1. Introduction

The wavefront coding technique was introduced by Dowski and Cathey [1] for extending the depth of focus in incoherent optical imaging systems. In this approach, a properly designed phase mask is inserted into the optical system and modifies the point spread function (PSF) so that it is insensitive to misfocus. Although the resulting image is severely distorted, the image distortion remains constant with misfocus and can be restored by digital processing. The resulting optical-digital system produces an image that is comparable to a diffraction-limited system, but has a much longer depth of focus. In the initial work [1], a cubic phase mask where the phase varies across the mask as $\Phi (x)=\beta x^3$ was selected where x is a normalized parameter defined as $\left\{-1\le x\le 1\right\}$ and for best results, $\left|\beta \right|>>20$ rad. In practice, two-dimensional separable masks are used to simplify the digital processing step. Since that time, a large number of alternative mask designs have been proposed [2–5] to obtain improved performance. There have been very few experimental studies [5–7] of either the PSFs or the imaging properties of these masks to our knowledge.

This previous experimental and theoretical work does not clearly include the effects of the optical system. In many cases, the mask is inserted between an objective lens and an imaging lens where the beam is collimated [5–7]. In other cases, the mask is inserted at the exit pupil focal plane of a single lens system [1,2]. In many of the other papers, the optical system is not mentioned at all. As we will conclude, the optical system is critical to the success of these masks.

However the goal of an invariant PSF for the wavefront coding community has been demonstrated within the Airy beam community. Nondiffracting accelerated Airy and parabolic beams are solutions to the paraxial wave equation in Cartesian [8] and parabolic coordinates [9] whose shape remains constant over an extended distance. There are extensive experimental results [10–13] that demonstrate the properties of the Airy and parabolic beams where the appropriate masks are encoded onto spatial light modulators.

In the usual experimental configuration, an optical system takes the Fourier transform of this amplitude and phase mask. The required Fourier transform masks for these Airy and parabolic beams combine the cubic phase masks proposed by Dowski and Cathey with a Gaussian amplitude term and are written (in one-dimension) as $\exp \left(i\beta k_x^3\right)\exp \left(-a{\beta }^{2/3}{k}_x^2\right)$. Here $k_x$ is a normalized spatial frequency coordinate and the parameter a controls the Gaussian aperture function.

The accelerating beams are formed in the region past the focal plane of the Fourier transform lens. These beams show a transverse displacement that increases quadratically with the propagation distance z from the focal plane and decreases with β as

The range of these beams increases with β and is given roughly [14] by

In general, the size and nondiffracting range of these accelerating beams increase as the cubic phase parameter β increases. The nondiffracting range also increases as the Gaussian aperture parameter a decreases. The deflection decreases as both the cubic phase parameter and the focal length of the lens increase and it deflects in the opposite direction when the complex conjugate of the phase mask is used. Note that all of these results agree with the observations and comments regarding the wavefront coding masks. While amplitude and phase masks are required for the exact Airy beams, excellent experimental results for the Airy beams have been obtained with phase-only masks [13].

Because these Airy beams are nondiffracting solutions to the paraxial wave equation, they would appear to be the perfect solution to the wavefront coding problem as envisioned by Dowski and Cathey where a propagation invariant PSF is required. Consequently the optimal mask should be a cubic phase mask with a Gaussian amplitude function. It is important to repeat that this cubic phase mask represents the Fourier transform of the nondiffracting Airy beam. So the optical system must be capable of performing the perfect Fourier transform of the cubic phase mask.

More critically, the optical system must present a collimated flat phase profile (with an infinite radius of curvature R) at the output plane. Otherwise, both the range and deflection of the Airy and Parabolic beams are adversely affected [13,15] and Eq. (1) is modified as

Reference [13] showed experimental results where the deflection and range of these accelerating beams were severely affected by this radius of curvature.

So, those results form the major focus of this paper and stress that a major factor in the performance of these cubic phase masks for wavefront coding is their implementation into the optical imaging system. The existence of a quadratic phase profile at the output plane will dramatically limit the distance over which the PSF remains constant. This aspect of the wavefront coding problem has not been previously addressed to our knowledge.

2. Optical systems

Figure 1(a) shows the general optical system that is used for both the Airy and parabolic beams [10–13] and for some of the wavefront coding applications [5–7]. The object plane is located a distance $d_1=f_1$ from the first lens having a focal length of $f_1$ and is illuminated by a Gaussian beam having an infinite radius of curvature. The mask plane is located at a distance $d_A$ from the first lens and at a distance $d_B$ from a second objective lens having a focal length of $f_2$. The two lenses are separated by a distance $d_2=d_A+d_B$. The image plane is located at a distance $d_3=f_2$ from the second lens. For the Airy beams, the propagation is measured at distances z away from the image plane and depends on the focal length $f_2$ as in Eq. (1). The radius of curvature R of the beam at the output plane can be derived using the ABCD optical system [13,15] as

 

Fig. 1 Representative optical systems.

