Van Cittert-Zernike theorem in space-frequency domain

The van Cittert-Zernike theorem describes the spatial coherence function that propagates from the spatially incoherent planer source. The optical system is shown in Fig. 2.1. The mutual coherence function at the two points Q₁ and Q₂ on the observation plane is given by

$$J(Q_1, Q_2) = \left(\displaystyle\frac{\bar{\nu}}{cz}\right)... ...\boldmath$\rho$ }_2) \cdot \mbox{\boldmath$r$ } \right] dr \:,$$ (2.20)

  

Figure 2.1: Optical system considered.

where $\bar{\nu}$ is the central frequency of the quasi-monochromatic source, $\psi = \pi\bar{\nu}(\rho_1^2 - \rho_2^2)$, and $I_P(\mbox{\boldmath$r$ })$is the intensity at $P(\mbox{\boldmath$r$ })$ on the source plane. In this subsection, let us consider the relationship between the spectral density distribution and the spectral coherence in the paraxial far field. The paraxial far field implies the approximation $s_l \simeq z + \vert\mbox{\boldmath$r$ } \cdot \mbox{\boldmath$\rho$ }_l\vert / z \: (l = 1,2)$.[77]

With reference to Fig. 2.1, we assume that the spectrum of the source is broad, the spectral profile $S_P(\mbox{\boldmath$r$ }; \nu)$ depends on the position of the source plane, and the observation plane is located in the far field. The analytic signals $V_P(\mbox{\boldmath$r$ };t)$ and $V_Q(\mbox{\boldmath$\rho$ };t)$ at the source plane and the observation plane are expressed by the Fourier integral

$$V_P(\mbox{\boldmath$r$ };t) = \displaystyle\int^{\infty}_{0} U_P(\mbox{\boldmath$r$ };\nu)\exp(-2\pi i\nu t)d\nu \:,$$ (2.21)

$$V_Q(\mbox{\boldmath$\rho$ };t) = \displaystyle\int^{\infty}_{0} U_Q(\mbox{\boldmath$\rho$ };\nu)\exp(-2\pi i\nu t)d\nu \:.$$ (2.22)

By using the impulse response $h(\mbox{\boldmath$r$ },\mbox{\boldmath$\rho$ };t)$ from $P(\mbox{\boldmath$r$ })$ to $Q(\mbox{\boldmath$\rho$ })$, the analytic signal at $Q(\mbox{\boldmath$\rho$ })$is represented by the convolution integral

$$V_Q(\mbox{\boldmath$\rho$ };t) = \displaystyle\int\!\!\!\dis... ...boldmath$\rho$ };t-t') V_P(\mbox{\boldmath$\rho$ };t')dt' \:.$$ (2.23)

By introducing the transmission function $H(\mbox{\boldmath$r$ },\mbox{\boldmath$\rho$ };\nu)$ in the space-frequency domain, $h(\mbox{\boldmath$r$ },\mbox{\boldmath$\rho$ };t)$ and $H(\mbox{\boldmath$r$ },\mbox{\boldmath$\rho$ };\nu)$ are related by the Fourier transform relationship:

$$h(\mbox{\boldmath$r$ }, \mbox{\boldmath$\rho$ };t) = \displa... ...$r$ },\mbox{\boldmath$\rho$ };\nu) \exp(-2\pi i\nu t) d\nu \:.$$ (2.24)

Under the paraxial approximation, $H(\mbox{\boldmath$r$ },\mbox{\boldmath$\rho$ };\nu)$ can be represented by

$$H(\mbox{\boldmath$r$ },\mbox{\boldmath$\rho$ };\nu) = -\disp... ...tyle\frac{2\pi i\nu s_l}{c}\right) \hspace{5mm} (l = 1, 2)\:.$$ (2.25)

Substituting Eqs. (2.21) and (2.24) into Eq. (2.23) gives

$$V_Q(\mbox{\boldmath$\rho$ };t) = \displaystyle\int\!\!\!\dis... ...\nu) U_P(\mbox{\boldmath$r$ };\nu) \exp(-2\pi i\nu t)d\nu \:.$$ (2.26)

Next, let us consider the mutual coherence function defined by

$$\Gamma_Q(\mbox{\boldmath$\rho$ }_1, \mbox{\boldmath$\rho$ }_2... ...h$\rho$ };t) V_Q(\mbox{\boldmath$\rho$ }_2;t+\tau) \rangle \:.$$ (2.27)

