Second-order spatial coherence

First, let us consider the analytic signal $V(\mbox{\boldmath$r$ }_1;t)$ and $V(\mbox{\boldmath$r$ }_2;t)$ that fluctuate with time t at different two points $\mbox{\boldmath$r$ }_1$ and $\mbox{\boldmath$r$ }_2$, and the field is assumed to be stationary. Their cross-correlation function is defined by

$$\Gamma(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2; \tau) ... ...dmath$r$ }_1 ;t) V(\mbox{\boldmath$r$ }_2; t+\tau) \rangle \:,$$ (2.1)

where the angular bracket denotes the ensemble average, the asterisk denotes the complex conjugate, and $\tau$ is the time difference. This cross-correlation function is called the mutual coherence function in statistical optics. The mutual coherence function with $\tau = 0$ is named as the mutual intensity, and the definition is

$$J(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2) = \Gamma(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2, 0).$$ (2.2)

The complex degree of coherence is defined as the normalized mutual coherence:

$$\gamma(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2, \tau) ... ...\sqrt{I(\mbox{\boldmath$r$ }_1)I(\mbox{\boldmath$r$ }_2)}} \:,$$ (2.3)

where

$$I(\mbox{\boldmath$r$ }_j) = \Gamma(\mbox{\boldmath$r$ }_j, \m... ...math$r$ }_j;0) \:, \hspace{5mm} (j = 1 \:\mathrm{or}\: 2) \:.$$ (2.4)

The complex degree of coherence, which is defined by Eq. (2.3), is related to the visibility of the spatial interference fringes that appear in the two-beam interference of the light from $\mbox{\boldmath$r$ }_1$ and $\mbox{\boldmath$r$ }_2$. The relationship between the complex degree of coherence and the visibility V is represented by

$$V = \displaystyle\frac{I_{\mathrm{max}}(\mbox{\boldmath$\rho$... ...(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2;\tau)\vert \:,$$ (2.5)

where $I_{\mathrm{max}}(\mbox{\boldmath$\rho$ })$ and $I_{\mathrm{min}}(\mbox{\boldmath$\rho$ })$are the maximum and the minimum intensities that are seen around $\mbox{\boldmath$\rho$ }$. While the modulus of the complex degree of coherence represents the visibility of the interference fringes, the argument of the complex degree of coherence is related to the location of the interference fringes. The complex degree of coherence describes the spatial correlation of the quasi-monochromatic wavefields in the space-time domain. On the other hand, the dependence of the correlation property on the optical frequency is essential in the case where the spectral bandwidth of the wavefields is broad. The cross-spectral density and the spectral degree of coherence play a critical role in this case.

The analytic signal $V(\mbox{\boldmath$r$ };t)$ at $\mbox{\boldmath$r$ }$ is a scalar variable if we consider one of the orthogonal components, and the analytic signal and the Fourier spectrum $U(\mbox{\boldmath$r$ };\nu)$ that represents a particular frequency component are related by the Fourier transform pair:

$$V(\mbox{\boldmath$r$ };t) = \displaystyle\int^{\infty}_{0} U(\mbox{\boldmath$r$ };\nu) \exp(-2\pi i\nu t) d\nu \:,$$ (2.6)

and

$$U(\mbox{\boldmath$r$ };\nu) = \displaystyle\int^{\infty}_{-\infty} V(\mbox{\boldmath$r$ };t) \exp(2\pi i\nu t) dt \:.$$ (2.7)

Let us consider the ensemble average of the product of two Fourier spectra with different frequencies at two points $\mbox{\boldmath$r$ }_1$and $\mbox{\boldmath$r$ }_2$:

 

$$\langle U^*(\mbox{\boldmath$r$ }_1; \nu) U(\mbox{\boldmath$r$ }_2; \nu') \rangle \displaystyle\int\!\!\!\displaystyle\int^{\infty}_{-\infty} \lang... ...}_1;t)V(\mbox{\boldmath$r$ }_2;t') \rangle \exp[2\pi i(\nu' t' - \nu t)] dt dt'$$ =

$$\displaystyle\int\!\!\!\displaystyle\int^{\infty}_{-\infty} \lang... ...}_2;t+\tau) \rangle \exp[2\pi i(\nu' - \nu)t] \exp(2\pi i\nu' \tau) dtd\tau \:,$$ = (2.8)

