Currently, there are two primary approaches to addressing nonlinear electromagnetic inverse scattering problems. One method involves linearizing these issues, while the other treats them as iterative optimization problems. However, a significant challenge arises in real-world applications, where achieving uniform antenna coverage around the target proves to be exceptionally difficult. To surmount this challenge, limited aperture imaging methods have been proposed. Although limited aperture imaging provides more significant flexibility, it amplifies the nonlinearity of electromagnetic inverse scattering problems, resulting in relatively limited research on this front.This paper introduces a novel method founded on the limited aperture Rytov integral approximation for the purpose of quantitative inversion imaging of high relative permittivity and large-sized target objects. That is, the antennas are distributed only on one side of the region of interest or on some specific angles; limited aperture imaging provides greater flexibility. To begin, we introduce the concept of the phaseless Rytov approximation and meticulously analyze the complex refractive index within low-loss media. By analyzing the reflection and transmission of the incident field from the free space to the lossy media, the concept of effective refractive index is introduced and combined with Snell's theorem to solve the relationship between the effective refractive index and the actual refractive index, and the relationship between the actual refractive index and the dielectric constant is established according to the relationship between the dielectric constant and the refractive index in the low-loss media. Under the conditions of high frequency and low loss, we estimate the contrast function by taking into account the interplay between the effective refractive index, the actual refractive index, and the dielectric constant. We then employ mathematical corrections to approximate the scattered field and its gradient within the scattering object, thereby enhancing the traditional Rytov approximation. This enhancement results in the development of a phaseless limited aperture Rytov integral approximation model.In the simulation section, three different scattering objects with varying strengths are selected, namely weak scatterers, medium scatterers, and strong scatterers. This model is capable of providing quantitatively better reconstruction results for weak and medium scatterers with different shapes, characterized by high permittivity and large target size. For strong scatterers, it accurately reconstructs the target shape by considering the imaginary component of the contrast function. Additionally, for multi-target scenarios, both medium and strong scatterers are well-reconstructed in terms of object shapes. Furthermore, the model also yields favorable results when altering the operating frequency and antenna placement layout. Lastly, this model exhibits strong noise resistance capabilities.The proposed method is expected to be widely used in medical imaging, non-destructive testing and ground penetrating radar.