By using Eq. (2.26) and the cross-spectral density across the source plane

$$W_P(\mbox{\boldmath$r$ }', \mbox{\boldmath$r$ };\nu) = \lang... ...boldmath$r$ }', \nu) U_P(\mbox{\boldmath$r$ };\nu) \rangle \:,$$ (2.28)

we obtain

 

$$\Gamma_Q(\mbox{\boldmath$\rho$ }_1,\mbox{\boldmath$\rho$ }_2;\tau) \displaystyle\int\!\!\!\displaystyle\int\!\!\!\displaystyle\int\!\!\!\displaystyle\int^{\infty}_{-\infty} d^2r d^2r'$$ =

$$\times \displaystyle\int^{\infty}_{0} W_P(\mbox{\boldmath$r$ }',\mbox{\b... ...mbox{\boldmath$r$ },\mbox{\boldmath$\rho$ }_2;\nu) \exp(-2\pi i\nu\tau)d\nu \:.$$ (2.29)

Note that Eq. (2.29) holds without any restriction on the source correlation. Let us then consider the spatially incoherent source in the following. The cross-spectral density is thus represented by using the spectral density of the source:

$$W_P(\mbox{\boldmath$r$ }', \mbox{\boldmath$r$ };\nu) = S_P(\... ...;\nu) \delta(\mbox{\boldmath$r$ } - \mbox{\boldmath$r$ }') \:.$$ (2.30)

Substituting Eq. (2.30) into Eq. (2.29) gives

$$\Gamma_Q(\mbox{\boldmath$\rho$ }_1,\mbox{\boldmath$\rho$ }_2;... ...,\mbox{\boldmath$\rho$ }_2;\nu) \exp(-2\pi i\nu\tau) d\nu \:.$$ (2.31)

Then we obtain the mutual intensity across the observation plane by substituting $\tau = 0$ into Eq. (2.31) as

$$J_Q(\mbox{\boldmath$\rho$ }_1,\mbox{\boldmath$\rho$ }_2) = \... ...(\mbox{\boldmath$r$ }',\mbox{\boldmath$\rho$ }_2;\nu) d\nu \:.$$ (2.32)

Under the paraxial approximation with respect to the propagation distances s₁ and s₂ (see Fig. 2.1), namely,

$$s_1 - s_2 = \displaystyle\frac{\mbox{\boldmath$\rho$ }^2_1 -... ... \mbox{\boldmath$\rho$ }_2) \cdot \mbox{\boldmath$r$ }}{z} \:,$$ (2.33)

substituting Eq. (2.25) into Eq. (2.32) gives

 

$$\left(\displaystyle\frac{1}{cz}\right)^2 \displaystyle\int\!\!\!\... ...-\infty}d^2r \displaystyle\int^{\infty}_{0} \nu^2 S_P(\mbox{\boldmath$r$ };\nu)$$ J_Q(Q₁,Q₂) =

$$\times \exp\left[\displaystyle\frac{-i\pi\nu(\mbox{\boldmath$\rho$ }^2_1... ... }_1 - \mbox{\boldmath$\rho$ }_2)\cdot\mbox{\boldmath$r$ }}{cz}\right] d\nu \:.$$ (2.34)

Equation (2.34) holds for the mutual intensity of the wavefields that propagates from the source with an arbitrary spectral profile.

Next, let us consider the propagation of the cross-spectral density. The Fourier transform of Eq. (2.29) with respect to $\tau$ and Eq. (2.25) give the following equation:

$$W_Q(\mbox{\boldmath$\rho$ }_1,\mbox{\boldmath$\rho$ }_2;\nu) ... ...\displaystyle\frac{2\pi i\nu(s_1-s_2)}{c}\right]d^2r d^2r' \:.$$ (2.35)

Since we assume that the source is spatially incoherent, substituting Eq. (2.30) into Eq. (2.35) gives

 

$$W_Q(\mbox{\boldmath$\rho$ }_1,\mbox{\boldmath$\rho$ }_2;\nu) = \l... ...1 - \mbox{\boldmath$\rho$ }_2) \cdot \mbox{\boldmath$r$ }} {cz}\right] d^2r \:.$$ (2.36)

Equation (2.36) implies that the cross-spectral density at the far field is given by the two-dimensional Fourier transform of the spectral density across the spatially incoherent source. It should be noted that this relationship corresponds to the van Cittert-Zernike theorem described by Eq. (2.20).