Optical system having a large focal depth for distant object tracking

Xinping Liu; Xianyang Cai; Shoude Chang; Chander P. Grover

doi:10.1364/OE.11.003242

1. Introduction

In an imaging system used to track distant objects, e.g., a star tracker, the image of the distant object is focused within one pixel of a CCD camera. For a highly accurate tracking system, the image plane is commonly defocused slightly from the best focal plane, spreading the spot image over several pixels [1–6]. Thus, it is possible to determine the position of the distant object by calculating the centre of the image (the centroid) and interpolating to a small fraction of one pixel. On the other hand, it is not desirable to smear the object image on too large an area, because this will decrease the SNR. The required spot size and algorithms for calculating the centroid of image an mass have been presented for spacecraft attitude determination [1–3], but the allowed defocus depth is usually very short for a conventional imaging system, due to the small F-number. The tracking accuracy will be affected unexpectedly by large defocus caused by a temperature variation or a vibration. Increasing the focal depth of an imaging system will make the tracking system more resistant to environmental change.

In this paper, we analyze the performance of a perfect lens comprising a quartic phase plate which has an effective focal length of 29 mm, an F-number of 1.6 and a primary wavelength of 0.62 µm, and we find a suitable quartic phase plate for a lens used in a distant object tracking system having a field of view of 20 degrees. We analyze the performance of the lens system with and without the quartic phase plate, and show that the focal depth of the tracking system is extended more than threefold after the quartic phase plate is inserted.

2. Theoretical analysis

The spot size and its energy distribution in the image plane are an important aspect of an optical system used as a star tracker [1–6]. For a diffraction-limited optical system, the diameter of the Airy disk is given by:

d_{0} = 2.44 λ F .

If F=1.6 and λ=0.62 µm, the value of d₀ is about 2.4 µm. Since the pixel size of currently available CCD area sensors is in the order of 10 µm, the value of d₀ is only about 1/4 pixel. This means that the centroid technique of calculating star positions cannot be applied to diffraction-limited images. In fact, a spot size in the order of 3–5 pixels in the focal plane would require an F-number of 20~33, but this would definitely compromise light gathering, and affect the performance of the tracking system. Another method of increasing image size is to slightly defocus the image plane so that the spot image covers the required number of pixels. The defocus value can be calculated using the following equation:

z = DF,

where D is the spot size required for tracking purposes. For an aberration-free optical system having an F-number of 1.6 and a required spot size range of 30~50 µm, the value of the defocus range is 50~80 µm. The defocusing depth would be about 30 µm. For a real lens system involving aberrations, the defocusing depth would be smaller than this number. Inserting a quartic phase plate in the pupil plane of the imaging system is an attempt to have the spot image meet the requirements of distant object tracking in an extended focal depth range. In the following analysis, we discuss the method of selecting a suitable phase plate to enhance the focal depth of the optical system used to track distant objects.

A quartic phase pupil function derived from a Wigner distribution function can be described by [7]

ϕ (ρ) = - π \cdot α \cdot (\frac{ρ^{4}}{{ρ_{0}}^{4}} - \frac{ρ^{2}}{{ρ_{0}}^{2}}),

where the value of α is related to the desired focal depth and a constant term is omitted. The intensity distribution of the PSF for a lens having this phase pupil function is written as follows:

I (r, z) = {∣ 2 π \int_{0}^{ρ_{0}} J_{0} (\frac{2 π}{λ_{0}} \cdot \frac{r}{f_{0}} \cdot ρ) • \exp [- i \cdot π \cdot α \cdot (\frac{ρ^{4}}{{ρ_{0}}^{4}} - \frac{ρ^{2}}{{ρ_{0}}^{2}})] • \exp (i \frac{π}{λ_{0} \cdot {f_{0}}^{2}} \cdot z \cdot ρ^{2}) • ρ • d ρ ∣}^{2} .

Using the above equation, we can describe the on-axis intensity by

I (0, z) = {∣ 2 π \int_{0}^{ρ_{0}} \exp [- i \cdot π \cdot α \cdot (\frac{ρ^{4}}{{ρ_{0}}^{4}} - \frac{ρ^{2}}{{ρ_{0}}^{2}})] • \exp (i \frac{π}{λ_{0} \cdot {f_{0}}^{2}} \cdot z \cdot ρ^{2}) • ρ • d ρ ∣}^{2} .

We apply the above equation to an aberration-free system having an effective focal length of 29 mm, an F-number of 1.6 and, a primary wavelength of 0.62µm. Figure 1 shows the variations in axial intensity associated with defocusing from the focal plane. The result indicates that the focal depth is extended when the value of α increases.

Fig. 1. On-axis intensity for different values of z.

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For tracking applications, we wish to find a suitable value of α for the phase plate so that the spot image size containing 80% energy is within 30–50 µm in the extended focal depth. Using this criterion, we calculated the enclosed energy of a circle of radius c in the focal plane by means of the following equation:

RED (c) = \frac{\int_{0}^{c} 2 π \cdot I (r) \cdot r \cdot d r}{\int_{0}^{\infty} 2 π \cdot I (r) \cdot r \cdot d r} .

Fig. 2. Fraction of enclosed energy for a diameter of 30 µm and different values of α.

