A laser microphone has recently been focused on acquiring the distant target speech without unnecessary sounds. It measures the vibration of the object near the target sound source by irradiating the object surface with the laser beam. However, the speech acquired with this microphone degrades the sound quality. For instance, the speech components at higher frequencies are attenuated by vibration characteristics of the object, and the speech is collapsed by stationary noise due to lower power of the laser beam from the object. To improve the sound quality of the degraded speech, deep neural network (DNN) has recently been proposed. It is trained by using a set of acoustic features extracted from the degraded speech and the clean speech. However, the speech components at higher frequencies are still attenuated after processed by DNN. Therefore, we propose the method to reconstruct the harmonic structure of the speech after processed by DNN. The notice point is that the harmonic structure composes the higher frequencies of the speech. The proposed method complements the spectral amplitudes on positive integer multiples of the fundamental frequency which are the components at harmonics. We evaluated the effectiveness of the proposed method though perceptual evaluation of speech quality.