Improved Exponential Phase Mask for Generating Defocus Invariance of Wavefront Coding Systems

Abstract

Wavefront coding is an effective way to extend the depth of field of optical imaging systems. The invariant defocusing imaging feature can be obtained by adding a phase mask with a suitable form to the aperture of a typical optical system. Traditional exponential phase mask defocusing optical characteristics exhibit strong invariance in the frequency domain, but the point spread function (PSF) variation is significant in the image plane To reduce PSF position deviation, we presented an improved exponential phase mask. The phase function of the improved mask is obtained by analyzing the relationship between Taylor expansion and wave aberration. Numerical analysis and imaging simulation are used to evaluate the performance of the proposed phase mask to that of other standard phase masks. The simulation results show that the improved exponential phase mask has a stronger defocus invariance of the modulation transfer function (MTF), and the position deviation of the PSF has been effectively controlled.

1. Introduction

The wavefront coding technology proposed in 1995 has become an effective way for increasing the depth of field of an optical system [1]. The wavefront coding system is to add a designed phase mask to the pupil plane of the traditional optical system as Figure 1 shows. The phase mask is used to make the point spread function of the optical system have the characteristics of defocus invariance. However, the size of the point spread function will get larger and imaging results will be blurred. So, it requires an image restoration process to make blurred images clear. The higher the degree of defocus invariance, the better the image restoration result of the system is. When we need a system with a large depth of focus such as long-distance photography, we can add a designed phase mask to the aperture of the traditional phase mask. The phase mask function which can generate the defocus invariance is from the mathematical derivation based on the Fourier optics. We want that there is no defocus variable in the MTF function, and then solve the equation to get the phase mask function. It may effectively ease the conflict between the improvement of the depth of field and the decline of imaging resolution by integrating the optical imaging method with an image restoration algorithm. The defocusing invariance character of an optical system can be obtained by adding an appropriate phase mask to the aperture. The corresponding image restoration algorithm is designed to restore the blurred image taken by optical systems [2] based on defocusing invariance, and the image in a wide range of depth of field may be obtained in the end. As a result, designing a phase mask with a suitable shape to modify the optical system is one of the key points of wavefront coding technology.

Many new phase masks have been proposed since 1995, including extended cubic phase mask [3], logarithmic phase mask [4,5,6], exponential phase mask [7], fractional power phase mask [8], extended polynomial phase mask [9], sinusoidal phase mask [10], tangent phase mask [11], square root phase mask [12], anti-sinusoidal phase mask [13,14] and Jacobi–Fourier phase mask [15]. The most common non-rotationally symmetric phase mask is the cubic phase mask. The numerical analytical solutions of point spread function and optical transfer function can be calculated by using the stable phase method [2] or Fresnel integral method [16], which is convenient for theoretical analysis. The cubic phase mask characteristics have been studied extensively in the space domain [17] and frequency domain [18] and applied to the iris recognition system [19]. The sharpness and clarity of the restored images of cubic wavefront mask systems will, however, be lowered in practice. In the extended depth of field imaging, Diaz et al. examined the performance of phase masks that were optimized using an image quality criterion after deconvolution, finding that the exponential phase mask and fractional power phase mask outperform the cubic phase mask and logarithmic phase mask [20]. Sinusoidal phase masks have smaller imaging translations than other phase masks, although the modulation transfer function (MTF) fluctuates substantially with defocusing [10]. The logarithmic phase mask and its improved functions have low MTF defocusing invariance [7]. The change in MTF with defocusing is one of the most critical elements in determining the imaging results under the same reconstruction settings. Although the MTF of an exponential phase mask has a high invariance, the position translation of its PSF on the image plane is considerable, leading the mask to leave the imaging area and resulting in imaging findings that are inappropriate.

