## Abstract

A method of designing a lens with an extended focal depth is studied. The lens is a cemented doublet composed of a birefringent lens and a conventional lens. The crystal optical axis of the birefringent lens is perpendicular to the axis of the optical system. By properly selecting the parameters of the birefringent lens and the conventional lens, we can flexibly configure an imaging system that simultaneously has a large focal depth and high resolution. We also provide a theoretical analysis which shows that the focal depth of the lens is approximately 1.5 times that of a conventional lens.

©2005 Optical Society of America

## 1. Introduction

Optical systems that simultaneously have large focal depth and high modulation transfer function (MTF) are very useful for various applications, including precision alignment, profile measurements, and industrial automation. The focal depth of an imaging system can be extended by means of a number of methods [1–3]. Several types of technologies, such as phase masks and aspherical elements, give rise to well-focused energy distributions in 3-D space in the neighborhood of the focal plane; many of these techniques introduce an extra pupil function in the imaging system and reduce the resolution of the system to some extent. While the quality of final images can be improved by means of proper post-processing, this may not be suitable for many applications.

The polarization properties of a light beam may add some new dimensions to an optical system and can be used to enhance the performance of the optical system [4–7]. It was reported that a single birefrigent lens could produce a large focal depth when the parameter of the birefringence lens satisfies some specific conditions [7]. The limitation of using a single birefringent lens for achieving a large focal depth is that a specific value of the birefringence has to be carefully selected to match lens parameters including focal length, pupil aperture and wavelength. To increase the flexibility, we propose a cemented doublet, comprised of a birefringent lens and a conventional lens to configure an imaging system with a large focal depth. The crystal optical axis of the birefringent lens is perpendicular to the principal axis of the lens, such that the lens system may have two different foci for the two orthogonal vibrations (i.e., extraordinary and ordinary ray vibrations). The parameters of the birefringent lens have to be specifically chosen based on the requirement of the focal depth and a conventional lens can be selected to obtain the required power for an imaging system. This also gives more freedom to choose the birefringence material and balance the aberration of the doublet. We analyze the intensity point spread function (PSF) for such a lens system followed by a polarizer at a proper angle, and determine the optimum parameters of the lens required to achieve a large focal depth.

## 2. Intensity distribution in the image plane

Figure 1 shows a configuration where the proposed doublet is used as an imaging lens for a distant object. The input beam is a collimated beam polarized at an angle of 45° to the vertical axis. The crystal optic axis of the birefringent lens is along the vertical direction and is perpendicular to the optical system axis. A polarizer is placed behind the doublet with its transmission axis at an angle of 45° to the vertical axis. To simplify analysis, it is assumed that the two lenses are thin and the aperture stop is at the position of second surface of the conventional lens. Under the paraxial approximation, the optical path difference introduced by the lens for ordinary vibration and extraordinary vibration are given by

and

respectively. In Eqs. (1) and (2), *n*
_{1o} and *n*
_{1e} are refractive indices of ordinary and extraordinary vibrations of the birefringent lens, *f*
_{1o} and *f*
_{1e} are the corresponding focal lengths for the two vibrations, D_{1} is the thickness of the first lens; and *n*
_{2}, *f*
_{2} and *D*
_{2} are the refractive index, focal length and thickness of the second lens. Using Gaussian optics, *f _{1o}* and

*f*

_{1e}can be written as

and

where *R*
_{1} and *R*
_{2} are the radii of curvatures of the two surfaces of the birefringent lens, and *δC*
_{1} is the difference between the corresponding curvatures. The amplitude point spread functions (PSF) of the system for ordinary vibration and extraordinary vibration are as follows

and

where *z* represents the distance from the image plane to the aperture stop, *k*=2*π*/*λ* is the propagation constant, and *p*
_{1}(*x*
_{1},*y*
_{1}) is the amplitude pupil function and is given by

where *ρ* represents the half diameter of the aperture stop. The overall intensity PSF of the proposed system can be expressed as

Given the average focal length, *f*
_{1}, of the birefringent lens as defined as

the system focal length is then determined by

Assuming *ρ*≪*f*, in the polar coordinate system, the intensity PSF is obtained by

where *c*
_{1} is a constant, *r* is the radial coordinate in the image plane, Δ*z* is the axial distance from the focal plane and ξ is the normalized radial pupil coordinate. *A* and *B* are two constants related to the parameters of the birefringent lens, which are defined as,

and

respectively. Equation (12) can be further expressed by a summation of an integer N and a value of *α*(0.5≤*α*≤1.5), i.e., *A*=*N*+*α*. From Eq. (11), the on-axis intensity distribution, *I*(*r*, Δ*z*), is independent to N. Hence, it is given by

The intensity PSF in the focal plane is as follows

and the axial intensity in the focal plane is

where,

Equation (16) indicates that the axial intensity in the focal plane, *I*(0,0), is the function of parameters *α* and *B*.