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Figure 2 gives the calculated results for the enclosed energy RED(c), where c is specified as 15 µm. It can be seen that the fraction of enclosed energy decreases as the value of α increases, and that the spot size containing 80% energy is 30 µm in the focal plane when the value of α is ±28. Therefore, the phase-retardation function of the most suitable quartic phase plate for the specified optical system is as follows:

ϕ (ρ) = \pm π \cdot 28 \cdot (\frac{ρ^{4}}{{ρ_{0}}^{4}} - \frac{ρ^{2}}{{ρ_{0}}^{2}}) .

3. Optical system comprising a quartic phase plate

Fig. 3. Configuration of an optical system comprising no phase plate.

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Fig. 4. Spot size containing 80% energy in an optical system comprising no quartic phase plate for different defocus values.

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We have discussed the effects of a quartic phase plate on an aberration-free optical system as a special requirement for distant object tracking. In the following section, we apply the plate to a real lens system that can be used for distant object tracking. Figure 3 shows the lens structure for this analysis. The lens has an effective focal length of 29 mm, an F-number of 1.6, a field of view of 20 degrees and a wavelength range of 0.5~0.75 µm. Figure 4 gives the variations of spot size containing 80% energy at different field angles when there is a shift of the image plane. It can be seen that the plot is a little asymmetrical due to aberrations. The defocus range, when the spot image has the required size of 30~50 µm, is -52~-76 µm at the negative side and +52~+74 µm at the positive side of the plot. Therefore the allowable defocus depth for this lens is only 24 µm.

Fig. 5. OPD curves of an optical system with a quartic phase plate.

Maximum Scale: ±10 waves

0.50 0.62 0.75

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Next, we design a quartic phase plate for extending the focal depth of the lens system shown in Fig. 3. The design is based on the requirement that the OPD of new lens with a phase plate has an approximate distribution as that described by Eq. (7). Comparing the phase distribution given by Eq. (7) with the axial OPD of the lens, we obtain the optimum value of α=22.3. Therefore, the phase plate used for extending the focal depth of the lens system is given by

ϕ (ρ) = - π \cdot 22.3 \cdot [(\frac{ρ^{4}}{{ρ_{0}}^{4}}) - (\frac{ρ^{2}}{{ρ_{0}}^{2}})] .

Figure 5 gives the OPD of the new system with the designed phase plate. Figure 6 gives the variations in spot size containing 80 % energy at different field angles associated with axial distance from the focal plane after inserting the phase plate. It can be seen from the plot that the defocus range, when the spot image size containing 80% energy is within 30–50µm, is in the range of -22~+97µm for a 0 degree field angle and it falls within the range of -22~+66µm for a 20 degree field angle. The allowable defocus depth for the entire filed of this lens is equal to 88 µm, which corresponds to more than threefold extension of the focal depth as compared to the lens that has no quartic phase plate.

Figure 6 shows that the optimum image plane of the new lens is shifted due to the presence of the lens aberrations and the phase plate. In order to maintain same position of the optimum image plane, an additional quadric term can be introduced for finer adjustments. The new phase plate for the lens is then described by

ϕ (ρ) = - π \cdot [α_{1} \cdot (\frac{ρ^{4}}{{ρ_{0}}^{4}}) - α_{2} \cdot (\frac{ρ^{2}}{{ρ_{0}}^{2}})] .

For the lens system shown in Fig. 3, the optimum value of α₁ and α₂ are 22.3 and 24, respectively.

The lens’ design described above limits the quartic phase plate in the aperture stop plane of the actual lens. However, it is possible to design a phase plate to be placed in the entrance pupil plane of a lens system for extending its focal depth. We note that for a simple lens having a narrow field of view, it is possible to introduce the phase plate directly in front of the lens.

Fig. 6. Spot size containing 80% energy in an of the optical system comprising a quartic phase plate for different values of axis distance

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4. Conclusion

In this paper we analyzed the effects of a quartic phase pupil function on an imaging system used specifically to track distant objects like stars. Using selected criterion for this tracking application, we designed an optimum quartic phase plate for a real lens system, and the focal depth of the lens system was increased more than threefold.

References and links

1. Carl Christian Liebe, “Accuracy performance of star tracking-a tutorial,” IEEE Transactions on Aerospace and Electronic Systems 38, 587–599 (April 2002). [CrossRef]

2. G. Borghi, D. Procopio, and M. Magnaniet al., “Stellar reference unit for CASSINI mission,” Proc. SPIE 2210, 150–161 (1994). [CrossRef]

3. Giancarlo Rufino and Domenico Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronautica, 53, 135–147 (2003). [CrossRef]

4. Carl Christian Liebe, “Star trackers for attitude determination,” IEEE AES Systems Magazines, 10–16 (June 1995). [CrossRef]

5. Kazuhide Noguchi and Koshi Satoet al., “CCD star tracker for attitude determination and control of satellite for space VLBI mission,” Proc. SPIE 2810, 190–200 (1996). [CrossRef]

6. Joseph F. Kordas and Isabella T. Lewiset al., “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83 (1995). [CrossRef]

7. Dobryna Zalvidea and Enrique E. Sicre, “Phase pupil functions for focal-depth enhancement derived from a Wigner distribution function,” Appl. Opt. 37, 3623–3627 (1998). [CrossRef]

Optical system having a large focal depth for distant object tracking

Abstract

1. Introduction

2. Theoretical analysis

3. Optical system comprising a quartic phase plate

4. Conclusion

References and links

Cited By

Figures (6)

Equations (9)

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