The wavefront coding optical system with higher invariance of defocus has better defocus imaging capability. Due to the high MTF defocusing invariance of the exponential phase mask, the phase mask with better defocusing invariance can be obtained if the image plane translation of the PSF is improved. Therefore, an improved exponential phase mask is proposed in this paper. Through the Taylor expansion of the exponential phase mask function, the relationship between the expansion term and the wave aberration type is analyzed, and a new phase mask function is devised to eliminate the offset aberration generated by the exponential phase mask. Fisher information is used to construct the optimization equation to optimize the surface parameters of the improved phase mask. By using a numerical analysis method, the performance of the improved phase mask is compared to that of the exponential phase mask, cubic phase mask, sinusoidal phase mask, logarithmic phase mask, and tangent phase mask. To illustrate the phase mask’s defocusing imaging capabilities, the invariance of MTF and the shift of PSF are calculated. In addition, imaging simulations were performed to evaluate the suggested phase mask.

2. Improved Exponential Phase Mask and Optimization Method

The traditional surface equation of exponential phase mask is [1]

$$f(u,v)=\alpha u{e}^{\beta u^2}+\alpha v{e}^{\beta v^2}$$

(1)

where u and v are the normalized pupil plane coordinates in x and y directions, respectively. The range of normalized pupil coordinates is −1 to 1, which equals the true pupil coordinates, which are split by the greatest pupil coordinates. The following description is condensed to a one-dimensional situation because the two directions are independent of each other and have the same change condition. Figure 2 shows its PSF in one dimension.

The peak position of the point spread function of the exponential phase mask has an obvious translation on the image plane, as shown by the red dotted line in Figure 2, and its maximum value is not at zero, resulting in a mismatch between the object shooting range and the imaging range during the imaging process, resulting in the imaging error. The exponential phase mask equation’s Taylor expansion is shown as

$$f(u)=\alpha u{e}^{\beta u^2}=\alpha u(1+\frac{\beta u^2}{1!}+\frac{{\beta }^2{u}^4}{2!}+\frac{{\beta }^3{u}^6}{3!}+\cdots )$$

(2)

Term $\alpha u$ in Equation (2) is the first-order expansion of the pupil coordinate. The first-order term of the coordinate corresponds to the optical system’s translational aberration from the standpoint of wave aberration. As a result, by adding one term to the equation to eliminate the first-order component, the phase mask shape may be created, which weakens the picture plane translation of the point spread function. As the Taylor expansion of $\alpha u{e}^{-\beta u^2}$ is

$$\alpha u{e}^{-\beta u^2}=\alpha u(1-\frac{\beta u^2}{1!}+\frac{{\beta }^2{u}^4}{2!}-\frac{{\beta }^3{u}^6}{3!}+\cdots )$$

(3)

The first-order term has been removed when Equation (2) is subtracted from Equation (3). As a result, the improved exponential phase mask equation given in this research is as follows:

$$f_{improved-exponential}(u,v)=\alpha u(e^{\beta u^2}-e^{-\beta u^2})+\alpha v(e^{\beta v^2}-e^{-\beta v^2})$$

(4)

The surface parameters of the phase mask must be optimized after getting the phase mask’s surface equation. Fisher information is chosen as the phase mask optimization approach here [1]. When a phase mask function f(u) is added to the pupil of the optical system without aberrations, the exit pupil equation of the system is

$$p(u)=\{\begin{array}{lll}\frac{1}{\sqrt{2}}{e}^{jf(u)}& & \mathrm{for}\hspace{0.17em}\left|u\right|\le 1\\ 0& & \mathrm{for}\hspace{0.17em}\left|u\right|>1\end{array}$$

(5)

The optical transfer function (OTF) of the optical system with different defocusing coefficients can be calculated by the exit pupil equation of the system, as

$$H(\rho ,\psi )={\displaystyle {\int }_{-\infty }^{\infty }p(u+\frac{\rho }{2})e^{j\psi {(u+\rho /2)}^2}\times p^{*}(u-\frac{\rho }{2})e^{-j\psi {(u-\rho /2)}^2}du}$$

