Collins formulae in both space and frequency domains for ABCD optical systems with small deformations

Generalized Collins formulae for arbitrary imperfect ABCD optical systems with small deformations (misalignments and/or deviations from ideal optical operations) are obtained in both the space domain and the frequency domain. These formulae can provide a unified way to analyse the performance of a practical optical system in both domains, including the ideal as its special case. Particularly, it shows that a reciprocally symmetrical ABCD system with small deformations can implement so-called almost-fractional Fourier transformation (FRT) simultaneously in both domains with the same order. Some other applications in practice are also discussed to verify the effectiveness of our proposed method.