## Abstract

An analysis of the depth of field (DOF) of the wavefront coding imaging system with a cubic phase mask is presented. A necessary condition on the base of that MTF of wavefront coding system is defocus-independent is described. Then the extension ratio of the DOF relative to that of traditional optical system is calculated. And the conclusion is also verified by the simulation results.

© 2008 Optical Society of America

## 1. Introduction

The resolution and the DOF of imaging optical systems are related to the exit pupil of the system, both are identified by the size and location of the exit pupil. The resolution enhances with the increase of the exit pupil, but DOF will be reduced simultaneously. Extending the DOF of imaging optical system with a fixed exit pupil has been an active research topic for many years. The wavefront coding imaging technology [1] invented by W. T. Cathey and Edward Dowski of the University of Colorado has solved this problem in a certain degree. This technology not only can extend the DOF of imaging optical systems without losing the system resolution, but also can decrease many kinds of aberrations and the errors which caused by the temperature change and the assembly, while reducing the size, weight, and cost of imaging systems also [2–3]. Currently, wavefront coding technology is widespreadly used in many fields [4–9]. The cubic phase mask being introduced in the exit pupil of traditional optical system changes the shape of the wavefront and makes the MTF of the optical system defocus-independent. Although the phase mask makes the point spread function (PSF) deviate from the δ function, which worsen the imaging performance of the in-focus image, but it makes the out-of-focus PSF is similar to the in-focus PSF. This makes the images have the same blur characteristics in a very large range of DOF. As long as the receiver has sufficient resolution to the blur image signal including all the original image information, which can be received and converted into digital image signal, the signal processing of the system can decode the information and then the final images can be reconstructed. In other words, whether in focus or defocus, the original image can be restored through an appropriate digital filter, thus the DOF of the system is extended.

The wavefront coding imaging system has advantages of analog coding method and digital decoding technique by combining them, which has solved the inner contradiction in traditional optical imaging system. But, how much can this technique extend the DOF of optical system? This is a question that must be answered in the engineering application of this technique. Laboratory and simulation results from the relative literature [1–2] show that a wavefront coding imaging system can make an order of magnitude improvement in DOF. However, the theoretical model of wavefront coding can not answer this question up to now. In this paper, by investigation of the OTF of wavefront coding imaging system with cubic phase mask, it is discovered that MTF of wavefront coding system is defocus-independent when a special condition is satisfied. Based on the condition, an expression of the DOF of wavefront coding imaging system and the extension ratio of the DOF relative to that of traditional optical system are derived. Then the calculation shows that the DOF of wavefront coding imaging system can be extended more than 60 times when α=20π. And the conclusion is also verified by the digital simulation.

## 2. A theoretical analysis of extension ratio of the DOF for wavefront coding imaging system

#### 2.1 The basic principles of wavefront coding system

The traditional optical systems are sensitive to defocus. There is little change of MTF within a permissive defocus, so it obtains good image in the image plane; But the MTF drops quickly to zero as the defocus goes beyond the permissive range and the image is getting worse. The zero value of MTF brings the loss of components of corresponding frequency; as a result, the traditional optical system can’t transfer the imaging information corresponding to these spatial frequencies.

Wavefront coding system places an aspheric surface phase mask at pupil plane. As an example, a cubic phase mask is chosen here. When the optical pupil is a square with side length L, the two dimension pupil phase modulation function of the cubic phase mask can be expressed as follows:

where *x* and *y* are normalized coordinates.

Experimental MTFs along x direction of traditional imaging system (dashed lines), wavefront coded imaging system before filtering (fine real lines), and wavefront coded imaging system after filtering (thick real lines) are shown in Fig. 1[9], the traditional imaging system is just the optical system that the cubic phase mask of wavefront coding system with a square aperture is taken out. Each set of MTFs contains two curves representing an in-focus and a defocus image plane position respectively. The MTF of the traditional imaging system is seen to drastically change with defocus. A zero or null at approximately 25 lp/mm is also introduced in the traditional system MTF with the particular amount of defocus shown. However, the wavefront coding MTFs are very insensitive to defocus. Further more, the spatial frequency at which the MTFs of the wavefront coding system drop to zero at the first time when the system is defocus is over doubled the traditional optical system on the same condition.

