Propagation of cross-spectral density

In this subsection, let us review the propagation law of the angular spectrum and the four-dimensional Fourier transform of the cross-spectral density. With reference to Fig. 2.2, we assume a planar, quasi-monochromatic light source on a source plane $\Sigma_s$ at z = 0 and a wavefield propagating toward the right of the figure in free space. A reference plane $\Sigma _r$ is located at z=z_r parallel to $\Sigma_s$. Let a particular point P at a reference plane be specified by a three-dimensional position vector $\mbox{\boldmath$r$ } = (x,y,z)$. The origin of the coordinate system is set on $\Sigma_s$.

  

Figure 2.2: Geometry of the optical system.

For the wavefield propagation, the spectral amplitude $U(\mbox{\boldmath$r$ };\nu)$at a particular point on $\Sigma _r$ satisfies the Helmholtz equation

$$(\nabla^2 + k^2)U(\mbox{\boldmath$r$ };\nu) = 0 \:,$$ (2.37)

where $k = 2\pi\nu/c$ and c is the speed of light in free space. Since the z coordinate can be regarded as a parameter that expresses the location of each plane, we rewrite $\mbox{\boldmath$r$ } = (\mbox{\boldmath$r$ }_{\bot},z)$. Thus the wavefield at $\Sigma _r$ is also rewritten as $U(\mbox{\boldmath$r$ }_{\bot},z_r;\nu)$. At any location, the field $U(\mbox{\boldmath$r$ }_{\bot},z_r;\nu)$ may be represented by a Fourier integral:

$$U(\mbox{\boldmath$r$ }_{\bot},z_r;\nu) = \displaystyle\frac{... ... }_{\bot} \cdot \mbox{\boldmath$r$ }_{\bot}) d^2 k_{\bot} \:,$$ (2.38)

and obviously the correspondence is

$$\tilde{U}(\mbox{\boldmath$k$ }_{\bot},z_r;\nu) = \displaysty... ... }_{\bot} \cdot \mbox{\boldmath$r$ }_{\bot}) d^2 r_{\bot} \:,$$ (2.39)

where $\mbox{\boldmath$k$ } = (\mbox{\boldmath$k$ }_{\bot},k_z) = (k_x,k_y,k_z)$. Substitution of Eq. (2.38) into Eq. (2.37) gives a differential equation in $\tilde{U}(\mbox{\boldmath$k$ }_{\bot},z_r;\nu)$:

$$\left( \displaystyle\frac{\partial^2}{\partial z_r^2} + k_z^2 \right) \tilde{U}(\mbox{\boldmath$k$ }_{\bot},z_r;\nu) = 0 \:,$$ (2.40)

where

 

$$k_z = \left\{ \begin{array}{l} [k^2 - \mbox{\boldmath$k$ }_{\bot}... ..._{\bot}^2 - k^2]^{1/2} \hspace{5mm}(k_{\bot} \geq k) \\ \end{array}\right. \:.$$ (2.41)

Although it is seen from Eq. (2.41) that the optical waves with $k_{\bot} \geq k$ are evanescent, we will take only the propagating wave into account. A propagation law of the field traveling from $\Sigma_s$ to $\Sigma _r$ in the positive direction of z is obtained as a solution of Eq. (2.40)[78,79]:

$$\tilde{U}(\mbox{\boldmath$k$ }_{\bot},z_r;\nu) = \exp(ik_z z_r) \tilde{U}(\mbox{\boldmath$k$ }_{\bot},0;\nu) \:.$$ (2.42)

The cross-spectral density across $\Sigma _r$ is represented by a cross correlation of wavefields at two points $P(\mbox{\boldmath$r$ }_{\bot},z_r)$ and $P(\mbox{\boldmath$r$ }_{\bot}',z_r)$[1]:

$$W^{(z_r)}(\mbox{\boldmath$r$ }_{\bot}',\mbox{\boldmath$r$ }_{... ...',z_r;\nu) U(\mbox{\boldmath$r$ }_{\bot},z_r;\nu) \rangle \:.$$ (2.43)

The z coordinate in parentheses will be attached to W and $\tilde{W}$ to specify the plane considered in the following formulas. The four-dimensional Fourier transform of Eq. (2.43) gives a definition of the cross-spectral density in the $\mbox{\boldmath$k$ }_{\bot}$ domain[79]:

$$\tilde{W}^{(z_r)}(\mbox{\boldmath$k$ }_{\bot}',\mbox{\boldmat... ...ot\mbox{\boldmath$r$ }_{\bot}')] d^2r_{\bot} d^2r_{\bot}' \:.$$ (2.44)

From Eqs. (2.39), (2.43) and (2.44), $\tilde{W}^{(z_r)}(\mbox{\boldmath$k$ }_{\bot}',\mbox{\boldmath$k$ }_{\bot};\nu)$ can also be represented by

$$\tilde{W}^{(z_r)}(\mbox{\boldmath$k$ }_{\bot}',\mbox{\boldmat... ...u) \tilde{U}(\mbox{\boldmath$k$ }_{\bot},z_r;\nu) \rangle \:.$$ (2.45)

Substitution of Eq. (2.42) into Eq. (2.45) leads to the propagation law of the cross-spectral density in the Fourier domain from $\Sigma_s$ to $\Sigma _r$[78,79]:

$$\tilde{W}^{(z_r)}(\mbox{\boldmath$k$ }_{\bot}',\mbox{\boldmat... ...mbox{\boldmath$k$ }_{\bot}',\mbox{\boldmath$k$ }_{\bot};\nu) .$$ (2.46)

This expression for the propagation law of the cross-spectral density is simple and enables us to treat the propagation of the spatial coherence easily. In addition, it should be mentioned that the derivation of Eq. (2.46) does not require any paraxial approximations nor any assumptions on the state of coherence.