## On the chromatic aberration of microlenses

Optics Express, Vol. 14, Issue 11, pp. 4687-4694 (2006)

http://dx.doi.org/10.1364/OE.14.004687

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### Abstract

The optical properties of plano-convex refractive microlenses with low Fresnel Number (typically FN <10) are investigated. It turns out that diffraction effects at the lens aperture limit the range of the effective focal length. The upper limit of the focal length is determined by the diffraction pattern of a pinhole with equal diameter. In addition achromatic microlenses can be realized because refraction and diffraction have opposing effects on the focal length. Gaussian beam propagation method has been used for simulation. The presented results are of relevance for applications, where microlenses with small apertures and long focal lengths are used, for example, Shack Hartmann wavefront sensors or confocal microscopes.

© 2006 Optical Society of America

## 1. Introduction

2. Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. **39**, 211 (1981). [CrossRef]

3. Y. Li and H. Platzer, “An experimental investigation of diffraction patterns in low Fresnel-number focusing systems,” Optica Acta **30**, 1621 (1983). [CrossRef]

4. W. Wang, A. T. Friberg, and E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A. **12**, 1947 (1995). [CrossRef]

_{p}) for different microlenses diameters as function of ROC. The focal length is increased by increasing the ROC. However at a certain value for ROC, the focal length cannot be further increased. The limit is fixed by the position of the peak irradiance of the light diffracted by an aperture equal to the lens diameter. Secondly we study the position of the peak irradiance at different wavelengths. We observe two phenomena influencing Z

_{p}. One is the focal shift due to diffraction and the second is the chromatic aberration due to the material dispersion. Because these two phenomena have opposing influence on Z

_{p}, a choice for the ROC can be found where Z

_{p}is the same at two wavelengths. This property could be used to design achromatic lenses in one single material.

## 2. Basic considerations

*f*

_{E}of a plano-convex refractive lens is derived from the radius of curvature ROC and the refractive index n of the lens material

_{2})<n(λ

_{1}) for λ

_{2}>λ

_{1}. The longitudinal primary chromatic aberration, corresponds to a chromatic shift of the focal length,

*f*

_{E}(λ

_{2})>

*f*

_{E}(λ

_{1}) for λ

_{2}>λ

_{1}, [1]. The Fresnel number FN of a lens with a lens diameter

*Ø*=2ρ is defined by

5. J. Arnaud, “Representation of Gaussian beams by complex rays,” Appl. Opt. **24**, (1985). [CrossRef] [PubMed]

## 3. Plano-convex refractive microlens

*Ø*=635 µm, ROC=2.03 mm, made of fused silica with n (633 nm)=1.456 has been chosen. From paraxial geometric optics, the focal length is

*f*

_{E}=4.45 mm. The Fresnel number of the lens is FN=35.7. As shown in Fig. 2, diffraction analysis [5

5. J. Arnaud, “Representation of Gaussian beams by complex rays,” Appl. Opt. **24**, (1985). [CrossRef] [PubMed]

_{p}at 4.36 mm. The difference between the position of the peak irradiance Z

_{p}derived from diffraction theory and the focal length

*f*

_{E}obtained from geometrical optics is defined as focal shift δ [6

6. U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. **157**, 218–224 (1998). [CrossRef]

_{p}=162.93 mm is the maximum obtainable focus spot position for a microlens of

*Ø*=635 µm diameter.

## 4. Refractive and diffractive regime

_{p}function of ROC for different microlens diameters. For low ROC (typically ROC <1 mm for

*Ø*=635 µm microlens) the peak irradiance corresponds to the focal length obtained by Eq. (1). For increasing values of ROC, the peak irradiance converges to a maximum value Z

_{p}

_{max}illustrated by vertical dash lines in Fig. 6. This value corresponds to the peak irradiance position obtained for a pinhole with no optical power, i.e. ROC=∞. To check the validity of the present approach, the values of Z

_{p}

_{max}obtained by the Gaussian beam decomposition algorithm are now compared to the values derived from the Rayleigh-Sommerfeld integral.

