September 15, 1998 / Vol. 23, No. 18 / OPTICS LETTERS 1477Information, resolution, and space–bandwidth productMark A. NeifeldDepartment of Electrical and Computer Engineering, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721Received June 11, 1998The information capacities of two-dimensional optical low-pass channels are discussed. Coherent andincoherent systems operating under finite optical power and area constraints are characterized in terms of twocriteria: space–bandwidth product (SBP; the number of pixels required for achieving maximum informationcapacity) and resolution (Gmin; the smallest spot size capable of supporting positive capacity gain). A coherentsystem operating with an initial signal-to-noise ratio (SNR) of 5 can achieve a 48% capacity gain by operatingat an optimal SBP that is 3.4 times that of the nominal system. The same system has a resolution that is 0.31times nominal. Incoherent systems experience additional SNR loss, and with an initial SNR of 5 they achievecapacity gains of 29% at the optimal SBP of 2.8 times nominal. The incoherent system resolution is found tobe 0.4 times nominal.  1998 Optical Society of AmericaOCIS codes: 350.5730, 200.3050, 070.2580, 110.4280, 100.3020, 100.6640.Optical resolution is commonly defined in terms of ei-ther the Rayleigh or the Sparrow criteria.1,2Thesetraditional definitions are important because theyoffer convenience for visual or heuristic purposes; how-ever, they do not offer significant insight into questionsof reliability or fidelity. Severe blur, for example,does not necessarily prohibit high fidelity, as evidencedby various inverse methods and superresolution tech-niques whose performance is ultimately limited bynoise.2,3A meaningful definition of resolution shouldtherefore depend on the signal-to-noise ratio (SNR). Asimilar argument can be applied to another importantoptical system metric, the space–bandwidth product(SBP). SBP is often taken to represent the infor-mation content of an optical image and is commonlydefined with respect to the (traditional) system reso-lution sGd as SBP  AyG2, where A is the total imagearea.4Although this definition can be useful for somepurposes, it does not appear to be adequate as a mea-sure of information capacity, primarily because it doesnot incorporate the effects of noise.The fidelity of images captured through optical chan-nels is influenced by many factors, such as power loss,interpixel interference, and detector bandwidth. In-formation theory offers a framework for understand-ing the effects of these various factors, in particularthe relationship between SNR and the information ca-pacity of a system.5Such an information-theoreticalapproach is useful in establishing bounds on the ca-pabilities of optical low-pass systems. The resultingbounds are important for two reasons. First, owingto the ubiquity of optical low-pass systems, these in-formation capacity bounds will determine the ulti-mate performance limitations within a wide varietyof application domains. These bounds can be appliedto optical systems for storage, computing, signal pro-cessing, and communications. Second, owing to recentadvances within the traditional communications com-munity, techniques now exist for approaching theseinformation-theoretical bounds in serial channels (e.g.,turbo codes and likelihood-based methods), and simi-lar techniques can be adapted for highly parallel two-dimensional (2D) optical channels.6This Letter introduces fidelity-based definitionsof both resolution and SBP and establishes theinformation-theoretical capacities of coherent andincoherent optical systems. The total informationcapacity of a pixelated 2D optical imaging systemsCTd is defined as the capacity of a single pixel5fCp1/2 log2s1 1 SNRdg times the number of pixelssNd, where SNR is defined in the electrical signal(e.g., current) domain or equivalently the opticalpower domain. Given the constraint of constant totaloptical power, the SNR per pixel must decrease as thenumber of pixels is increased sSNR ~ 1yNd. Giventhe additional constraint of constant image area, anincrease in the number of pixels must be accompaniedby a decrease in pixel area G2sG2~ 1yNd. Thisreduction in pixel area also causes the SNR per pixelto decrease by virtue of additional optical power fallingoutside the fixed passband of the system. Becauseinformation capacity is a monotonically increasingfunction of SNR, there is a SNR-versus-G trade-offthat can be quantified as CT NCp. N increases asG decreases; however, SNR decreases with decreasingG owing to the two sources of power loss indicatedabove. This SNR cost causes Cpto decrease, givingrise to an optimum value of G sGoptd at which CTis maximized. The information-theoretical SBP ofan optical system can therefore be defined as thenumber of pixels required for maximizing total capac-ity so that SBP  Nopt AyGopt2. From the sameSNR-versus-G trade-off it is also possible to defineinformation-theoretical resolution as the smallestvalue of G sGmind for which the increase in N remainsgreater than the corresponding loss in Cp. Noticethat Gminneed not equal Gopt. The remainder ofthis Letter will discuss this capacity optimization indetail, along with the characteristics of the resultinginformation-theoretical SBP and Gmin.Both the input and the detector planes of the opti-cal low-pass system studied here comprise 2D arraysof square pixels, and the system point-spread functionwill be defined by a square Fourier plane aperture. Ibegin with the case of coherent illumination, for whichthe nominal system will be defined by a normalized0146-9592/98/181477-03$15.00/0  1998 Optical Society of America1478 OPTICS LETTERS / Vol. 23, No. 18 / September 15, 1998pixel size G  1, such that only the main lobe of thefield spectrum of each pixel is passed by the Fourierplane aperture. It is assumed that the detector ar-ray is dominated by additive white Gaussian noise andthat the various system parameters (input power, de-tector bandwidth, receiver electronics, etc.) have beenselected so that the nominal system operates at somespecified value of SNR. Note that this initial SNR in-cludes the effects of spatial light modulator contrastand throughput, power loss from higher diffracted or-ders, scattering loss, detector responsivity, noise, fillfactor, and many other loss and fidelity corruptingmechanisms. These effects will be assumed to remainunchanged. In particular, the