## Dynamic compensation of chromatic aberration in a programmable diffractive lens

Optics Express, Vol. 14, Issue 20, pp. 9103-9112 (2006)

http://dx.doi.org/10.1364/OE.14.009103

Acrobat PDF (219 KB)

### Abstract

A proposal to dynamically compensate chromatic aberration of a programmable phase Fresnel lens displayed on a liquid crystal device and working under broadband illumination is presented. It is based on time multiplexing a set of lenses, designed with a common focal length for different wavelengths, and a tunable spectral filter that makes each sublens work almost monochromatically. Both the tunable filter and the sublens displayed by the spatial light modulator are synchronized. The whole set of sublenses are displayed within the integration time of the sensor. As a result the central order focalization has a unique location at the focal plane and it is common for all selected wavelengths. Transversal chromatic aberration of the polychromatic point spread function is reduced by properly adjusting the pupil size of each sublens. Longitudinal chromatic aberration is compensated by making depth of focus curves coincident for the selected wavelengths. Experimental results are in very good agreement with theory.

© 2006 Optical Society of America

## 1. Introduction

*f*(

*λ*)=(

*λ*

_{0}/

*λ*)

*f*

_{0}, where

*λ*is the illumination wavelength and

*f*

_{0}is the focal length for the design wavelength

*λ*

_{0}. Some solutions, based on hybrid diffractive-refractive configurations consisting of a number of properly separated lenses, have been proposed to obtain dispersion-compensated imaging systems and correlators [1, 2

2. P. Andrés, V. Climent, J. Lancis, G. Mínguez-Vega, E. Tajahuerce, and A. W. Lohmann, “All-incoherent dispersion- compensated optical correlator,” Opt. Lett. **24**, 1331–1333 (1999). [CrossRef]

3. V. Laude, “Twisted-nematic liquid-crystal pixilated active lens,” Opt. Commun. **153**, 134–152 (1998). [CrossRef]

5. M. S. Millán, J. Otón, and E. Pérez-Cabré, “Chromatic compensation of programmable Fresnel lenses,” Opt. Express **14**, 6226–6242 (2006) [CrossRef] [PubMed]

## 2. Design of the set of sublenses

*l*, placed at the plane of rectangular coordinates (

*x*,

*y*), with a quadratic phase function given by

*f*

_{0}is the focal length of design for the wavelength

*λ*. Let us consider that this lens function is sampled with a sampling period given by the pixel space (or pixel pitch). The lens is displayed on a

*M*×

*M*pixel array SLM, with square pixel pitch Δ and fill factor less than unity. For the sake of simplicity, the active area of a pixel is represented by a rectangle of dimensionsΔ

*x*’,Δ

*y*’, that is, by rect

*x*’<Δ andΔ

*y*’<Δ. Under broadband illumination a quasimonochromatic filter selects a narrow bandwidth centred in

*λ*. The lens of Eq. (1) has a circular aperture of radius

*R*and is only active for a time

*T*. Taking into account all these circumstances, the sublens

*l*

_{i}(

*x*,

*y*,

*t*) can be expressed by

*t*

_{i}, has been considered to limit the time

*T*

_{i}to which the sublens is active;

*τ*

_{i}(Δ

*λ*) is the amplitude transmittance of the quasimonochromatic filter that selects a narrow bandwidth around

*λ*

_{i}(consequently,

*τ*

_{i}(Δ

*λ*)≈0 except for Δ

*λ*=|

*λ*-

*λ*

_{i}|≈0); the circ function corresponds to a circular pupil of radius

*R*

_{i}; the summation corresponds to a 2D-comb function that establishes the positions of the sampling points and the convolution (⊗) by the active area of the pixel ensures that there is a uniform single value throughout its extension. We assume that the sublens pattern reaches, at most, the Nyquist frequency at the circular contour of the aperture and, consequently, no secondary lenses appear. This implies that the focal length

