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Waveguide-based optical parametric amplification (OPA) is advantageous in generating broadband or pulsed squeezed vacuum. To enhance the squeezing level, it is crucial to clarify its limiting factors. In this study, we develop a numerical method which allows us to derive the squeezing level in waveguide OPA. Based on the developed method, we investigate the effect of the higher-order mode of the pump light, which generates noise due to the OPA between the fundamental mode and the higher-order mode. We also investigate the effect of optical loss in the waveguide. Specifically, we formulated coupled equations that take into account the degenerate OPA between the pump and signal both in the fundamental mode, and the non-degenerate OPA between the pump in the higher-order mode, the signal in the fundamental mode, and the idler in the higher-order mode. By solving the coupled equations, we can express the quantum state as a linear combination of annihilation and creation operators, and the squeezing level can be derived from its coefficients. The analysis allowed us to quantify the effect of OPA on the fundamental and higher-order modes, and the requirement for mode matching of pump light with the fundamental mode. We also analyzed the effect of loss in the waveguide based on the same procedure. The results show the loss in the waveguide to have less effect than the same loss at the output of the waveguide. These results provide important design criteria of waveguide OPA for squeezing.
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