Using the convolution form of the Fresnel diffraction integral (cf. Fig. 2), we can represent the signal complex field $US(x2,y2)$ incident on the FPA as Display Formula
$US(x2,y2)=ejkfj\lambda f\u222b\u2212\u221e\u221e\u222b\u2212\u221e\u221eUS+(x1,y1)exp{jk2f[(x2\u2212x1)2+(y2\u2212y1)2]}dx1\u2009dy1,$(21)
where $US+(x1,y1)$ is the signal complex field leaving the exit-pupil plane. Specifically, Display Formula$US+(x1,y1)=US\u2212(x1,y1)TP(x1,y1),$(22)
where $US\u2212(x1,y1)$ is the signal complex field incident on the exit-pupil plane, and Display Formula$TP(x1,y1)=cyl(x12+y12D1)exp[\u2212jk2f(x12+y12)]$(23)
is the complex transmittance function of the exit-pupil plane (i.e., a circular aperture placed against a thin lens). In Eq. (23), Display Formula$cyl(\rho 1)={10.500\u2264\rho 1<0.5\rho 1=0.5\rho 1>0.5$(24)
is a cylinder function where $\rho 1=x12+y12$, $D1$ is the exit-pupil diameter, $k=2\pi /\lambda $ is the angular wavenumber, $\lambda $ is the wavelength, and $f$ is the focal length. Substituting Eq. (22) into Eq. (21) we arrive at the following result: Display Formula$US(x2,y2)=ejkfj\lambda f\u2009exp[jk2f(x22+y22)]\u2062F{US(x1,y1)}\nu x=x2\lambda f,\nu y=y2\lambda f,$(25)
where Display Formula$US(x1,y1)=US\u2212(x1,y1)cyl(x12+y12D1)$(26)
is the signal complex field that exists in the exit-pupil plane of the imaging system (cf. Fig. 2), and $F{\u2218}vx,vy$ denotes a 2-D Fourier transformation, such that Display Formula$V\u02dc(\nu x,\nu y)=F{V(x,y)}\nu x,\nu y=\u222b\u2212\u221e\u221e\u222b\u2212\u221e\u221eV(x,y)e\u2212j2\pi (x\nu x+y\nu y)dx\u2009dy.$(27)
A 2-D inverse Fourier transformation then follows as Display Formula$V(x,y)=F\u22121{V\u02dc(\nu x,\nu y)}x,y=\u222b\u2212\u221e\u221e\u222b\u2212\u221e\u221eV\u02dc(\nu x,\nu y)ej2\pi (x\nu x+y\nu y)d\nu x\u2009d\nu y.$(28)
With Fig. 2 in mind, we can also represent the reference complex field $UR(x2,y2)$ incident on the FPA as resulting from the Fresnel approximation to a tilted spherical wave. Here, Display Formula$UR(x2,y2)=AR\u2009exp[jk2f(x22+y22)]exp(\u2212j2\pi xRx2\lambda f)exp(\u2212j2\pi yRy2\lambda f),$(29)
where $AR$ is a complex constant and $(xR,yR)$ are the coordinates of the off-axis local oscillator, which is located in the exit-pupil plane.