#### 2.2 The DOF of Traditional Optical System

For comparion,, first of all, it is helpful to review the defocus analysis of traditional optical system. The generalized pupil function of defocus is expressed as

$P(x,y)=\{\begin{array}{cc}\mathrm{exp}\left[j\frac{k\epsilon}{2}\left({x}^{2}+{y}^{2}\right)\right]\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}& \mathrm{inside}\mathrm{the}\mathrm{pupil}\\ 0\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}& \mathrm{otherwise}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\end{array}$

where ε is the amount of focusing error of the system and defined as

where d_{i} is the distance from the pupil plane of the lens to the image plane; Δ is the distance from actual image plane to the image plane.

According Rayleigh criterion, the maximum optical path difference on the edge of the aperture should be within a quarter of wavelength for a sharp and clear image. For a square optical pupil with side length L, the maximum optical path difference W_{20} is

For a sharp and clear image, the defocus, needs satisfy the following expression :

2Δ is conventionally regarded as the DOF of traditional optical system.

#### 2.3 The DOF of Wavefront Coding System

From Eq. (1), the pupil phase modulation function of the wavefront coding system is a separable function. Since OTFs in x and y directions are exactly the same, one-dimensional analysis could be easily generalized to two-dimension.

The one-dimension OTF of a defocus system can be expressed as

where spatial coordinates and spatial frequency are not normalized value but actual value. Then the one dimensional pupil phase modulation function which described in Eq. (1) can be rewritten as

where

According to Eq. (5) and Eq. (6), the OTF of wavefront coding system with cubic plate can be written as

In order to calculate Eq. (8) conveniently, let

In order to simplify *f*(*x*) in the exponential function, the point satisfying
$\frac{df\left(x\right)}{dx}=0$
is denoted as x_{0}, then

The expansion of function *f*(*x*) in Taylor series at point x_{0} includes only two terms because the three and above all derivatives are zeros, it is simplified as

According to Eq. (5) and (10), the value of g(x_{0}) is 1 as long as x is in the overlapping region of the optical pupil, otherwise the value is 0, so g(x) could be replaced by g(x_{0}) if x is in the overlapping region of the optical pupil. Substituting Eq. (12) into Eq. (8) yields the following expression:

Neglecting the constant phase term $\left(\frac{\pi}{4}j\right)$ , the OTF of the Wavefront Coding system with a cubic phase mask can be written as

Equation (14) shows that the modulus of the OTF is insensitive to the defocus parameter W_{20}. In the two phase terms, the first is insensitive to W_{20}, however, the second term is related with W_{20}, but if α≫_{20}, it can be regarded as insensitive to W_{20}.

To insure x in the overlapping region of the optical pupil, the following equation must be satisfied:

From Eqs. (7), (11) and (15), we get the maximum wave aberration

As a result, a formula of the DOF of the wavefront coding system can be expressed as

where 2Δ’ is regarded as the DOF of wavefront coding optical system.

#### 2.4 The Extension Ratio of the DOF for Wavefront Coding Imaging System

A comparison of Eq. (17) and Eq. (4) yields the extension ratio of the DOF for wavefront coding optical system compared with traditional optical system, which is expressed as

Equation (18) shows that the extension ratio of the DOF depends on the range of spatial frequency and α, which is the coefficient of the phase term of the cubic phase mask system. When the spatial frequency reach or exceed the incoherent cutoff frequency $\frac{\lambda {d}_{i}\mid {f}_{x}\mid}{L}$ , from Eq. (18), the extension ratio of the DOF will be reduced to zero. In the situation,, the information of two dimensional object of this frequency cannot be restored by the optical system completely, so the DOF is reduced to zero. Eq. (18), can be rewritten as

This expression means that the greatest spatial frequency with which the DOF can be extend have been determined by the coefficient α and the extension ratio M. That means, essentially what the equation tells us is that given a conventional F-no. system and a targeted improvement in depth-of-field, any upper bound placed on the alpha-parameter effectively bounds the spatial resolution of the system. Knowledge of such a relationship between the spatial resolution of the system and the alpha-parameter of a wavefront coding system is very important for designing of wavefront coding imaging system.

