Multidimensional detection enabled by twisted black arsenic–phosphorus homojunctions

Supplementary Note 4. Reconstruction of polarization angle and intensity information using the double twisted b-AsP homojunctions The relationship between responsivity (R) of HJ-1 and HJ-1 and polarization angle () can be described by the following equations: 𝐼𝐷𝑆,1 𝑃 = 𝑅1 = 𝐴cos (2𝜃− 𝜋 4) + 𝐵 (1) 𝐼𝐷𝑆,2 𝑃 = 𝑅2 = 𝐶cos (2𝜃+ 𝜋 4) + 𝐷 (2) where IDS,1 and IDS,2 are the measured photocurrent of HJ-1 and HJ-2 under MIR illumination, and R1 and R2 are the responsivity of HJ-1 and HJ-2, respectively. P is the incident intensity. A, B, C, and D are constants that can be obtained by fitting the polarization angle-dependent responsivity, which are listed in Supplementary Table 1. By combining equations (1) and (2), we can get the following equations: 𝑎cos 2𝜃+𝑏sin 2𝜃= 𝑐 (3) 𝑎= √2 2 (𝐴𝐼𝐷𝑆,2 −𝐶𝐼𝐷𝑆,1) (4) 𝑏= √2 2 (𝐴𝐼𝐷𝑆,2 + 𝐶𝐼𝐷𝑆,1) (5) 𝑐= 𝐷𝐼𝐷𝑆,1 −𝐵𝐼𝐷𝑆,2 (6) The following new equations can be obtained by transforming the relationship of trigonometric functions: (𝑐+ 𝑎)(tan 𝜃)2 −2𝑏tan 𝜃+ 𝑐−𝑎= 0 (7) Based on the equation (7), the polarization angle () can be obtained as follows: θ = tan−1 𝑏+√𝑏2−𝑐2+𝑎2 𝑐+𝑎 (8) or θ = tan−1 𝑏−√𝑏2−𝑐2+𝑎2 𝑐+𝑎 (9) Two  values will be obtained, but only one of the two values is the correct polarization angle. To get this correct polarization angle, we can plug these two values into the equation (1) or (2) and solve for the power value. According to our calculations, the solved power values will be a positive value and a negative value. This positive intensity value (P) corresponds to the correct polarization angle (). Based on these, the