*f*

_{0}has to be longer than or equal to the reference focal

*f*

_{r}[7

7. D. M. Cottrell, J. A. Davis, T. R. Hedman, and R. A. Lilly, “Multiple imaging phase-encoded optical elements, written as programmable spatial light modulators,” Appl. Opt. **29**, 2505–2509 (1990). [CrossRef] [PubMed]

*M*) and the wavelength

*λ*

_{i},

*R*

_{i}≤

*R*=

*M*Δ/2, where

*R*is the circular aperture of maximum radius placed against the SLM screen. In particular, we demonstrated in a previous work [5

5. M. S. Millán, J. Otón, and E. Pérez-Cabré, “Chromatic compensation of programmable Fresnel lenses,” Opt. Express **14**, 6226–6242 (2006) [CrossRef] [PubMed]

*τ*

_{i}(Δ

*λ*)

*λ*

_{i}=constant(Fig. 1), which represents the condition to obtain PSF profiles of the same maximum height for all the wavelengths

*λ*

_{i}. The condition was referred to as the PSF

_{I}-condition.

*L*(

*x*,

*y*,

*t*)displayed in the SLM can be described by

*ε*

_{i}represents the time necessary to refresh both the spectral filter and the lens displayed on the SLM from sublens

*l*

_{i}to

*l*

_{i}

_{+1}. The sublenses are successively displayed, thus satisfying the following relationships between the temporal constants:

*T*

_{S}in Eq. (5), lasts a time given by

*T*

_{S}=

_{=1}(

*T*

_{i}+

*ε*

_{i}). Although the refreshing time

*ε*

_{i}could be dependent on the sublenses, we will assume that it is approximately constant (

*ε*) for all the sublenses of the set. Since the whole set of sublenses has to be displayed during the integration time of the sensor

*T*

_{0}, the condition

*T*

_{0}≥

*T*

_{S}=

*Nε*+

_{=1}

*T*

_{i}has to be satisfied. In such a case,

*L*(

*x*,

*y*,

*t*)periodically repeats identical sequences of

*N*sublenses designed to have a unique focal plane for all the selected wavelengths. In this way, the strong chromatic aberration of a diffractive lens can be compensated. Thus, in the instant

*T*

_{i}, the phase Fresnel lens function of Eq. (1) is displayed on the SLM while a uniform plane wave of

*λ*

_{i}impinges the aperture with radius

*R*

_{i}. We calculate the Fresnel propagation of the amplitude distribution from the plane behind the lens to the focal plane following a procedure similar to that applied in Ref. [5

5. M. S. Millán, J. Otón, and E. Pérez-Cabré, “Chromatic compensation of programmable Fresnel lenses,” Opt. Express **14**, 6226–6242 (2006) [CrossRef] [PubMed]

*U*

_{00i}is

*d*

_{i}=

*R*

_{i}/

*λ*

_{i}

*f*

_{0}. Equation (6) is the convolution of a wavelength dependent term by a wavelength independent rectangle function. The variation of the central lobe of

*U00*

_{i}with

*λ*can therefore be analyzed through the variation of the first term of Eq. (6) with

*λ*. The width of the central lobe of the Bessel

*J*

_{1}function in Eq. (6) is 1.22

*λ*

_{i}

*f*

_{0}/

*R*

_{i}and its height is weighted by the precedent factor. Although the focal plane is the same for all

*λ*

_{i}, the central lobe of

*U*

_{00i}shows different sizes and heights. As it was aforementioned, the PSF

_{I}-condition (

*τ*

_{i}(Δ

*λ*)

*λ*

_{i}=constant) represents the condition to obtain PSF profiles of the same maximum height for all the wavelengths

*λ*

_{i}. We demonstrated in Ref. [5

**14**, 6226–6242 (2006) [CrossRef] [PubMed]

**14**, 6226–6242 (2006) [CrossRef] [PubMed]

*R*

_{i}), the time multiplexing scheme keeps two complementary advantages of the mosaic and rotating multisector apertures [5

