Imaging Systems Laboratory, Department of Electrical Engineering University of Colorado, Boulder, Colorado 80309 Aberration Invariant Optical/Digital Incoherent Optical Systems

Edward R. Dowski, Jr., W. Thomas Cathey, and Joseph van der Gracht Abstract Control of optical aberrations is a principal objective of any optical design. High performance optical designs with well-corrected aberrations typically have a very low tolerance to fabrication and alignment errors and are composed of very specific optical materials. Such systems are almost always more costly than equivalent systems with less well-corrected aberrations. By optimum combination of optical pre-processing and digital post-processing, or optical coding and digital decoding of the image information, incoherent optical systems invariant to numerous aberrations can be formed. The theory of aberration invariance can also be used with low-cost, low-precision optics to produce systems that image with the performance of high-cost, high-precision, or near diffraction-limited, spatial resolution. We can show that the central aberration to be controlled is second order, or misfocus. Systems that are invariant to misfocus are also invariant to chromatic aberration, astigmatism, thermal effects, and spherical aberration. This paper therefore describes, with experimental evidence, a focus-invariant optical system. Research The most common approach to approximating a focus-invariant imaging system is to stop down the pupil aperture. Although stopping down the aperture does increase the amount of focus invariance, or depth of field, there is an attendant loss in optical power at the image plane, as well as a reduction of the diffraction-limited image resolution. The aperture can be viewed as a simple absorptive mask in the pupil plane of an optical imaging system; however, this absorptive nature results in a loss of light. We have designed a nonabsorptive rectangularly separable phase mask that produces an optical point spread function (PSF) that is highly invariant to misfocus, and a corresponding optical transfer function (OTF) that has no regions of zeros within its passband [1]. 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 PSF is insensitive to misfocus, all values of misfocus can be restored through a single, fixed, digital filter. 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. The phase mask necessary to produce this focus-invariance is a simple cubic phase structure. One dimension of this mask is given by

The constant a controls the phase deviation of the mask. Our derivation of this mask relies on the ambiguity function representation of the OTF as a function of misfocus [2], and the method of stationary phase [3]. Focus-invariant systems can be recognized 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 focus-invariant qualities. We have produced a first-generation 12mm by 12mm cubic phase element via a continuous relief laser pattern generator [4]. This element was constructed for a design phase deviation of a = 20p at a wavelength of 632.8 nm. Due to the challenging nature of manufacturing nonrotationally symmetric aspheres, we exploited diffractive optics fabrication technology to produce a modulo one wavelength element of the form where mod(a,b) is a modulo b. Preliminary results of narrow-band imaging look promising. Figure 1 shows experimental results of misfocus imaging with a standard optical system compared with our cubic phase method. An off-the-shelf 8-bit video camera was used to produce these results. The digital filter was a simple least-squares inverse filter. The standard optical system is seen to be very sensitive to misfocus. The focus-invariant images, after simple digital filtering, are seen to be nearly invariant to misfocus, as desired. Future experiments will use 10 and 11 bit digital cameras and more sophisticated digital processing algorithms. In addition, we are looking at ways to fabricate a continuous phase mask, and experimentally exploring broadband performance and general aberrations-invariance of our cubic-phase method.

Figure 1: Experimental images from standard incoherent imaging system (a, b, & c,) and from focus-invariant optical/digital system (d, e, & f). (a,d) Geometrically in-focus, (b, e) small mis-focus, and (c,f) large mis-focus. Even at large misfocus, the focus-invariant system images are as clear as the standard system when infocus. References 1) R. Dowski, Jr. and W. T. Cathey, ``Extended Depth of Field Through Wavefront Coding,'' Applied Optics 34, 1859-1866 (1995). 2) K. Brenner, A. Lohmann, and J. Ojeda-Castañeda, ``The Ambiguity Function as a Polar Display of the OTF,'' Optics Communications 44, 323-326 (1983). 3) M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1989) . 4) J. van der Gracht, E. Dowski, W. T. Cathey, and J. P. Bowen, SPIE Proceedings on Novel Optical System Design and Optimization (San Diego, 1995) , Vol. 2537, pp. 279-288.