## Abstract

A new bifocal optical system used for distant object tracking is proposed. This system combines a birefringent element with a conventional glass lens so that the spot image size and its variation with the axial distance can be controlled according to the requirement of a distant object tracker. The lens design for the tracking application is discussed and an example is given. The new lens system provides a more uniform spot image size and an extended focal depth compared to a conventional lens with one focus.

©2005 Optical Society of America

## 1. Introduction

In highly accurate tracking systems, e.g., a star tracker, the image plane is commonly defocused slightly from the focal plane, spreading the spot image over several pixels of a CCD camera [1–6]. The required spot image size for calculating the centroid of image has previously been reported [1–3]. For a CCD camera having a pixel size of 10 µm, the spot image must cover over 3 pixels of the sensor, meaning that the image size should be 30 µm or larger. On the other hand, smearing the object image on a too large area will decrease the signal-to-noise ratio. For a conventional tracking lens system, control of the smearing of the image size to a designated value can be difficult due to the short defocusing depth for a conventional tracking lens - the spot size increases or decreases quickly with a shift of the image plane. The tracking accuracy will be affected by the shift, which may be caused by small perturbations including temperature variations or vibrations. Keeping the required spot size in the image plane and decreasing the variation of the spot size with the axial distance are high desirable for distant object tracking.

It is well-known that the focal depth of imaging lens can be modified by using aspheric lenses or additional phase pupil plates [7–11], e.g., Jorge Ojeda-Castaneda presented a zone plate for arbitrarily high focal depth [11]. However, this method requires fabrication of high precision aspheric elements or zone plates and the high cost associate with it. In this paper, we study the performance of a bifocal lens and present an approach for designing a bifocal optical system that could be specifically used for distant object tracking. The lens system combines a bifocal element with a conventional lens. Separation between two foci is determined based on the requirement of the tracking application. A bifocal system which comprises a birefringent lens is specifically studied. We have designed a sample lens with an effective focal length of 30 mm, an *f*-number of 2, a field of view (FOV) of 20 degrees, and a working wavelength range of 0.5~0.7 µm, which demonstrates the efficiency of this lens system for highly accurate tracking applications.

## 2. Intensity spread function of the spot image

For an aberration free lens system suffering from defocusing error, the intensity spread function (ISP) of a spot image has radial symmetry. The ISP is the inverse Fourier transform of the optical transfer function (OTF) of the lens. The OTF of the system, *D _{H}(Ω,),* can be approximately expressed as the product of the OTF of a diffraction limited system and the OTF of geometric optics, which is given by [12]

Here *O* represents the spatial frequency, *D*
_{0} (*O*) represents the diffraction limited OTF, and *D _{g}* (

*O*) represents the OTF predicated by geometric optics. If

*I*

_{0}(

*r*) represents the diffraction limited intensity point spread function and

*I*(

_{g}*r*) represents the ISP of the geometrical spot diagram of a defocused image, the overall ISP of the spot image,

*I*(

*r*), can be written as

where *F*
^{-1} represents inverse Fourier transform, * represents convolution, and *r* is the coordinate of any radial direction in the image plane.

The normalized *I*
_{0} (*r*) is expressed as

where *λ* represents the working wavelength and *F _{n}* represents the

*f*-number of the lens.

For a bifocal lens, the intensity distribution of the spot diagram is related to the separation of the two foci. Figure 1 shows a schematic of a lens having two foci, *P*
_{1} and *P*
_{2}. *S*
_{1} and *S*
_{2} represent axial distances from the exit pupil plane of the lens to the two foci, respectively, and Δ*s* represents the separation between two foci, and Δ*s*≪*S*
_{1}. A reference plane of this lens is set in the middle between two foci. *δ _{z}* represents the axial distance of the image plane from the reference plane and |

*δ*|<Δ

_{z}*s*/2. Assuming that the incident beam is unpolarized, the intensity distribution of spot image in the image plane shifted

*δ*from the reference plane can be approximately expressed by

_{z}Here *I _{g}*

_{1}(

*r*) and

*I*

_{g}_{2}(

*r*) represent the ISP of the geometrical spot diagram for two incoherent beams, which are given by

Applying Eqs. (3), (5) and (6) to (4), we get

The encircled energy of a bifocal lens for a circle of radius *c* is

The spot size in the image plane and its variation with the shifting distance of the image plane are two important aspects for a tracking lens. It is required that the spot image size has a specific value and its variations within the focal range remain minimum. As an example, we study a tracking lens that has an *f*-number of 2, a wavelength of 0.59 µm and a spot image size containing 80% energy of 35 µm. Using Eq (10), the variations of the spot size in the reference plane with the value of Δ*s* is calculated and shown in Fig. 2. It can be seen that the spot size becomes large as the value of Δ*s* increases. The spot size in the reference plane is 35 µm when Δ*s*=0.154mm. For a bifocal lens with Δ*s*=0.154mm, the variation of spot size with the defocusing value of *δz* is shown in Fig. 3, in which the variation of spot size is 4.2 µm when the value of *δz* varies from -0.02 to +0.02 mm.

For a conventional lens having one focus, the variation of the spot size with the axial distance can be calculated by using Eq. (10) with Δ*s*=0. The spot size varies from 35 µm to 48.6 µm when the defocusing ranges from 0.077mm to 0.107mm. The variation of the spot size is 13.6 µm when the image plane shifts 0.04 mm. When compared to the conventional lens, the bifocal lens provides a larger focal depth for tracking application if a proper separation of two foci is selected.

