Correlation-induced spectral changes

In 1996, E. Wolf pointed out that the spectrum propagates from the spatially, partially coherent source is not invariant even though the light propagates through free space. In this section, let us briefly review the fundamental concept of the correlation-induced spectral changes.

First, we assume the following conditions:

1. Sources are planar and quasi-homogeneous.

2. Spectral profile is uniform across the source.
The quasi-homogeneous sources are characterized by the following conditions:

1. Source is large enough compared with the effective correlation length,

2. Intensity is almost uniform within the effective correlation length,

3. Spatial coherence function is space invariant.

  

Figure 2.3: Optical system considered.

With reference to Fig. 2.3, we consider a spatially, partially coherent source, two particular points $P_1(\mbox{\boldmath$r$ }_1)$ and $P_2(\mbox{\boldmath$r$ }_2)$ on the source plane, and the unit vector $\mbox{\boldmath$u$ }$. If the source is quasi-homogeneous, the spectral degree of coherence across the source plane is represented by

$$\mu(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2;\nu) = \mu(\mbox{\boldmath$r$ }_1 - \mbox{\boldmath$r$ }_2; \nu) \:.$$ (2.47)

In addition, if the spectral profile is uniform over the source area, the cross-spectral density across the source plane is represented by

$$W(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_1;\nu) = W(\mbox{\boldmath$r$ }_2, \mbox{\boldmath$r$ }_2;\nu) = S_p(\nu) \:,$$ (2.48)

and

$$W(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2;\nu) = \eps... ...) \mu(\mbox{\boldmath$r$ }_1 - \mbox{\boldmath$r$ }_2;\nu) \:,$$ (2.49)

where $\epsilon(\mbox{\boldmath$r$ }_1)$ and $\epsilon(\mbox{\boldmath$r$ }_2)$ are 1 within the source area and 0 outside the source area.

The radiant intensity $J_{\nu}(\mbox{\boldmath$u$ })$ is proportional to the Fourier transform of the spectral degree of coherence across the source plane and $\cos^2\theta$, namely,

$$J_{\nu}(\mbox{\boldmath$r$ }) = k^{2}A S_P(\nu)\tilde{\mu}(k\mbox{\boldmath$u$ }_{\bot};\nu) \cos^{2}\theta \:,$$ (2.50)

where A is the area of the source and $\mbox{\boldmath$u$ }_{\bot} = (u_x, u_y)$. By using the equation

$$\tilde{\mu}(\mbox{\boldmath$f$ };\nu)=\displaystyle\frac{1}{(... ...ath$r$ }' = \mbox{\boldmath$r$ }_2-\mbox{\boldmath$r$ }_1) \:,$$ (2.51)

the normalized spectra in the far field is represented by

$$S_o(\mbox{\boldmath$u$ };\nu) =\displaystyle\frac{J_{\nu}(\m... ...k^2 S_P(\nu)\tilde{\mu}(k\mbox{\boldmath$u$ }_{\bot},\nu)d\nu}$$ (2.52)

Equation (2.52) implies that the normalized spectrum depends on $\mbox{\boldmath$u$ }$, namely, the spectral profile depends on the location of the observation point. When the Fourier transform of the spectral degree of coherence is represented by a function of the product of the spatial-frequency and $\mbox{\boldmath$u$ }_{\bot}$:

$$\tilde{\mu}(k\mbox{\boldmath$u$ }_{\bot};\nu)=F(\nu)\tilde{H}(\mbox{\boldmath$u$ }_{\bot}) \:,$$ (2.53)

then Eq. (2.52) can be rewritten to the form

$$S_o(\mbox{\boldmath$u$ };\nu) =\displaystyle\frac{k^2 S_P(\n... ...} {\displaystyle\int_{0}^{\infty} k^2 S_P(\nu)F(\nu)d\nu} \:.$$ (2.54)

In contrast to Eq. (2.52), Eq. (2.54) implies that the spectrum is independent of $\mbox{\boldmath$u$ }_{\bot}$. Taking the Fourier transform of Eq. (2.53) gives

 

$$\mu(\mbox{\boldmath$r$ }';\nu) F(\nu)\displaystyle\int_{-\infty}^{\infty} \tilde{H}(\mbox{\boldm... ...ldmath$u$ }_{\bot} \cdot\mbox{\boldmath$r$ }')d^2(k\mbox{\boldmath$u$ }_{\bot})$$ =

$$k^2 F(\nu)H(k\mbox{\boldmath$r$ }') \:.$$ = (2.55)

If $\mu(\mbox{\boldmath$r$ }';\nu)$ is 0 only when $\mbox{\boldmath$r$ }'=\mbox{\boldmath$r$ }_1 - \mbox{\boldmath$r$ }_2$, $\mu(0;\nu) = 1$ for all frequencies, and the following equations hold with the help of Eq. (2.55)

 

$$\left. \begin{array}{ll} \mu(0;\nu) = k^2 F(\nu)H(0)=1 \\ k^2 F(\nu) = \left\{ H(0) \right\}^{-1} \end{array}\right\} \:.$$ (2.56)

Since both sides of Eq. (2.56) must be a constant, let the constant be $\alpha$, and

$$F(\nu)=\displaystyle\frac{\alpha}{k^2} \:.$$ (2.57)

We obtain the following equation by substituting Eq. (2.57) into Eq. (2.54):

$$S_o(\mbox{\boldmath$u$ };\nu) = S_o(\nu) = \displaystyle\frac{S_P(\nu)} {\displaystyle\int_{0}^{\infty} S_P(\nu)d\nu} \:.$$ (2.58)

Equation (2.58) means that the observed spectral profile is

1. independent of the observation location
and

2. same as that of the source spectrum.
Moreover, substituting Eq. (2.57) into Eq. (2.55), and rewriting $\mbox{\boldmath$r$ }'=\mbox{\boldmath$r$ }_1 - \mbox{\boldmath$r$ }_2$ and $\alpha H = h$gives

$$\mu(\mbox{\boldmath$r$ }_2-\mbox{\boldmath$r$ }_1;\nu) =h[k(... ...}{c}(\mbox{\boldmath$r$ }_2-\mbox{\boldmath$r$ }_1)\right] \:.$$ (2.59)

Equation (2.59) is called the scaling law and this means that the propagated spectrum is invariant if the spectral degree of coherence across the source takes the form of Eq. (2.59). In other words, the propagated spectrum changes if the spectral degree of coherence across the source does not obey the scaling law.