**14**, 6226–6242 (2006) [CrossRef] [PubMed]

*Nε*≪

_{=1}

*T*

_{i}. Otherwise, the result would be a rather noisy signal. Secondly, to ensure a white focal spot, the possibility of fine tuning the chromaticity of the focal spot is needed. This must be optimized for color imaging tasks such as white balance and analysis of the color content conveyed by the system. The problem is not trivial because it depends at least on the following factors: the wavelength sampling, the efficiency of the spectral filter and the bandwidth for each selected wavelength, the efficiency and the phase and amplitude modulation configuration of the liquid crystal display for each selected wavelength, the spectral power distribution of the white light source and the spectral sensibility of the sensor. In fact, Márquez et al. already dealt with the problem in Ref. [4

4. A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Achromatic diffractive lens written onto a liquid cristal display,” Opt. Lett. **31**, 392–394 (2006). [CrossRef] [PubMed]

*T*

_{i}, for which the sublens

*l*

_{i}is displayed, as another degree of freedom and adjust it to be its weight within the sequence of

*N*sublenses.

*T*

_{i}is the time that the rotating filter sector with transmittance in

*λ*

_{i}is covering the whole aperture of the liquid crystal display (Fig. 2). For sublenses of different weights, their corresponding sectors will accordingly have different angular amplitudes in the design of the multisector spectral filter.

_{R}=450nm, λ

_{G}=550nm, λ

_{R}=650nm). The integration time of the sensor is

*T*

_{0}=1/20=0.05s that is nine times as much as the time corresponding to the maximum frame rate of the SLM. At this maximum rate, the SLM needs

*ε*=(1 180)s to refresh the image. If we take the integration time

*T*

_{0}equal to the time of the sublens sequence

*T*

_{S}and that each sublens can be displayed with the same weight, it results

*T*

_{i}=(2/180)s. The ratio (

*T*

_{i}/

*ε*)=2 should predict an acceptable signal with relative low noise.

## 3. Experimental results

*f*

_{0}=1250mm. This value is the longest reference focal that corresponds to the short extreme of visible spectrum (400nm). We displayed sublenses corresponding to λ=500, 550, 600, 650 nm. We made another assumption for the depths of the phase modulations for these wavelengths [Fig. 3(b)]: they were linear extensions of the plot obtained for λ=543nm. This assumption was not acceptable for

*λ*≤480nm, for which the behaviour of the SLM showed some significant deviations.

9. R. D. Juday, “Optical realizable filters and the minimum Euclidean distance principle,” Appl. Opt. **32**, 5100–5111 (1993). [CrossRef] [PubMed]

10. I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. **34**, 6423–6432 (1995). [CrossRef]

*R*and the same focal plane as before but only for the wavelength of λ=550nm. We kept this lens constant for the successive illumination with the set of four wavelengths. Fig. 4(b) shows the cross section curves of the experimental PSF obtained in this case in the focal plane (it is only a focal plane of design for λ=550nm). As we can see in this figure, only the light of wavelength λ=550nm focuses in this plane. The rest of wavelengths contributed with very blurred and low intense figures due to the presence of the strong and non-compensated chromatic aberration. Figures 4(c) and 4(d) show 2D-pseudocolored images of the focal spots corresponding to the respective figures above [Figs. 4(a) and 4(b)]. They have been obtained by representing the four PSF images in the R, G, B channels according to the following proportions(

*R*,

*G*,

*B*)=(PSF

_{500nm}, PSF

_{550nm}, 0.2PSF

_{600nm}+0.8PSF

_{650nm}) and merging them for visualization. In the case of Fig. 4(c), a spot close to white is obtained. The relative weights (1, 1, 0.2, 0.8) can be maintained in first approximation to compute the lifetimes

*T*

_{i}of the sublenses in the total sequence time

*T*

_{S}. This numbers must be considered just as an illustrative example of the feasibility of the procedure. Depending on the set of selected wavelengths, many different combinations of the relative weights can be found and they should be more finely tuned to obtain a white spot. Finally, Figs. 4(e) and 4(f) show the distribution of intensity along the optical axis for each wavelength. In Fig. 4(e), the time multiplexed programmable lens provides a nearly common depth of focus for the four wavelengths. Thus, it reaches a high compensation of longitudinal chromatic aberration. In contrast, the lens programmed just for the wavelength λ=550nm with maximum aperture

*R*and no compensation by time multiplexing, obtains a distribution of intensity along the axis strongly variant with wavelength, as it was aforementioned, and according to the expression

*f*(

*λ*)=(

*λ*

_{0}/

*λ*)

*f*

_{0}. Results of Figs. 4(b), 4(d), and 4(f) are consistent with the high chromatic aberration, typical of diffractive lenses, that motivated this work.