where $\tau = t' - t$. Substitution of Eq. (2.1) into Eq. (2.8) gives

$$\langle U^*(\mbox{\boldmath$r$ }_1;\nu) U(\mbox{\boldmath$r$ ... ...math$r$ }_1, \mbox{\boldmath$r$ }_2;\nu)\delta(\nu - \nu') \:,$$ (2.9)

where $W(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2;\nu)$ is the cross-spectral density. The mutual coherence function and the cross-spectral density are also related by the Fourier transform pair:

$$W(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2;\nu) = \dis... ..., \mbox{\boldmath$r$ }_2; \tau) \exp(2\pi i\nu\tau) d\tau \:,$$ (2.10)

and

$$\Gamma(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2; \tau) ... ..._1, \mbox{\boldmath$r$ }_2;\nu) \exp(-2\pi i\nu\tau) d\nu \:.$$ (2.11)

Note that Eq. (2.9) implies that the Fourier spectra at different frequencies $\nu$ and $\nu'$ are uncorrelated. Therefore, the cross-spectral density represents the spatial correlation of the wavefields at a particular frequency.

As stated above, the complex degree of coherence is defined as the normalized mutual coherence function. The normalization of the cross-spectral density defines the spectral degree of coherence in a similar way:

$$\mu(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2; \nu) = \... ...\nu)W(\mbox{\boldmath$r$ }_2,\mbox{\boldmath$r$ }_2;\nu)}} \:.$$ (2.12)

The modulus of the spectral degree of coherence is limited to the range [0,1], namely

$$0 \leq \vert\mu(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2; \nu)\vert \leq 1 \:.$$ (2.13)

The modulus of the complex degree of coherence represents the visibility of the spatial interference fringes, and on the other hand the modulus of the spectral degree of coherence represents the interference efficiency of the spectral interference.

As described in Eqs. (2.10) and (2.11), the mutual coherence function and the cross-spectral density are related by the Fourier transform relationship. However, the relationship between their normalization forms, namely the complex degree of coherence and the spectral degree of coherence, is more complicated. Now, let us define the Fourier transform of the complex degree of coherence by

$$\beta(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2;\nu) = ... ...1, \mbox{\boldmath$r$ }_2;\tau) \exp(2\pi i\nu\tau) d\tau \:.$$ (2.14)

As is obvious from Eq. (2.3), the representation of $\beta(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2;\nu)$ with the cross-spectral density is

$$\beta(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2;\nu) = ... ...t{I_1(\mbox{\boldmath$r$ }_1)I_2(\mbox{\boldmath$r$ }_2)}} \:.$$ (2.15)

Equation (2.15) can be rewritten when $\mbox{\boldmath$r$ }_1 = \mbox{\boldmath$r$ }_2$ as

$$\beta(\mbox{\boldmath$r$ }_1,\mbox{\boldmath$r$ }_1;\nu) = \... ... }_1,\mbox{\boldmath$r$ }_1;\nu)}{I(\mbox{\boldmath$r$ })} \:,$$ (2.16)

and $I(\mbox{\boldmath$r$ })$ can be also expressed by

$$I(\mbox{\boldmath$r$ }) = \Gamma(\mbox{\boldmath$r$ },\mbox{\... ...{0} W(\mbox{\boldmath$r$ },\mbox{\boldmath$r$ },\nu) d\nu \:.$$ (2.17)

Then the following equation holds:

$$\displaystyle\int^{\infty}_{0} \beta(\mbox{\boldmath$r$ }_1,\mbox{\boldmath$r$ }_2;\nu)d\nu = 1 \:.$$ (2.18)

Equations (2.12), (2.15), and (2.16) gives the representation of the spectral degree of coherence with $\beta(\mbox{\boldmath$r$ },\mbox{\boldmath$r$ };\nu)$ as

$$\mu(\mbox{\boldmath$r$ }_1, \mbox{\boldmath$r$ }_2; \nu) = \... ...\nu)W(\mbox{\boldmath$r$ }_2,\mbox{\boldmath$r$ }_2;\nu)}} \:.$$ (2.19)

Equation (2.19) means that the complex degree of coherence and the spectral degree of coherence are not related by the Fourier transform pair unlike the relationship between the mutual coherence function and the cross-spectral density.