(6)

where $\rho$ is the normalized spatial frequency from −2 to 2, which equals the normalized frequencies are also the real frequencies divided by the maximum frequencies. $\psi$ represents the phase difference caused by defocusing, and its calculation formula is [2]

$$\psi =\frac{\pi D^2}{4\lambda }(\frac{1}{f}-\frac{1}{d_0}-\frac{1}{d_i+\Delta d})$$

(7)

where D is the pupil diameter, λ is the working wavelength, f is the focal length, d₀ and d_i are the object distance and image distance of the optical system when focusing, and Δd is the difference between the image distance of the optical system when focusing and the image distance of the actual imaging. MTF can be used to assess the optical system’s imaging quality. The greater the region between the MTF function and the coordinate axis, the higher the optical system’s imaging quality. Without considering noise, a wavefront coding system with higher invariance of OTF function in full space-frequency under different defocusing coefficients have better imaging quality. In the whole frequency domain, the Fisher information of OTF with defocusing is determined as follows:

$$FI(\psi )={{\displaystyle {\int }_{-2}^2\left|\frac{\partial }{\partial \psi }H(\rho ,\psi )\right|}}^{2}d\rho$$

(8)

For Equation (8), the smaller the value of the function is, the smaller the relationship between the OTF and the defocusing parameters is, which is equivalent to the higher defocusing invariance of the optical system. By setting the maximum defocusing range of the optical system ψ₀ the Fisher information of the whole working range of the system is calculated as

$$F={\displaystyle {\int }_{-{\psi }_0}^{{\psi }_0}FI(\psi )}d\psi$$

(9)

Based on pursuing the defocusing invariance of the system and ensuring the imaging quality of the system, the optimization equation is

$$\begin{array}{ll}\underset{Para}{\min }& I(Para)=F(Para,{\psi }_0)\\ s.t.& {\displaystyle {\int }_{-2}^2\left|H(\rho ,0)\right|d\rho \ge th}\\ & Para>Par{a}_0\end{array}$$

(10)

where Para is the surface parameter vector of the phase mask to be optimized, and th is the lower limit of the MTF function area of the system under the non-defocusing condition which can determine different image qualities. Para₀ is the lower limit of phase mask surface parameters. The optimization was carried out using a solver that can identify the interior point of a restricted nonlinear multivariable function with the algorithm. When tuning the six phase masks, a unified th threshold should be utilized to compare the defocusing performance of different phase masks under the same image quality. When the threshold value is set high, the system’s imaging quality is high, but the invariance of the optical characteristic function in the expanded range of depth of field is low; when the threshold value is set low, the system’s imaging quality is low, but the invariance of the optical characteristic function in the extended range of depth of field is high. As a result, the threshold selection should be considered in the optimization design.

3. Simulated Results

The parameters of the improved exponential phase mask and other phase masks are optimized in one dimension. The optimum threshold value is th = 0.3, with a defocus range of 0 to 60, to achieve both defocusing imaging invariance and imaging quality.

Table 1. Optimization results of six kinds of phase masks.

	Function	Parameters
Improved exponential	$\alpha u\left(e^{\beta u^2}-e^{-\beta u^2}\right)$	22.744, 1.645
Exponential	$\alpha u{e}^{\beta u^2}$	29.924, 1.832
Cubic	$\alpha u^3$	89.757
Sinusoidal	$\alpha u^4\sin \left(\beta u\right)$	211.174, 1.856
Improved logarithmic	$sgn\left(u\right)\alpha u^4\log \left(\left|\left|u\right|+b\right|\right)$	162.286, 1.990
Tangent	$\alpha u^2\tan \left(\beta u\right)$	40.593, 1.342

shows the optimization results; the first parameter is alpha, and the second parameter is beta.

According to the surface equations of the phase masks and their corresponding optimization parameters in Table 1, the phase profiles of the above six kinds of phase masks are drawn in Figure 3a. Figure 3b shows the two-dimensional phase profile of the improved exponential phase mask at $\alpha =22.744$, $\beta =1.645$.