## 3. Bifocal optical system comprising a birefringent lens

H_{2}: the 1^{st} principal point of lens 2

F_{1o}′: the 2^{nd} focus of lens 1 for o ray

F_{2}: the 1^{st} focus of lens 2

F_{o}′: the 2^{nd} focus of lens system for o ray

*x*
_{2}: distances of point F_{1e}′ from point F_{2}

Δ*d*
_{1}: distance of point F_{1o}′ from point F_{1e}′

*f*
_{1e}′: distance of the point F_{1e}′ from lens 1

*f*
_{2}: distance of the point F_{2} from point H_{2}

H_{2}′: the 2^{nd} principal point of lens 2

F_{1e}′: the 2^{nd} focus of lens 1 for e ray

F_{2}′: the 2^{nd} focus of lens 2

F_{e}′: the 2^{nd} focus of lens system for e ray

*x*
_{2}′: distances of point F_{e}′ from point F_{2}′

Δ*d*: distance of point F_{o}′ from point F_{e}′

*f*
_{2}′: distance of point F_{2}′ from point H_{2}′

*a*: distance of point F_{2} from lens 1

These calculations predicate that the performance of a distant optical tracking lens can be enhanced by selecting a proper bifocal lens. The bifocal lens can be made in different ways and one of them is to use a birefringent element [13]. One of the two foci is generated from ordinary rays while other is from extraordinary ones. However, there are limitations when one uses a single birefringent lens as a tracking lens. For real lens design applications, the bifocal lens system can be built by combining a birefringent lens with a glass lens and setting the aperture stop in the surface of the birefringent lens. This arrangement gives the system design more flexibility in the control of the system parameters and aberrations. The function of the birefringent lens is to provide two foci for the lens system, while the conventional lens provides the required lens power and balance of lens aberrations. Figure 4 shows a schematic of a bifocal lens system. Lens 1 is a thin birefringent lens whose crystal optical axis is perpendicular to the optical system axis and lens 2 can be a group of glass lenses.

For simplicity, assuming lens 1 is plano-convex, the focal length for ordinary rays and extraordinary rays can be written as follows

Here *r*
_{1} represents the radius of curvature of the first surface, and *n*
_{o} and *n*
_{e} represent the refractive indices of ordinary rays and extraordinary rays, respectively. The separation of two foci for lens 1 then can be approximately calculated as follows

Applying Eqs. (11) and (12) to (13), we have

Then the two foci for the lens system is written as

where *x*
_{2} represents the distance from the first focus of lens 2 to the second focus of lens 1, and *x*
_{2}′ represents the distance from the second focus of lens 2 to the second focus of the lens system. For lens 2, we have the following relation:

where *f*
_{2}′ represents the focal length of lens 2. For the bifocal lens system, we have

where *f*′ represents the focal length of the lens system and *a* represents the distance from the principal plane of lens 1 to the first focus of lens 2. According to the structure of the lens system shown in Fig. 4, the value of *a* can be expressed as

From Eqs. (12), (14), (15), (16), (17) and (18), we can get the expression of Δ*d* as follows

Equation (19) shows that the value of Δ*d* is related to the refractive indices of the birefringent material, the effective focal length of the system and the value of *r*
_{1}. It decreases when the value of *r*
_{1} increases for a specific focal length and selection of birefringent material. The required separation of two foci can always be obtained by selecting *r*
_{1} for a bifocal lens system. It should be mentioned that the refractive index of birefringent material for extraordinary rays varies with the direction of the propagation. We neglect this effect as a tracking system normally has a small or moderate FOV.

In previous sections, we obtained the best separation for a bifocus lens with an *f*-number of 2 and a working wavelength of 0.59 µm. The separation of 0.154 mm was shown to be most suitable for this tracking lens. In the following, we present a design of a real bifocal lens with the same system parameters.

Figure 5 shows the lens system where the aperture stop is at the first surface of the birefringent lens. Selecting uniaxial crystal quartz as the birefringent material of lens 1, we get *r*
_{1}=54.0 by using the Eq. (19), for the system focal length *f*′=30 mm and Δ*d*=0.154 mm. The lens system is optimized by varying the parameters of lens 2 for extraordinary rays. In the design process, the thickness of the birefringent lens was selected to be 3.5mm and the real radius of the curvature of *r*
_{1} was then adjusted to be 52.2 mm to achieved the required foci separation.

Variations of spot image size with focal shift for different view angles are calculated for the lens. Fig. 6 shows the variations for the spot image generated only by extraordinary rays, and Fig. 7 shows the variations for overall spot size. Comparison of the two figures demonstrates the extension of the focal depth by a bifocal lens. For the real lens in Fig. 5, we reached the same range of spot size and focal depth as predicted in previous section, e.g., the spot size varies about 5 µm when the image plane is shifted from -0.02 to 0.02 mm.

## 4. Conclusion

We have analyzed the intensity spread function of the spot image of a bifocal lens. The size of spot image and its variation with the axial distance are controlled by selecting a proper separation between the two foci to meet the requirement of a distant object tracking system. We have shown a practical configuration of a bifocal tracking lens. For this new lens system, the desired separation of the two foci can be calculated based on the requirement of tracking application and system parameters, and obtained by control lens power of a birefringent lens. An example is given for the design of a real bifocal tracking lens, which demonstrates the advantage of using the bifocal lens over a conventional lens system, with a small variation of the spot image size in a large focal depth.

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