The Fourier transform is commonly used in interferogram analysis. However, owing to the truncation effect, the direct application of Fourier transform for sampled data tends to cause spectral leakage. To address this problem, the apodization function is often adopted. In this paper, we first analyze the performances of different interferogram apodization functions. Then, the effects of the mainlobe width and sidelobe attenuation of the apodization functions on spectral leakage are studied. Finally, we propose an improved triangular window apodization function based on the zero-order Bessel function. It can accelerate the sidelobe attenuation by weighing the triangular window function. Results from extensive experiments show that the improved triangular window apodization function can effectively suppress energy leakage. When compared to the triangular window, this study suggests that the proposed method improves the mean of the peak-to-peak signal-to-noise ratio by 4.9% and the root mean square signal-to-noise ratio by 3.5%. In addition, these results are more accurate than that achieved using the Blackman window. Further, the mainlobe width of the improved triangular window apodization function is 0.043π, which is closed to that of the Hanning window. Therefore, our proposed method proffers a good frequency resolution.