Extending the depth of field of incoherent optical systems has been an
active research topic for many years. The majority of the literature on
this topic has concerned methods of employing an optical power absorbing apodizer, with possible
phase variations, on a standard incoherent optical system as a means to increase the
depth of field [5][4][3][2][1]. These methods have all suffered from two
significant
deficiencies; a decrease of optical power at the image plane, and a decrease of
image resolution. A unique method of achieving extended depth of field
without an apodizer [6] was described in 1972. The major
shortcoming of this method is that focus must be varied during exposure.
We describe a novel method for extending the depth of field of incoherent optical systems that does not suffer from the significant deficiencies of earlier methods. Our method employs a phase mask to modify the incoherent optical system in such a way that the point spread function (PSF) is insensitive to misfocus, while forming an optical transfer function (OTF) that has no regions of zero values within its passband. The PSF of the modified optical system is not directly comparable to that produced from a diffraction limited PSF. However, because the OTF has no regions of zeros, digital processing can be used to ``restore'' the sampled intermediate image. Further, because the OTF is insensitive to misfocus, the same digital processing restores the image for all values of misfocus. This combined optical/digital system produces a PSF that is comparable to that of the diffraction limited PSF, but over a far larger region of focus. We term the general process of modifying the incoherent optical system and received incoherent wavefront, by means of a phase mask, wavefront coding. By modifying only the phase of the received wavefront, general wavefront coding techniques maximize optical power at the image plane.
When designing extended depth of field systems, we make two main assumptions. The first is that the incoherent optical system is being modified by a rectangularly separable phase mask. This leads to a rectangularly separable PSF and OTF. Secondly, we assume that any resulting image will be an ``intermediate image''. This intermediate image will require digital processing. This second assumption follows from our belief that best performance is obtained by optimum preprocessing through optics, followed by optimum digital postprocessing [7]. These pre- and post processing stages are optimum in the sense that each is ``matched'' to the other in order to solve an interesting problem.
Our solution to extended depth of field systems relies on the theory of the ambiguity function [10][9][8], and the method of stationary phase [13][12][11]. The ambiguity function can be used as a polar display of the OTFs of a rectangularly separable incoherent optical system as a function of misfocus [10]. Extended depth of field systems can be noticed almost by inspection of their corresponding ambiguity functions. The method of stationary phase allows the design of phase masks whose corresponding ambiguity functions have desired extended depth of field qualities.
In the following section we outline this method of designing phase masks for extended depth of field systems. This design leads to the cubic-pm mask, or cubic phase modulation mask, which is introduced in section 3. Through simulated measurement of the width of the combined optical/digital PSF, as well as through simulations of imaging a spoke target, we show that this method can produce an incoherent optical system with a large depth of field with near diffraction limited imaging performance. A complete mathematical derivation of the cubic-pm mask function, based on the method of stationary phase, can be found in Appendix A.