New near-field and far-field attenuation models for free-space variable optical attenuators

Novel closed-form attenuation models have been developed for free-space variable optical attenuators (VOAs)-one deals with the near-field condition when the two single-mode fibers are aligned very close to each other while the other deals with the far-field condition. In both models, the relationship between mirror (i.e., shutter) position and attenuation is represented by a constant term and another term in the form of an extended error function. The constant term determines the insertion loss, while the error function defines the shape of the attenuation curve. Compared with the conventional model that employs time-consuming numerical integrals, these two models show clearly the physical picture of the attenuation mechanism and provide closed-form expressions of attenuation versus mirror position. They are computationally efficient for attenuator design and optimization. Numerical calculation and experimental study have also been carried out to verify the attenuation models developed in this paper.