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This paper looks at a particular form of optical processing, namely a form of cross-correlation, and demonstrates how the method measures certain beam profile features of a laser pulse. Beam profile is defined to mean a description of the electromagnetic field of a laser pulse in space and time. In this paper, I represent the laser pulse as a complete set of orthogonal modes and show that an appropriate spatial filter and a measurement system can provide information about the beam profile of the laser in terms of the individual eigenfunctions of this representation. First I trace what happens when a laser pulse is modified by the spatial filter. I then do a specific example which looks at the TEMOO laser beam pulse with beam tilt, beam curvature, beam width, and beam shift to show that these effects produce higher order Hermite modes in the measurement system. The spatial filter modifies the electric field distribution in the focal plane such that at known spatial locations, the magnitude of the intensity is proportional to the pulse power or energy in particular Hermite modes. Since the size of these locations is infinitesimal (without getting errors from the electromagnetic fields from other modes), I demonstrate the effect and errors associated with using finite size detectors for measuring the magnitude of the intensity at these locations. The purpose of this paper is to demonstrate the concept of using optical processing to measure laser beam profile. Hermite modes are used because they are similar to many actual laser beam profiles and because they can be simply expressed in analytical form which is convenient for a theoretical presentation. In practice it is probably desirable to choose a set of modes for a basis which more closely represents the actual characteristics of the laser beam. This choice of course determines the properties of the spatial filter. Subject to reasonable assumptions shown in the paper, it is possible to use the optical unit with nine detectors to measure the beam profile properties of a laser beam. The detectors, each with high speed electronics can be chosen to provide measurement speeds in the nanosecond range. Two of the detectors for each axis provide primary information about beam tilt and beam shift, two detectors for each axis give secondary information about beam shift and beam width, and one detector gives the power in the primary mode (the TEMOO mode for the example in this paper). Five detectors are located at selected points in the focal plane of the lens and four are located just before the spatial filter. Detectors of five micrometer diameter give a 23 percent ratio of desired to undesired signal and implies a lower limit in beam control of +/- 30 microradians. Some discussion is presented about reducing these values. The use of nine detectors applies to a single mode. More detectors are required if the measurement is to apply to multimode lasers. Also one must be careful to choose the most efficient basis for the particular measurement of interest. The optimum basis would be sensitive to the key features of interest and suppress the features of no interest.
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