From Eq. (11), the pupil function of the system can be expressed as

where (*x*, *y*) are the coordinates of the pupil plane. As the optical transfer function (OTF) of the lens can be expressed as the autocorrelation of the pupil function, the normalized OTF for the proposed system is given by

and the modulus of OTF, MTF, and can be expressed as

where *v* is the normalized spatial frequency (0≤*v*≤2).

## 3. Results and discussion

It is noted from Eq. (14) that the axial intensity *I*(0,Δ*z*) is the function of both coefficients *α* and *B* which are related to birefringence and lens power of the birefringent lens. For easy analysis, we shall define the normalized intensity *s*=*I*(0,0)/*I*(0,Δ*z*)* _{max}*, where

*I*(0,Δ

*z*)

*is the maximum intensity near the focus. The value of*

_{max}*s*ranges between 0 and 1. Obviously,

*s*=0 means there are two totally separated foci located at the two sides of the focal plane, and

*s*≈1means that the two foci are overlapped and the system has only one focus with a large focal depth. It is also noted that the value of

*I*(0,Δ

*z*)

*is dependent on the selections of*

_{max}*α*and

*B*. This implies that we can flexibly select values of

*α*and

*B*from a number of combinations to make

*s*≈1. Figure 2 shows the variations of

*s*versus

*α*(0.5≤

*α*≤1.5) and B(0≤B≤2).

The flat top hat edges of the plot give two lines (1 and 2) where α and B can be selected for a large focal depth.

It is noticed from Fig. 2 that the values of α and *B* are not independently selected for a large focal depth. The relation between α and *B* for line 1 is shown in Fig. 3. The optimum values for α and B can be determined by the requirement that *I*(0,0) reaches the maximum intensity.

As α and B is mutually related, the focus intensity *I*(0,0) satisfying *s*≈1can be expressed as *I*
* _{0}*(

*B*(

*α*)). A plot for variation of

*I*(

_{0}*B*(

*α*)) versus

*B*is shown in Fig. 4, in which the highest intensity happens when

*B*=0.9. Correspondently, the optimum value for α is obtained from relationship shown in Fig. 3, which is 0.67. Given

*B*=0.9 and α=0.67, the intensity PSFs and the MTFs of the cemented doublet lens at three defocusing positions, 0, 1.0

*d*and 1.5

*d*, where

*d*=

*λf*

^{2}/(

*2ρ*), are calculated and shown in Figs. 5 and 6, respectively.

^{2}MTFs of the lens with different values of *B* and α, (0,0), (0.5, 0.6) and (0.9, 0.67), are calculated, and shown in Fig. 7. The plot for (0,0) represents the MTF of a conventional lens and the other two curves correspond to the putative lens with an extended focal depth and high resolution. As the difference in last two curves is small, it implies that we can flexibly select the parameters for the lens to configure an imaging system, which has a large focal depth and required resolution.

Figure 8 plots the variations of relative axial intensity versus the value of defocusing. One curve is for *B*=0.9 and α=0.67, and another curve is for *B*=0 and α=0. It demonstrates that the focal depth of doublet is 1.5 times longer than that of the conventional lens.

## 4. Conclusions

We have described a cemented birefringent lens for an imaging system having a large focal depth. The parameters of the lens system including lens power, thickness, birefringence and focal depth can be selected flexibly to meet the requirement of different applications. To generate a maximum axial intensity in the focal plane and a focal depth 1.5 times longer than a conventional lens, the optimal values of *B* and α are 0.9 and 0.67, respectively. Actual lens parameters can be determined by Eqs. (12) and (13). It is worth to mention that if the value of the normalized intensity *s* is close to 1, the lens system can still be considered as one focus system. This diversifies the potential values of α and *B*, and the focal depth can be further extended based on applications.

Furthermore, it is noted that the index for the extraordinary rays is not precisely equal to the principal index but depends on the direction of the wave which gives rise to the issue of special aberrations caused by birefringence of the unaxial crystal [8–10]. For an approximately normal incidence beam on the lens, this effect is very weak and can be neglected.

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