of the aperture stop and aberration control

*θ*= 65°; the minimum focal length is

*f*=2.813mm and the travel of the moving element is restricted to 2.5 mm. These conditions dictate the focal lengths of the three groups in the zoom lens via Eqs. (2)–(3). The values for the focal lengths of the lens groups are given in Table 1 below.

_{min}### 3.2 Example of a miniaturized SME zoom lens with WFC

*W*

_{20}=-2.36λ(λ=550nm) at lens position 0.9mm and

*W*

_{20}=2.36λ at lens position 2.5mm (wide field of-view) and

*W*

_{20}=0.76λ at lens position 0.0mm (narrow field of view). In this article we report the mitigation of this defocus by use of cubic phase modulation at the aperture stop, although other EDOF techniques may also be used.

*W*

_{20}∣=3

*α*(1-

*v*) [16

_{N}16. G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. **30**, 2715–2717 (2005). [CrossRef] [PubMed]

*v*is the normalized Nyquist spatial-frequency. This corresponds to a peak-to-peak surface relief of 3.78 microns at 550 nm. The sampled on-axis PSFs (1.75 micron pixel size) and on-axis MTFs (up to Nyquist frequency 142 cycles/mm) with and without the phase mask are shown in Fig. 4 for zoom positions corresponding to focal lengths of 6.6mm, 4.5mm and 2.9mm. The MTFs for a conventional mechanically compensated zoom lens with two moving elements are also included in the MTF plots for comparison. Mechanical compensation involves movement of the third lens group to retain sharp focus.

_{N}*quid quo pro*however is a modest decrease in the SNR of the final image.

## 4. Conclusions

## Acknowledgment

## References and links

1. | K. Yamaji, “Design of zoom lenses,” in |

2. | W. J. Smith, |

3. | J. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. |

4. | S. Mezouari, G. Muyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A |

5. | S. Mezouari and A. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. |

6. | S. Mezouari, G. Muyo, and A. R. Harvey, “Amplitude and phase filters for mitigation of defocus and third-order aberrations,” in |

7. | W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. |

8. | D. Zalvidea and E. E. Sicre, “Phase pupil functions for focal-depth enhancement derived from a Wigner distribution function,” Appl. Opt. Vol. |

9. | E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Radial mask for imaging systems that exhibit high resolution and extended depths of field,” Appl. Opt. |

10. | Z. Zalevsky, A. Shemer, A. Zlotnik, E. B. Eliezer, and E. Marom, “All-optical axial super resolving imaging using a low-frequency binary-phase mask,” Opt. Express |

11. | S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization or extended focus, aberration corrected imaging systems,” Proc. SPIE |

12. | J. Ares García, S. Bará, M. Gomez García, Z. Jaroszewicz, A. Kolodziejczyk, and K. Petelczyc, “Imaging with extended focal depth by means of the refractive light sword optical element,” Opt. Express |

13. | I. A. Prischepa and J. E. R. Dowski, “Wavefront coded zoom lens system,” in |

14. | K. Kubala, E. Dowski, and W. Cathey, “Reducing complexity in computational imaging systems,” Opt. Express |

15. | G. Muyo, A. Singh, M. Andersson, D. Huckridge, and A. Harvey, “Optimized thermal imaging with a singlet and pupil plane encoding: experimental realization,” in |

16. | G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. |

**OCIS Codes**

(080.2740) Geometric optics : Geometric optical design

(080.3620) Geometric optics : Lens system design

**History**

Original Manuscript: November 3, 2008

Revised Manuscript: February 24, 2009

Manuscript Accepted: March 20, 2009

Published: April 1, 2009

**Citation**

Mads Demenikov, Ewan Findlay, and Andrew R. Harvey, "Miniaturization of zoom lenses with a single moving element," Opt. Express **17**, 6118-6127 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-8-6118

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### References

- K. Yamaji, "Design of zoom lenses," in Progress in Optics, E. Wolf, ed., (North Holland, Amsterdam, 1967), pp. 105-170.
- W. J. Smith, Modern Optical Engineering, Third edition, (McGraw-Hill, 2000).
- J. E. R. Dowski and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995). [CrossRef] [PubMed]
- S. Mezouari, G. Muyo, and A. R. Harvey, "Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism," J. Opt. Soc. Am. A 23, 1058-1062 (2006). [CrossRef]
- S. Mezouari and A. Harvey, "Phase pupil functions for reduction of defocus and spherical aberrations," Opt. Lett. 28, 771-773 (2003). [CrossRef] [PubMed]
- S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," in Optical Design and Engineering, 238-248 (SPIE, St. Etienne, France, 2004).
- W. Chi and N. George, "Electronic imaging using a logarithmic asphere," Opt. Lett. 26, 875-877 (2001). [CrossRef]
- D. Zalvidea and E. E. Sicre, "Phase pupil functions for focal-depth enhancement derived from a Wigner distribution function," Appl. Opt. 37, 3623-3627 (1998) [CrossRef]
- E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, "Radial mask for imaging systems that exhibit high resolution and extended depths of field," Appl. Opt. 45, 2001-2013 (2006). [CrossRef] [PubMed]
- Z. Zalevsky, A. Shemer, A. Zlotnik, E. B. Eliezer, and E. Marom, "All-optical axial super resolving imaging using a low-frequency binary-phase mask," Opt. Express 14, 2631-2643 (2006). [CrossRef] [PubMed]
- S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, "Pupil-phase optimization or extended focus, aberration corrected imaging systems," Proc. SPIE 5559, 335-345 (2004). [CrossRef]
- J. Ares García, S. Bará, M. Gomez García, Z. Jaroszewicz, A. Kolodziejczyk, and K. Petelczyc, "Imaging with extended focal depth by means of the refractive light sword optical element," Opt. Express 16, 18371-18378 (2008). [CrossRef] [PubMed]
- I. A. Prischepa and J. E. R. Dowski, "Wavefront coded zoom lens system," in Zoom Lenses III 83-93, (SPIE, San Diego, CA, USA, 2001).
- K. Kubala, E. Dowski, and W. Cathey, "Reducing complexity in computational imaging systems," Opt. Express 11, 2102-2108 (2003). [CrossRef] [PubMed]
- G. Muyo, A. Singh, M. Andersson, D. Huckridge, and A. Harvey, "Optimized thermal imaging with a singlet and pupil plane encoding: experimental realization," in Electro-Optical and Infrared Systems: Technology and Applications III, 63950-63959 (SPIE, Stockholm, Sweden, 2006).
- G. Muyo and A. R. Harvey, "Decomposition of the optical transfer function: wavefront coding imaging systems," Opt. Lett. 30, 2715-2717 (2005). [CrossRef] [PubMed]

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