Spatial and temporal coherence

The concepts of spatial and temporal coherence are important in discussing the phase characteristics of imaging systems. In general, the concept of coherence is related to the stability, or predictability, of phase.

Spatial coherence describes the correlation between signals at different points in space. Temporal coherence describes the correlation or predictable relationship between signals observed at different moments in time.

Spatial coherence is described as a function of distance, and is often presented as a function of correlation versus absolute distance between observation points $\rho_{AB}(\vert x_1-x_2\vert)$. If $\big{\langle} P(x_1)P(x_2) \big{\rangle} = 0$, there is no spatial coherence at $\vert x_1-x_2\vert$, i.e. $\rho_{AB}(\vert x_1-x_2\vert) = 0$.

The same operation can be performed in time with the results plotted as correlation versus relative delay $\tau$. For the echo signal over the ensemble, temporal correlation is periodic at the inverse of the pulse repetition frequency (1/PRF), as shown in Figure 7.1. This holds for both transmit and receive, and for a point target or a speckle target.

Figure 7.1: (Top) Under multiple interrogations of a target with coherent radiation, the temporal correlation of the echo has periodicity within a pulse length and also at the pulse repetition interval. (Bottom) Incoherent radiation typically exhibits some finite correlation, but only within the pulse length.

The temporal correlation length of pulsed ultrasound is typically on the order of 1 $\mu$s, and in the case of repeated interrogation, is periodic. Therefore the speckle target looks the same on each interrogation. For incoherent radiation, there is no temporal coherence on the order of the pulse repetition interval. All radiation has some finite correlation length.

If you capture an image within the correlation length, you get a single speckle pattern. Laser light has a temporal correlation length on the order of seconds, therefore one can observe stable speckle patterns from scattering functions illuminated with laser light. Images captured at different times with incoherent radiation are uncorrelated, and are averaged in the final image if viewed with a detector whose response time is longer than the finite correlation length.

The spatial correlation of echoes from a point target is constant. For a speckle target, the VCZ curve takes over. (If this were not the case, spatial compounding would have no effect.) Propagation is a low pass filter, therefore we expect the spatial correlation length to increase with distance from the source.

The spatial coherence of the radiation falling on a surface can be measured by changing the spacing between the two openings in the surface at $p_1$ and $p_2$, and observing the interference pattern that is generated on a screen beyond, as shown in Figure 7.2. If the pattern dies out after the first fringe, the temporal coherence is on the order of one wavelength. For laser light, the interference pattern is very wide. If intensity patterns of incoherent radiation could be detected within its very short temporal correlation length, then the fringe pattern would extend out to infinity. Over many events, averaging flattens out this extended pattern. Assuming some type of averaging is occurring, often in the detector, only waves with some finite coherence will interfere. In optics using an intensity detector, the number of averaged images approaches infinity. The width of the interference pattern tells us about the temporal and spatial coherence of the radiation.

Figure 7.2: Schematic of the Michelson interferometer, in which the intensity pattern cast on a screen by a dual-slit aperture is used to determine the temporal coherence length of the incident radiation. Only coherent radiation creates an interference pattern. Given an mean incident intensity at the screen of $I_{0}$, perfect constructive interference produces an intensity of $4 I_{0}$.

In optics, temporal coherence is also measured by combining beams from the same source but having a known path length difference, and observing the interference pattern produced. This path length difference is achieved using a beam splitter, as shown in Figure 7.3.

Figure 7.3: Schematic of a beam-splitter interferometer, in which the intensity pattern cast on a screen by interfering beams traveling along different path lengths can be used to determine the temporal coherence length of the incident radiation.

The correlation coefficient in k-space

The normalized correlation coefficient is the cross correlation function adjusted to remove effects related to the energy in the signals. This coefficient is often designated by the variable $\rho$. Given two random variables $x$ and $y$, the continuous time expression for $\rho_{x,y}$ is:

$$\rho_{x,y} = \frac{ \int^{\infty}_{-\infty} (x-\mu_{x}) (y-\mu_{y... ...^{\infty}_{-\infty}(x-\mu_{x})^2 dt \int^{\infty}_{-\infty}(y-\mu_{y})^2 dt } }$$ (7.1)

As reviewed in Section 5.3, we can simplify this expression using that for variance. Here $\langle \rangle$ denotes ``expected value of'':

$$% latex2html id marker 4980 \begin{split}\text{The mean of }... ...text{For } x,y \text{ independent, }\rho_{x,y} &= 0 \end{split}$$ (7.2)

The correlation coefficient goes as the cosine between two vectors. ``Half-way,'' the correlation has a value of 0.707 = $\sqrt{2}$.