Download Full Size | PDF

In an attempt by reduce the length of the optical system [13], we tried making $d_B=0$ and $d_2=f_1$. The experimental results [13] showed that the output beam was adversely affected because of the radius of curvature ($R=f_2$) that was introduced. The size of the beam increased while the range of the beam decreased and the deflection also decreased. When this extra distance was programmed onto the cubic phase mask, the system worked as before. However these results showed the importance of the optical system to the accelerating beam community. We note that the previously mentioned experimental systems [5–7] seemed to have values of $d_2$ that would introduce finite radii of curvature at the output plane.

Figure 1(b) shows the single lens approach mentioned in some of the wavefront coding literature [1,2]. In this case, the mask is placed in the Fourier plane of the lens that coincides with the aperture-stop/exit pupil. The object plane is located a distance $d_1$ in front of the first lens having a focal length f and the image is formed at a distance $d_2$.

Again using the ABCD approach, the radius of curvature at the image plane is given as $R=d_2-f$. Assuming for simplicity that $d_1=d_2=2f$, the radius of curvature will be $R=f$ and should produce the same effects as the shortened system in [13]. Note that for this experimental setup, the output plane will always have a finite radius of curvature. It is often incorrectly assumed that the output plane contains the Fourier transform of the mask, but the quadratic phase is not seen in intensity. However the radius of curvature will affect the subsequent propagation of the beam.

3. Computational results

Figure 2 shows computational results comparing the results from three optical systems. For these computations, we used parameters similar to those used in our previous experiments [12,13] with 512x512 arrays of pixels having $42\mu m$ sizes illuminated with the argon laser wavelength of $0.514\mu m$. The cubic phase mask has a value of $\beta =243rad$.

 

Fig. 2 Computer simulated results showing output intensity. The left column uses the optical system of Fig. 1(a) where $d_2=f_1+f_2$. The center column uses the optical system of Fig. 1(a) where $d_2=f_1$. The right column uses the optical system of Fig. 1(b) where $d_1=d_2=2f$. The rows show results at distances of $d_3=\left\{2000,2400,2800,3200,3600\right\}mm$ from the image plane of the system.

Download Full Size | PDF

The left column shows the output intensity using the optical system from the Airy and parabolic beam communities where $d_2=f_1+f_2$. For this calculation, a collimated beam with an infinite radius of curvature was incident onto the amplitude and phase mask (we assumed that $d_A=d_B=f$). We then performed a Fresnel diffraction through a distance of $2000mm$, multiplied the result by a converging lens with a focal length of $2000mm$ and computed the Fresnel diffraction through distances of (each row) $d_3=\left\{2000,2400,2800,3200,3600\right\}mm$ from the lens. For the center column, we used the shortened system of [13] where$d_A=f$ and $d_B=0$. For this calculation, the collimated beam with an infinite radius of curvature was incident onto the converging lens with a focal length of $2000mm$. Again, we computed the Fresnel diffraction through distances of $d_3=\left\{2000,2400,2800,3200,3600\right\}mm$ from the lens. The right column shows the single lens imaging system for the case where$d_1=d_2=2f$. A diverging beam with a radius of curvature of $4000mm$ was incident onto the converging lens with a focal length of $2000mm$. We then computed the Fresnel diffraction pattern over a distance of $2000mm$ to the focal plane of the lens and multiplied this by the mask. Finally we computed the Fresnel diffraction through distances of $d_3=\left\{2000,2400,2800,3200,3600\right\}mm$ from the mask. This configuration allowed us to match the magnifications of the previous systems. In each case, Fig. 2 only shows a 64x64 pixel portion of the screen in order to see the results more clearly.

We see identical Airy beam patterns in the top row for all three systems at distances of 2000 mm corresponding to the image plane of the optical system. The left column shows results that are familiar to the Airy beam community. The PSF output remains uniform over the entire range and shows a quadratic shift. The center column shows the kinds of results seen experimentally [13] where the deflection decreases while the size of the PSF dramatically increases. The right column shows identical results as the center column, as expected because the incident beams onto the masks have the same radius of curvature. However most importantly, these results clearly show the effects of the optical system design on the propagation of the PSF. They emphasize the importance of the phase profile produced by the optical system at the image plane and clearly show that the conventional one lens imaging system will not perform as well as the optical system of Fig. 1(a).

4. Analysis and conclusions

Based on this analysis, we would expect improved performance for the wavefront coding experiments if the optical systems in [5–7] were designed as an exact Fourier transform system where $d_A=f_1$ and $d_B=f_2$. The alternate optical system where the mask is inserted at the focal plane of a single lens system [1,2] cannot satisfy the requirements for generating an infinite radius of curvature at the output plane and will result in a reduced distance over which the PSF remains invariant.

Next, we suggest that the zero order parabolic accelerating beam solutions [9,12,13] should be explored because of the extremely simple shape of the PSF.

In conclusion, we point out a connection between the wave front coding and the accelerating Airy beam communities. We conclude that the cubic phase mask is optimal because it provides a diffraction-free solution to the paraxial wave equation. However this mask demands that the optical system create a plane wave phase profile $(R=\infty )$ at the image plane in order to achieve a maximum distance for a distortion free PSF. We look forward to additional progress in the wavefront coding community as a result of this work.