In order to compare wavefront coding system with traditional system under the same condition, it is required to select the same spatial frequency as a standard. The experimental result in Fig. 1 shows that, in general technical conditions, MTFs of wavefront coding system after filtering are much higher than MTFs of traditional optical system in-focus if spatial frequency is under the cutoff coherent spatial frequency. However, MTFs decrease rapidly if spatial frequency is above half the cutoff incoherent spatial frequency. For a square optical pupil with side length L, half the cutoff incoherent spatial frequency can be expressed as

Substituting Eq. (20) into Eq. (18) yields the following expression:

In order to compare with Ref [4], we choose same parameter α, substitute α=20π into Eq. (21), and calculate the extension ratio of DOF. Then, we conclude that the DOF of wavefront coding system can be extended 60 times than that of traditional imaging system when α=20π.

## 3. The Simulation of Extension Ratio of the DOF

In order to verify the conclusion above, a simulation is shown below. A Sector Star Target is taken as original image, which is given in Fig.2.

In the simulation, the wavefront coded system is identical to the traditional optical system only except a cubic phase modulation plate at the aperture stop. Images of the Sector Star Target from the traditional imaging system are shown in Fig. 3, and Fig.3(a) shows the best focus image, Fig. 3(b) is with defocus of 2 depth of focus, and Fig. 3(c) is with defocus of 30 depth of focus. Notice that the images of defocus are very bad; it is especially bad when defocus is severe.

Figures 4 and 5 show images of the Sector Star Target by the wavefront coded system before filtering and after filtering respectively. Figures 4(a) and 5(a) show the best focus images, Fig.4(b) and Fig.5(b) show the defocus images of defocus of 30 DOF, and Fig.4(c) and Fig.5(c) show the defocus images of defocus of 60 DOF. The images reveal that a wavefront coded system may produce a blur image before filtering, even in best focus and these images are similar within a certain range of defocus. In other words, images are similar when defocus is within 60 depth of focus. Decoding the blur images by the same PSF, which is in best focus, can acquire Fig. 5. These images are all identical to the original image. In a word, a wavefront coded system can make clear images in the range of defocus that expressed in Eq. (21).

Figure 6 shows images of the Sector Star Target by the wavefront coded system before filtering and after filtering when the defocus exceed the range above. Figures 6(a) and 6(b) show the defocus images of defocus of 65 DOF, and Fig.6(c) and Fig.6(d) show the defocus images of defocus of 100 DOF from best focus before filtering and after filtering respectively. Figures 6(a) and 6(c) reveal that images before filtering of defocus beyond the range above are different from Fig. 4, and Fig. 6(b) and Fig.6(d) reveal that wavefront coded system can’t make clear images beyond the range of defocus that expressed in Eq. (21).

## 4. Conclusion

The extension ratio of the DOF relative to that of traditional optical system by wavefront coding system is discussed in this paper. We find that wavefront coding system can extend the DOF of optical system to a certain range, which is dependent on the parameter α of cubic phase mask. MTF of wavefront coding system is defocus-independent only when a special condition is satisfied, which is obtained by an analysis of the OTF of wavefront coding imaging system with cubic phase mask. The extension ratio of the DOF relative to that of traditional optical system is determined by the condition. For example, the DOF of wavefront coding imaging system can be extended up to 60 times as the coefficient of phase term α equals to 20π. The theoretical analysis has been verified by digital simulation.

As the MTFs of a wavefront coding system whether in-focus or defocus are dropped compared with the in-focus traditional system, the effect of noise at the detector is considerably more important than in conventional systems. So the reduction of signal-to-noise ratio of a wavefront coding system will reduce its extension ratio of the DOF, which is not discussed in the analysis above.

## Acknowledgments

This work is supported by the National Basic Research Program of China (2007CB935303, 2005CB724304), National Natural Science Foundation of China (60778031), Research Fund for the Doctoral Program of Higher Education (20050252004), and Omnivision Research Foundation.

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