The covariance definition of spatial coherence:

$$\Bbb{C}_{P_{1},P_{2}}(P_{1},P_{2},\tau) = \big{\langle} \mu(P_{1},t+\tau) \mu^{\ast}(P_{2},t) \big{\rangle}$$ (7.3)

Given stationarity:

$$\Bbb{C}_{P_{1},P_{2}}(P_{1},P_{2},\tau) = \Bbb{C}_{P_{1},P_{2}}\big{(}\vert P_{1}-P_{2}\vert,\tau\big{)}$$ (7.4)

In Goodman this is referred to as the ``mutual coherence function''[12].

The covariance definition of temporal coherence:

$$\Bbb{C}_{P_{1}}(P_{1},\tau) = \big{\langle} \mu(P_{1},t+\tau) \mu^{\ast}(P_{1},t) \big{\rangle}$$ (7.5)

For the averaging of intensity patterns, as in compounding:

$$I_{1+2} = I_{1}+I_{2}+2\rho_{1,2}\sqrt{I_{1}I_{2}}$$ (7.6)

As shown in Figure 7.2, given $I_{1}=I_{2}$,

$$\begin{split}\text{If }\rho=1&, I=4I_{1}\ \text{If }\rho=-1&, I=0\ \text{If }\rho=0&, I=2I_{1}\ \end{split}$$ (7.7)

Consider the vector notation $\overrightarrow{V}_{1}$ and $\overrightarrow{V}_{2}$ for the complex instantaneous amplitudes of two signals separated in phase by the angle $\phi$:

$$\vert\overrightarrow{V}_{1+2}\vert^2=\vert\overrightarrow{V}_{1}\... ...t^2 + 2\vert\overrightarrow{V}_{1}\vert\vert\overrightarrow{V}_{2}\vert\cos\phi$$ (7.8)

Using these equations to calculate the effect of spatial compounding on the signal-to-noise ratio (SNR). The compounding of two uncorrelated images produces an SNR improvement of $\sqrt{2}$.

$$\begin{split}\Bbb{G}(f) &= \text{lim}_{T\rightarrow\infty}\fr... ...e }\nu_{T}(f) &= \int^{T}_{T}x(t)\exp(j2\pi f t) dt \end{split}$$ (7.9)

In this terminology, Parseval's theorem is expressed:

$$\int^{\infty}_{-\infty}x(t)x^{\ast}(t) dt=\int^{\infty}_{-\infty}\vert\nu_{T}(f)\vert df$$ (7.10)

in which phase information has been lost through conjugate multiplication.

The power spectral density of the output of a random process is the squared modulus of the transfer function of linear system times the power spectrum density of the input random process:

$$\Bbb{G}_{output}(f) = \vert H(f)\vert^2\Bbb{G}_{input}(f)$$ (7.11)

The cross correlation function and cross spectral density function are Fourier transform pairs:

$$\Bbb{G}_(f) = \vert H(f)\vert^2\Bbb{G}_{input}(f)$$ (7.12)

Neither contains meaningful phase information, and each contains equivalent information.

If we know the k-space windows of a system and target function, or two different windows of the system under compounding, the product of these windows gives the correlation between their echo signals.

The cross spectrum density is a measure of the similarity of two signals at each complex frequency:

$$\Bbb{G}_{u,v}(f) = \int^{\infty}_{\infty}\Bbb{C}_{u,v}(f)\exp(j2\pi f t) dt$$ (7.13)

In terms of two complex signals $u(t)$ and $v(t)$ and their transforms $U(f)$ and $V(f)$,

$$\Bbb{G}_{u,v}(f) = _{T\rightarrow\infty}\frac{\big{\langle} U_{T}(f) V_{T}^{\ast}(f) \big{\rangle}}{T}$$ (7.14)

The limit as $T\rightarrow\infty$ is required to generalize this expression to include functions that do not have analytical Fourier transforms.

This function is a measure of spectral similarity at each frequency:

$$\Bbb{C}_{u,v}(\tau) = \int^{\infty}_{-\infty} \Bbb{G}_{u,v}(f)\exp(-j2\pi f\tau) df$$ (7.15)

For $u,v$ having equal phase profiles, $G_{uv}(f)$ is a purely real number due to the conjugate operation. For unequal phase profiles, this quantity will be complex and exhibit interference patterns.

Integration of the k-space overlap gives the correlation coefficient at $\tau=0$:

$$\rho_{u,v}(0) = \frac{ \int^{\infty}_{-\infty}\Bbb{G}_{u,v}(f) df... ...}_{-\infty}\vert U(f)\vert^2 df \int^{\infty}_{-\infty}\vert V(f)\vert^2 df } }$$ (7.16)

Envelope detection shifts the signal band to base band, losing the carrier frequency. Ideally, everything but the carrier frequency is preserved, e.g. axial and lateral bandwidth are preserved. The $\rho_{u,v}(0)$ of the detected signals is the square of $\rho_{u,v}(0)$ of the corresponding RF signals.