**r**,z) at the point P is given by [7],

_{0}=0. For a plane wave diffracted at the circular aperture and propagating along the optical axis we have

*A*

_{0}(r

_{0})=

*A*

_{0}and

*r*

_{0}=|

**r**

_{0}|. Furthermore, for paraxial approximation we assume that cos(

*θ*)≈1. Substituting these relations into Eq. (4) and introducing polar co-ordinates yields to

*ρ*

_{0}is the radius of the limiting aperture at z

_{0}=0. Integrating Eq. (5) and approximating the square root in the phase by the first two terms of the Taylor series yields to

_{p}of pinholes with

*Ø*=2ρ

_{0}diameters obtained from Rayleigh-Sommerfeld, are shown as dashed lines in Fig. 6. The Gaussian beam decomposition algorithm corresponds well to the Rayleigh-Sommerfeld approach.

## 5. Achromaticity

_{p}respectively the focal length

*f*

_{E}is inversely proportional to the refractive index n(λ). The dispersion curve of a material approximated by the Cauchy Formula [1] Eq. (9),

_{p}) is equal to the focal length given by Eq. (1).

*is the difference of the peak irradiance positions for different wavelengths δ*

_{λ}*=Z*

_{λ}_{p}(λ

_{1}) - Z

_{p}(λ

_{2}), expressed along the optical axis [1]. In classical optics, an achromatic lens is designed to have the same focal length for two well separated wavelengths, i.e. δ

*=0.*

_{λ}*Ø*=635 µm, the peak radiance at 248 nm and 1550 nm are equal z

_{p}(248 nm)=z

_{p}(1550 nm)=21.7mm for ROC=10.8 mm, Fig. 7.

_{p}is required. It is now possible to plot the relative variations of this Z

_{p}against the wavelength of a defined spectrum. For example, Fig. 9 shows the variations as function of the wavelength of Z

_{p}expressed in percent of the Z

_{p}calibrated at 550 nm. This is done for different ROC while for each lens Z

_{p}at 550 nm is taken to normalize. The effect of diffraction appears when ROC increases. Correction of chromatic aberrations is possible for curve shapes that show a maximum in the desired wavelengths region. It would seem that small curvature (large radius of curvature) of the plot at the maximum would lead to small chromatic aberration in Fig. 9. Curves for ROC=20.2 mm (FN=4), 12.6 mm (FN=6) and 10.1 mm (FN=8) in Fig. 9 show a maximum. Note, that the FN are calculated at 550 nm. The maximum shifts to shorter wavelengths for increasing ROC. The microlens with a ROC fixed to 10.1 mm shows less than 1 ‰ variation on Z

_{p}from 530 nm to 690 nm. For comparison the microlens with ROC=2 mm is not influenced by diffraction effects and shows more than 8 ‰ variations on Z

_{p}inside the same range. For this wavelengths range we see from Fig. 9 that the lens with ROC=10.1 mm represents an optimum design for a lens of 635 µm diameter. The position of the peak irradiance is 21.5 mm.

## 6. Conclusion

## References

1. | D. Malacara and Z. Malacara, |

2. | Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. |

3. | Y. Li and H. Platzer, “An experimental investigation of diffraction patterns in low Fresnel-number focusing systems,” Optica Acta |

4. | W. Wang, A. T. Friberg, and E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A. |

5. | J. Arnaud, “Representation of Gaussian beams by complex rays,” Appl. Opt. |

6. | U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. |

7. | J. W. Goodman, |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: March 30, 2006

Revised Manuscript: May 9, 2006

Manuscript Accepted: May 10, 2006

Published: May 29, 2006

**Virtual Issues**

Vol. 1, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Patrick Ruffieux, Toralf Scharf, Hans Peter Herzig, Reinhard Völkel, and Kenneth J. Weible, "On the chromatic aberration of microlenses," Opt. Express **14**, 4687-4694 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-11-4687

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### References

- D. Malacara, and Z. Malacara, Handbook of lens design, (Dekker, New York, 1994).
- Y. Li, and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39,211 (1981). [CrossRef]
- Y. Li, and H. Platzer, "An experimental investigation of diffraction patterns in low Fresnel-number focusing systems," Optica Acta 30,1621 (1983). [CrossRef]
- W. Wang, A. T. Friberg, E. Wolf, "Structure of focused fields in systems with large Fresnel numbers," J. Opt. Soc. Am. A. 12,1947 (1995). [CrossRef]
- J. Arnaud, "Representation of Gaussian beams by complex rays," Appl. Opt. 24, (1985). [CrossRef] [PubMed]
- U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, "Unconventional treatment of focal shift," Opt. Commun. 157,218-224 (1998). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, 2nd ed., (MacGraw-Hill, New York, 1968), Chap. 4, pp. 63-69.

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