The invariance of MTF curves is examined first to verify the defocusing imaging performance of the improved exponential phase mask proposed in this paper. Figure 4 shows the MTF curves of the optimized six types of phase masks with defocusing phase parameters of 0, 20, 40, and 60. The improved exponential phase mask has the highest curve invariance under different defocusing values, as evidenced in the defocusing MTF curves of six different phase masks. The Hilbert angle is used to calculate the similarity level of two functions in Hilbert space, which can be calculated as

$$\theta (\psi )={\cos }^{-1}(\frac{\langle MTF(u,\psi ),MTF(u,0)\rangle }{\Vert MTF(u,\psi )\Vert \Vert MTF(u,0)\Vert })$$

(11)

The symbol $\langle \cdot \rangle$ stands for calculating the inner product of two vectors, and the symbol $\Vert \cdot \Vert$ stands for calculating the 2-norm of vectors. Equation (11) shows that when the Hilbert angle is less, the difference between the two functions is smaller, and the phase mask’s defocus invariance is stronger. The appropriate Hilbert space angles for the MTF curves of six types of phase masks are determined in the defocusing range [0, 60], and the results are displayed in Figure 5. The angle of the improved exponential phase mask is bigger than that of the exponential phase mask and the tangent phase mask when the defocusing value is less than 30, but its value is still low, so the invariance is maintained. When the defocus value is more than 30, however, the Hilbert angle of the improved exponential phase mask is much lower than the other phase masks, demonstrating that the improved exponential phase mask can perform better when the defocus value is large. In addition, the system should have similar imaging quality under different defocusing parameters to facilitate image restoration, so the area between the MTF curve and coordinate axis under different defocusing parameters is also one of the parameters to consider when evaluating phase mask imaging performance.

The region between the MTF and the two coordinate axes can be utilized to assess the system’s imaging quality. The following formula can be used to compute it:

$$\mathrm{Area}(\psi )={\displaystyle {\int }_{-2}^2\mathrm{MTF}(\rho ,\psi )}d\rho$$

(12)

The larger the area is, the better the imaging quality is. As a result, it may be used to assess the imaging quality of phase masks using various defocus values. Figure 6 depicts the area change values of the MTF curve and the coordinate axis of six different types of phase masks in the [0, 60] defocusing range, which shows that the improved exponential phase mask’s area change is very sluggish, and the overall area change is smaller than that of previous phase masks. As a result, the improved exponential phase mask provides a more stable imaging quality in the defocusing region.

We also use the Fisher information to examine the OTF curve. The simulation results of six phase masks in the defocus range of 0 to 60 are shown in Figure 7.

Under varied defocus values, the improved exponential phase mask exhibits low Fisher information, as shown in Figure 7. It is just slightly higher than the sinusoidal phase mask, but it has a greater MTF invariance. As a result, the improved exponential phase mask outperforms the other five phase masks in terms of defocus invariance. The modified exponential phase mask’s imaging invariance in the picture plane is also taken into account. As a result, the one-dimensional PSF curves of six distinct types of phase masks are simulated under various defocusing settings, with the defocusing parameters set to 0, 20, 40, and 60. PSF curves are chosen for comparison to observe the changes, and the results are displayed in Figure 8.

The improved exponential phase mask effectively suppressed the offset of the peak position, as shown in the simulation results in the red boxes in Figure 8a,b. The mismatch of imaging areas between objects and images caused by the offset of PSF’s peak location results in a considerable mistake in the system’s actual use. When compared to the other six phase masks, the improved phase mask’s peak intensity is as stable as the cubic phase masks and more stable than the other four phase masks. The PSF will change by defocusing on the image plane for the non-rotationally symmetric phase mask, introducing artifacts for image restoration, and causing image quality reduction. As a result, the peak shift changes of these six types of phase masks are compared in Figure 9. The improved exponential phase mask’s peak shift amount is second only to the cubic phase mask and better than the other four types of phase masks. As a result, the improved exponential phase mask exhibits strong defocus invariance in both spatial frequency domain and image plane analysis when compared to existing phase masks. As a result, it may perform better in extended depth of field imaging.