## 4. Conclusions

## Acknowledgments

## References and links

1. | D. Faklis and G. M. Morris, “Broadband imaging with holographic lenses,” Opt. Eng. |

2. | P. Andrés, V. Climent, J. Lancis, G. Mínguez-Vega, E. Tajahuerce, and A. W. Lohmann, “All-incoherent dispersion- compensated optical correlator,” Opt. Lett. |

3. | V. Laude, “Twisted-nematic liquid-crystal pixilated active lens,” Opt. Commun. |

4. | A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Achromatic diffractive lens written onto a liquid cristal display,” Opt. Lett. |

5. | M. S. Millán, J. Otón, and E. Pérez-Cabré, “Chromatic compensation of programmable Fresnel lenses,” Opt. Express |

6. | J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, “Apodizing filters in colour imaging,” J. Optics (Paris) |

7. | D. M. Cottrell, J. A. Davis, T. R. Hedman, and R. A. Lilly, “Multiple imaging phase-encoded optical elements, written as programmable spatial light modulators,” Appl. Opt. |

8. | |

9. | R. D. Juday, “Optical realizable filters and the minimum Euclidean distance principle,” Appl. Opt. |

10. | I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. |

**OCIS Codes**

(090.1000) Holography : Aberration compensation

(090.1970) Holography : Diffractive optics

(100.1160) Image processing : Analog optical image processing

(230.3720) Optical devices : Liquid-crystal devices

(230.6120) Optical devices : Spatial light modulators

**ToC Category:**

Holography

**History**

Original Manuscript: July 21, 2006

Manuscript Accepted: September 8, 2006

Published: October 2, 2006

**Citation**

María S. Millán, Joaquín Otón, and Elisabet Pérez-Cabré, "Dynamic compensation of chromatic aberration in a programmable diffractive lens," Opt. Express **14**, 9103-9112 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9103

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### References

- D. Faklis, G. M. Morris, "Broadband imaging with holographic lenses," Opt. Eng. 28, 592-598 (1989).
- P. Andrés, V. Climent, J. Lancis, G. Mínguez-Vega, E. Tajahuerce, and A. W. Lohmann, "All-incoherent dispersion- compensated optical correlator," Opt. Lett. 24, 1331-1333 (1999). [CrossRef]
- V. Laude, "Twisted-nematic liquid-crystal pixilated active lens," Opt. Commun. 153, 134-152 (1998). [CrossRef]
- A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, "Achromatic diffractive lens written onto a liquid cristal display," Opt. Lett. 31, 392-394 (2006). [CrossRef] [PubMed]
- M. S. Millán, J. Otón, E. Pérez-Cabré, "Chromatic compensation of programmable Fresnel lenses," Opt. Express 14, 6226-6242 (2006) [CrossRef] [PubMed]
- J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, "Apodizing filters in colour imaging," J. Optics (Paris) 17, 91-96 (1986). [CrossRef]
- D. M. Cottrell, J. A. Davis, T. R. Hedman, and R. A. Lilly, "Multiple imaging phase-encoded optical elements, written as programmable spatial light modulators," Appl. Opt. 29, 2505-2509 (1990). [CrossRef] [PubMed]
- http://www.cri-inc.com">http://www.cri-inc.com.
- R. D. Juday, "Optical realizable filters and the minimum Euclidean distance principle," Appl. Opt. 32, 5100-5111 (1993). [CrossRef] [PubMed]
- I. Moreno, J. Campos, C. Gorecki, M. J. Yzuel, "Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition," Jpn. J. Appl. Phys. 34, 6423-6432 (1995). [CrossRef]

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