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In this paper a new method of analysis for the far zone radiation pattern of the antenna with modulated aperture field is developed. The result of the conventional method is in terms of an infinite series of Bessel and sin functions and is not suitable for the prediction of sidelobes, beamwidth and the effect of various parameters. The purpose of our new method is to avoid the above disadvantages by writing the radiation pattern in simple closed form involving only the Anger function times a simple periodic modifying function. Using this method, the relationship between the aperture field parameters and the radiation pattern is clear. The type of modulation considered is cosinusoidal amplitude across the aperture and sinusoidal phase modulation across the aperture. The case of constant amplitude is discussed in detail and is applied to the modulated slow wave structure. The end of the visible region is chosen at the second peak of the total radiation function and the endfire pattern is considered in detail in terms of fundamental slow wave speed, and the magnitude, sign, and frequency of the phase modulation. Conditions are derived for excitation phase velocity, magnitude of phase modulation and the tilted main beam.