Relation between Fresnel transform of input light field and Radon transform of Wigner function of the field

We prove a new theorem about the relationship between optical field Wigner function’s Radon transform and optical Fresnel transform of the field, i.e., when an input field ψ (x′) propagates through an optical [D (−B) (−C)A] system, the energy density of the output field is equal to the Radon transform of the Wigner function of the input field, where the Radon transform parameters are D,B. We prove this theorem in both spatial-domain and frequencydomain. In optical communication theory every signal or image can be uniquely and indirectly described by a Wigner distribution function (WDF) [1, 2, 3]. The WDF (or named Wigner transform) of an optical signal field ψ (x′) is defined as Wψ(ν , x) = ∫ +∞ −∞ du 2π e ′uψ∗ ( x + u 2 ) ψ ( x − u 2 ) , (1) Wψ(ν ′, x′) involves both spatial distribution information and space-frequency distribution information of the signal. ν is named space frequency. Wψ(ν ′, x′) is said to be bilinear in the signal because the signal enters twice in its definition. The WDF undergoes certain variations if something happens to the signal. For examples, passage through a lens corresponds to a vertical shearing of the WDF, propagation in free space means a horizontal shearing of the WDF [4]. However, the WDF preserves space and space frequency marginal properties of any signal, ∫ +∞ −∞ dνWψ(ν , x) = |ψ (x) |, (2) ∫ +∞ −∞ dxWψ(ν , x) = |ψ̃ (ν) |, (3) where ψ̃ (ν) = ∫ +∞ −∞ dx √ 2π ψ (x) e . If one wants to reconstruct the Wigner function by using various probability distribution, obviously the position density |ψ (x′) | and the space-frequency density |ψ̃ (ν′) | were not enough, so the Radon transform [5, 6] of the Wigner function is introduced [7], R (x) ≡ ∞ ∫∫ −∞ dxdνδ (x−Dx +Bν)Wψ(ν, x), (4) R (x) is also a probability distribution along an infinitely thin phase space strip denoted by the real parameters D,B. The inverse relation of (4) is the foundation of optical tomographic imaging techniques (the techniques derive two-dimensional data from a three-dimensional object to obtain a slice image of the internal structure and thus have the ability to peer inside the object noninvasively.)