Use of Frequency Offset in Incoherent Optical Data Processing Albert Macovski

Electrical Engineering Department, Stanford Universi­ty, Stanford, California 94305. Received 18 April 1973. In a recent publication1 Lohmann described a new tech­

nique of major significance for optical data processing. It involved a moving diffuser and an incoherent Fourier transform system for achieving real-time operation. I should like to suggest two additional features that might help this technique become even more useful in optical data processing applications.

First, incoherent techniques, as Lohmann has pointed out, are plagued with a large bias term that is dependent on the number of data points used. Second, existing op­tical data processing systems, including the one described by Lohmann, require that the input data be in the form of a transparency. A method is described in this letter where the input information can be that of a diffusely re­flecting surface.

In one of the systems described by Lohmann, two trans­parencies separated by a distance d, a(x0,y0) and b(x0 — d,ýo) are placed at the back focal plane of a lens.2 The amplitude at the front focal plane is given by

where A and B are the Fourier transforms of a and b. This amplitude pattern is applied to a moving diffuser.1

The resultant incoherent intensity pattern is then, in real time, applied to an incoherent Fourier transform system such as a Wavefront Folding Interferometer3 or one of a number of other two-path interferometry systems.4 The resultant intensity pattern, caused by the interference of each point only with itself is given by

Thus the desired cross-correlation terms appear on either side of the axis with the sum of the autocorrelation terms at the center.

The major problem is the poor contrast ratio, C1/C0, especially for the cases of relatively complex patterns. This can be greatly alleviated by inserting a dynamic op­tical phase or frequency shifter5'6 in one of the two paths of the incoherent Fourier transform interferometer creat­ing a final intensity pattern given by

where α is the phase shift between paths. Thus the de­sired C1 term of the output becomes modulated by cosα. If a frequency shifter is used, such as a sonic Bragg cell driven at a frequency ω, the desired term becomes modu­lated by cosωt and is thus electrically separable.7 The output intensity measuring device can be an array of pho-todiodes or an image dissector television camera8 with their outputs followed by a bandpass filter centered at ω, which rejects the large bias term and passes only the de-

August 1973 / Vol.12, No. 8 / APPLIED OPTICS 1745

sired signal. Of course the photodetector itself will have to deal with the poor contrast ratio due to the large bias term. However, the resultant signal-to-noise ratio of the filtered output becomes a function of the amount of light and the processing or scanning time. These can be readi­ly controlled in a data processing environment to provide almost any arbitrary level. In incoherent systems with­out frequency offset, the signal-to-noise ratio is governed solely by the data themselves.

Another alternative is to alternately use two static phases for α that are 180° apart so the desired C1 term reverses in polarity. A storage device, such as a storage tube or magnetic video disk, is used to store the first frame so it can be subtracted from the second frame to form the desired output. Of course the camera and stor­age devices must have sufficient dynamic range to accom­modate the large bias term.

In applications where cross-correlation peaks are antici­pated in specific regions, an array of photocells followed by filters at those regions would appear to be a good solu­tion. In this case, frequency offset, as previously de­scribed, would be used.

If desired, the cross-correlation terms can be made elec­trically separable from the autocorrelation terms by in­serting an additional phase or frequency shifter in the path of the a or b transparency.9 For example, if a fre­quency shifter at ω1 is put in that path, and an additional frequency shifter of ω2 is put in the incoherent interferom­eter, only the desired cross-correlation terms will be mod­ulated by cos(ω1 ± ω2)t. Thus a filter tuned to the sum and/or difference frequency will extract only the cross-correlation terms.

The second general method deals with the use of source material consisting of a diffuse reflector. As shown in Fig. 1, an incoherent intensity pattern is derived from the diffuse surface. This can either be accomplished by illu­mination with incoherent light or, as shown, imaging the surface onto a moving diffuser. A first incoherent inter­ferometer with a phase or frequency shifter α1 is used to create a Fourier transform of Ia(x0,y0), the intensity pat­tern of the diffuse reflector. Thus the intensity at the second moving diffuser is given by

where Ia is the Fourier transform of Ia. This intensity pattern is applied through a transparency having an in­tensity transmission τb. This is a Vander Lugt filter10

having the Fourier transform of the comparison image on a spatial frequency carrier. Thus τb is given by

Fig. 1. Real-time optical data processing system.

1746 APPLIED OPTICS / Vol. 12, No. 8 / August 1973

where, m is the frequency of the spatial frequency carrier, and B is the Fourier transform of the reference image B. For example, B might be the Fourier transform of an array of alphanumeric characters used to decode a docu­ment Ia. The product of I(u,υ) and τb is applied to the second incoherent interferometer providing a final intensi­ty pattern given by

where F is the Fourier transform operator.

If, as before, we use frequency shifters where α1 = ω1t and α2 = ω2t, the desired terms can be readily separated. For example, if the detector or camera output signals are fil­tered at the sum (or difference) frequency, the resultant reconstruction will consist of the original Ia image at the center with the desired cross correlation and convolution centered at x = ±m. If the B images are desired for ref­erence purposes, an output filter centered at ω2 can be used. A number of variations on this theme can be used.

References 1. A. W. Lohmann, Opt. Commun. 3, 73 (1971). 2. C. S. Weaver and J. W. Goodman, Appl. Opt. 5, 1248 (1966). 3. L. Mertz, Transformations in Optics 2 (Wiley, New York,

1965), Chapt. 4. 4. G. Cochran, J. Opt Soc. Am. 56, 1513 (1966). 5. A. Macovski, Ph.D. Thesis, University Microfilm, Order No.

69-258,(1968). 6. A. Kozma and N. Massey, Appl. Opt. 8, 393 (1969). 7. A. Macovski, Appl. Phys. Lett. 16, 166 (1969). 8. D. Fink, Television Engineering (McGraw-Hill, New York,

1942), p. 95. 9. A. Macovski and S. D. Ramsey, Opt. Commun. 4, 319 (1971).

10. A. VanderLugt, IEEE Trans. Inf. Theory IT10, 139 (1964).