Finally, we simulated the imaging process to evaluate the performance of phase masks. We calculate the 2D-PSF of the system in MATLAB, and downsample it to a 5 × 5 template. The “Image man” has a size of 256 × 256. We use this PSF to convolve the original image and add Gaussian noise with a variance of 0.0001 to this image to simulate the optical imaging result from the wavefront coding optical system. Then we use the Wiener filter to restore this image to get the final imaging result as Figure 10 shows. Gaussian noise with a variance of 0.0001 has been added to the simulation, and the SNR is 80 dB. The simulation results of the improved exponential phase mask of image “Cameraman” are shown in Figure 10. In Figure 10, the six lines represent the results of the improved exponential phase mask, exponential phase mask, cubic phase mask, sinusoidal phase mask, improved logarithmic phase mask, and tangent phase mask, respectively, and four rows represent the defocus value of 0, 20, 40 and 60, respectively.

To quantified compared the performance of six phase masks, the differences between restored images and reality images are calculated by the following formula [21]:

$$\mathrm{RIQ}=-10{\log }_{10}\overline{{\left|{\mathrm{RI}}_{ij}-{\mathrm{O}}_{ij}\right|}^2}$$

(13)

where RI is the reconstructed image and O the object, which have been normalized, i and j the coordinates of the pixels, and $\overline{{\left|{\mathrm{RI}}_{ij}-{\mathrm{O}}_{ij}\right|}^2}$ denote the average values over all the pix els. The better the imaging quality, the higher the RIQ. The results are presented in

Table 2. RIQ factors of six phase masks.

Defocus Parameter	0	20	40	60
Improved exponential	25.70 dB	26.70 dB	24.74 dB	21.88 dB
Exponential	27.48 dB	24.33 dB	22.78 dB	21.16 dB
Cubic	22.43 dB	23.36 dB	22.05 dB	19.50 dB
Sinusoidal	27.30 dB	26.04 dB	20.08 dB	19.93 dB
Improved logarithmic	23.89 dB	23.05 dB	22.54 dB	19.71 dB
Tangent	24.64 dB	25.55 dB	22.59 dB	20.76 dB

. It can be seen that the improved exponential phase mask offers the best image quality for defocus values ranging from 20 to 60, and is less expensive than the sinusoidal phase mask and the exponential phase mask at defocus values of 0.

4. Discussion

The improved exponential phase mask exhibits good defocus invariance of MTF curves, as shown in the simulation results in Section 3. The area of MTF represents the imaging quality of the system, and that of the exponential phase mask has the smallest change with defocus, which guarantee image quality at different defocus values. However, from the simulation of Fisher information, that of the improved exponential phase masks shows higher than the sinusoidal phase mask, which means the PSF invariance is higher. As the Fourier transform of the PSF contains MTF and phase transfer function (PTF), the improved exponential phase mask has high MTF invariance but the PTF has a larger change with defocus. The main feature is to make the peak position closer to the origin and retain the high invariance of the MTF curves. The image simulation results confirm this conclusion.

5. Conclusions

Finally, to improve the imaging depth of field of incoherent optical systems, a new improved exponential phase mask is developed. The parameters of the phase mask are optimized using Fisher information, and the MTF and PSF functions of the phase mask are simulated using numerical analysis. The Hilbert space angle of the MTF curve, the Fisher information, the change of the defocusing area, and the peak shift of the PSF curve are used as analytical elements to determine the phase mask’s invariance under various defocusing parameters. The simulation results reveal that the improved exponential phase mask has improved imaging quality, increased MTF invariance, and reduced tilt aberration compared to the classic exponential phase mask. The improved exponential phase mask offers the best image quality at high defocus parameters, according to simulation imaging experiments. However, while the improved exponential phase mask improved defocus performance, it still changed with defocus values, resulting in artifacts in the final imaging result.